Cambridge studies in advanced mathematics
96
Lie Algebras of Finite
and Affine Type
Lie Algebras of Finite and Affine Type

Lie Algebras of
Finite and Affine Type
R. W. CARTER
Mathematics Institute
University of Warwick
В CAMBRIDGE
UNIVERSITY PRESS
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Dedicated to Sandy Green

Contents
Preface	page xiii
1	Basic concepts	1
1.1	Elementary properties of Lie algebras	1
1.2	Representations and modules	5
1.3	Abelian, nilpotent and soluble Lie algebras
2	Representations of soluble and nilpotent	Lie algebras	11
2.1	Representations of soluble Lie algebras	11
2.2	Representations of nilpotent Lie algebras	14
3	Cartan subalgebras	23
3.1	Existence of Cartan subalgebras	23
3.2	Derivations and automorphisms	25
3.3	Ideas from algebraic geometry	27
3.4	Conjugacy of Cartan subalgebras	33
4	The Cartan decomposition	36
4.1	Some properties of root spaces	36
4.2	The Killing form	39
4.3	The Cartan decomposition of a semisimple	Lie algebra	45
4.4	The Lie algebra M„(C)	52
5	The root system and the Weyl group	56
5.1	Positive systems and fundamental systems	of roots	56
5.2	The Weyl group	59
5.3	Generators and relations for the Weyl group	65
vii
6	The Cartan matrix and the Dynkin diagram	69
6.1	The Cartan matrix	69
6.2	The Dynkin diagram	72
6.3	Classification of Dynkin diagrams	74
6.4	Classification of Cartan matrices	80
7	The existence and uniqueness theorems	88
7.1	Some properties of structure constants	88
7.2	The uniqueness theorem	93
7.3	Some generators and relations in a simple Lie algebra	96
7.4	The Lie algebras L(A) and L(A)	98
7.5	The existence theorem	105
8	The simple Lie algebras	121
8.1	Lie algebras of type A,	122
8.2	Lie algebras of type I),	124
8.3	Lie algebras of type	128
8.4	Lie algebras of type C,	132
8.5	Lie algebras of type G,	135
8.6	Lie algebras of type F4	138
8.7	Lie algebras of types E6, E7, Es	140
8.8	Properties of long and short roots	145
9	Some universal constructions	152
9.1	The universal enveloping algebra	152
9.2	The Poincare-Birkhoff-Witt basis theorem	155
9.3	Free Lie algebras	160
9.4	Lie algebras defined by generators and relations	163
9.5	Graph automorphisms of simple Lie algebras	165
10	Irreducible modules for semisimple Lie algebras	176
10.1	Verma modules	176
10.2	Finite dimensional irreducible modules	186
10.3	The finite dimensionality criterion	190
11	Further properties of the universal enveloping	algebra	201
11.1	Relations between the enveloping algebra
and the symmetric algebra	201
11.2	Invariant polynomial functions	207
11.3	The structure of the ring of polynomial invariants	216
11.4	The Killing isomorphisms	222
Contents	ix
11.5	The centre of the enveloping algebra	226
11.6	The Casimir element	238
12	Character and dimension formulae	241
12.1	Characters of L-modules	241
12.2	Characters of Verma modules	244
12.3	Chambers and roots	246
12.4	Composition factors of Verma modules	255
12.5	Weyl’s character formula	258
12.6	Complete reducibility	262
13	Fundamental modules for simple Lie algebras	267
13.1	An alternative form of Weyl’s dimension formula	267
13.2	Fundamental modules for A;	268
13.3	Exterior powers of modules	270
13.4	Fundamental modules for B, and t),	274
13.5	Clifford algebras and spin modules	281
13.6	Fundamental modules for C;	292
13.7	Contraction maps	295
13.8	Fundamental modules for exceptional algebras	303
14	Generalised Cartan matrices and Kac-Moody algebras 319
14.1	Realisations of a square matrix	319
14.2	The Fie algebra L(A) associated with a complex matrix 322
14.3	The Kac-Moody algebra L(A)	331
15	The classification of generalised Cartan matrices	336
15.1	A trichotomy for indecomposable GCMs	336
15.2	Symmetrisable generalised Cartan matrices	344
15.3	The classification of affme generalised Cartan matrices	351
16	The invariant form, Weyl group and root system	360
16.1	The invariant bilinear form	360
16.2	The Weyl group of a Kac-Moody algebra	371
16.3	The roots of a Kac-Moody algebra	377
X
Contents
17	Kac-Moody algebras of affine type	386
17.1	Properties of the affine Cartan matrix	386
17.2	The roots of an affine Kac-Moody algebra	394
17.3	The Weyl group of an affine Kac-Moody algebra	404
18	Realisations of affine Kac-Moody algebras	416
18.1	Loop algebras and central extensions	416
18.2	Realisations of untwisted affine Kac-Moody algebras	421
18.3	Some graph automorphisms of affine algebras	426
18.4	Realisations of twisted affine algebras	429
19	Some representations of symmetrisable Kac-Moody algebras 452
19.1	The category О of L(A)-modules	452
19.2	The generalised Casimir operator	459
19.3	Kac’ character formula	466
19.4	Generators and relations for symmetrisable algebras	474
20	Representations of affine Kac-Moody algebras	484
20.1	Macdonald’s identities	484
20.2	Specialisations of Macdonald’s identities	491
20.3	Irreducible modules for affine algebras	494
20.4	The fundamental modules for L (Aj	504
20.5	The basic representation	508
21	Borcherds Lie algebras	519
21.1	Definition and examples of Borcherds algebras	519
21.2	Representations of Borcherds algebras	524
21.3	The Monster Lie algebra	530
Appendix	540
Summary pages - explanation	540
Type A,	543
Type BI	545
Type C,	547
Type D,	549
Type E6	551
Type Ej	553
Type As	555
Type F4	557
Type G2	559
Contents	xi
Type Aj	561
ТуреЛ'=2Л;	(1st description)	563
(2nd description)	565
Type A,	567
Type B,	570
ТуреВ; = 2Л;/|	573
Type C,	576
Type q = 2Di+1	579
TypeC;' = 2A2/	(1st description)	582
(2nd description)	585
Type £>4	588
Type Д, / > 5	590
Type E6	593
Type Ё7	596
Type Es	599
Type F4	602
Type Л) = 2Л6	604
Type G2	606
Type G2 = 3D4	608
Notation	610
Bibliography of books on Lie algebras	619
Bibliography of articles on Kac-Moody algebras	621
Index	629

Preface
Lie algebras were originally introduced by S. Lie as algebraic structures
used for the study of Lie groups. The tangent space of a Lie group at the
identity element has the natural structure of a Lie algebra, called by Lie the
infinitesimal group. However, Lie algebras also proved to be of interest in
their own right. The finite dimensional simple Lie algebras over the complex
field were investigated independently by E. Cartan and W. Killing and the
classification of such algebras was achieved during the decade 1890-1900.
Basic ideas on the structure and representation theory of these Lie algebras
were also contributed at a later stage by H. Weyl. Since then the theory of
finite dimensional simple Lie algebras has found many and varied applications
both in mathematics and in mathematical physics, to the extent that it is now
generally regarded as one of the classical branches of mathematics.
In 1967 V. G. Kac and R. V. Moody independently introduced the Lie alge-
bras now known as Kac-Moody algebras. The finite dimensional simple Lie
algebras are examples of Kac-Moody algebras; but the theory of Kac-Moody
algebras is much broader, including many infinite dimensional examples.
The Kac-Moody theory has developed rapidly since its introduction and has
also turned out to have applications in many areas of mathematics, includ-
ing among others group theory, combinatorics, modular forms, differential
equations and invariant theory. It has also proved important in mathematical
physics, where it has applications to statistical physics, conformal field theory
and string theory. The representation theory of affine Kac-Moody algebras
has been particularly useful in such applications.
In view of these applications it seems clear that the theory of Lie algebras,
of both finite and affine types, will continue to occupy a central position
in mathematics into the twenty-first century. This expectation provides the
motivation for the present volume, which aims to give a mathematically
rigorous development of those parts of the theory of Lie algebras most relevant
xiii
xiv
Preface
to the understanding of the finite dimensional simple Lie algebras and the
Kac-Moody algebras of affine type. A number of books on Lie algebras are
confined to the finite dimensional theory, but this seemed too restrictive for the
present volume in view of the many current applications of the Kac-Moody
theory. On the other hand the Kac-Moody theory needs a prior knowledge of
the finite dimensional theory, both to motivate it and to supply many technical
details. For this reason I have included an account both of the Cartan-Killing-
Weyl theory of finite dimensional simple Lie algebras and of the Kac-Moody
theory, concentrating particularly on the Kac-Moody algebras of affine type.
We work with Lie algebras over the complex field, although any algebraically
closed field of characteristic zero would do equally well.
I was introduced to the theory of Lie algebras by an inspiring course of
lectures given by Philip Hall at Cambridge University in the late 1950s.
I have given a number of lecture courses on finite dimensional Lie algebras at
Warwick University, and also two lecture courses on Kac-Moody algebras.
The present book has developed as a considerably expanded version of the
lecture notes of these courses. The main prerequisite for study of the book is
a sound knowledge of linear algebra. I have in fact aimed to make this the
sole prerequisite, and to explain from first principles any other techniques
which are used in the development.
The most influential book on Kac-Moody algebras is the volume Infinite-
Dimensional Lie Algebras, third edition (1990), by V. Kac. That formidable
treatise contains a development of the Kac-Moody theory presupposing a
knowledge of the finite dimensional theory, and includes information on
several of the applications. The present volume will not rival Kac’ account
for experts on Kac-Moody algebras. About half of the theory covered in
the 3rd edition of Kac’ book has been included. However, for those new to
the Kac-Moody theory, our account may be useful in providing a gentler
introduction, making use of ideas from the finite dimensional theory developed
earlier in the book.
The content of the book can be summarised as follows. The basic definitions
of Lie algebras, their subalgebras and ideals, representations and modules,
are given in Chapter 1. In Chapter 2 the standard results are proved on the
representation theory of soluble and nilpotent Lie algebras. The results on
representations of nilpotent Lie algebras are used extensively in the subsequent
development. The key idea of a Cartan subalgebra is introduced in Chapter 3,
where the existence and conjugacy of Cartan subalgebras are proved. We
make use of some ideas from algebraic geometry to prove the conjugacy of
Cartan subalgebras. In Chapter 4 the Killing form is introduced and used
to describe the Cartan decomposition of a semisimple Lie algebra into root
Preface
xv
spaces with respect to a Cartan subalgebra. The well-known example of the
special linear Lie algebra is used to illustrate the general ideas. In Chapter 5
the Weyl group is introduced and shown to be a Coxeter group. This leads on
to the definition of the Cartan matrix and the Dynkin diagram. The possible
Dynkin diagrams and Cartan matrices are classified in Chapter 6, and in
Chapter 7 the existence and uniqueness of a semisimple Lie algebra with a
given Cartan matrix are proved. In Chapter 8 the finite dimensional simple
Lie algebras are discussed individually and their root systems determined.
Chapters 9 to 13 are concerned with the representation theory of finite
dimensional semisimple Lie algebras. We begin in Chapter 9 with the intro-
duction of the universal enveloping algebra, of free Lie algebras and of Lie
algebras defined by generators and relations. The finite dimensional irre-
ducible modules for semisimple Lie algebras are obtained in Chapter 10 as
quotients of infinite dimensional Verma modules with dominant integral high-
est weight. In Chapter 11 the enveloping algebra is studied in more detail. Its
centre is shown to be isomorphic to the algebra of polynomial functions on a
Cartan subalgebra invariant under the Weyl group, and to the algebra of poly-
nomial functions on the Lie algebra invariant under the adjoint group. This
algebra is shown to be isomorphic to a polynomial algebra. The properties of
the Casimir element of the centre of the enveloping algebra are also discussed.
These are important in subsequent applications to representation theory. Char-
acters of modules are introduced in Chapter 12, and Weyl’s character formula
for the irreducible modules is proved. The fundamental irreducible modules
for the finite dimensional simple Lie algebras are discussed individually in
Chapter 13. Their discussion involves exterior powers of modules, Clifford
algebras and spin modules, and contraction maps.
This concludes the development of the structure and representation theory
of the finite dimensional Lie algebras. This development has concentrated
particularly on the properties necessary to obtain the classification of the
simple Lie algebras and their finite dimensional irreducible modules. Among
the significant results omitted from our account are Ado’s theorem on the
existence of a faithful finite dimensional module, the radical splitting theorem
of Levi, the theorem of Malcev and Harish-Chandra on the conjugacy of
complements to the radical, and the cohomology theory of Lie algebras.
The theory of Kac-Moody algebras is introduced in Chapter 14, where the
Kac-Moody algebra associated to a generalised Cartan matrix is defined. In
fact there are two slightly different definitions of a Kac-Moody algebra which
have been used. There is a definition in terms of generators and relations
which appears the more natural, but there is a different definition, given by
Kac in his book, which is more convenient when one wishes to show that a
xvi
Preface
given Lie algebra is a Kac-Moody algebra. I have used the latter definition,
but have included a proof that, at least for symmetrisable generalised Cartan
matrices, the two definitions are equivalent.
The trichotomy of indecomposable generalised Cartan matrices into those
of finite, affine and indefinite types is obtained in Chapter 15. The Kac-Moody
algebras of finite type turn out to be precisely the non-trivial finite dimensional
simple Lie algebras, and a classification of those of affine type is given.
The important special case of symmetrisable Kac-Moody algebras is also
introduced. This class includes all those of finite and affine types, and some of
those of indefinite type. In Chapter 16 it is shown that symmetrisable algebras
have an invariant bilinear form, which plays a key role in the subsequent
development. The Weyl group and root system of a Kac-Moody algebra are
also discussed. The roots divide into real roots and imaginary roots, and a
remarkable theorem of Kac is proved which characterises the set of positive
imaginary roots. Kac-Moody algebras of affine type are singled out for more
detailed discussion in Chapter 17. In Chapter 18 it is shown how some of
them can be realised in terms of a central extension of a loop algebra of a
finite dimensional simple Lie algebra, whereas the remainder can be obtained
as fixed point subalgebras of these under a twisted graph automorphism.
Chapters 19 and 20 are devoted to the representation theory of Kac-Moody
algebras. The representations considered are those from the category О intro-
duced by Bernstein, Gelfand and Gelfand. In Chapter 19 the irreducible mod-
ules in this category are classified, and their characters are obtained in Kac’
character formula, a generalisation to the Kac-Moody situation of Weyl’s
character formula. In Chapter 20 the representations of affine Kac-Moody
algebras are discussed. The remarkable identities of LG. Macdonald are
obtained by specialising the denominator of Kac’ character formula, interp-
reted in two different ways; one as an infinite sum and the other as an infinite
product. The phenomenon of strings of weights with non-decreasing multi-
plicities is investigated inside an irreducible module for an affine algebra.
Many of the applications of the representation theory of affine Kac-Moody
algebras use the theory of vertex operators. This theory lies beyond the scope
of the present volume. However, we have introduced the idea of a vertex
operator in Chapter 20 with the aim of encouraging the reader to explore the
subject further.
A theory of generalised Kac-Moody algebras was introduced in 1988 by
R. Borcherds. These Lie algebras were introduced as part of Borcherds’
proof of the Conway-Norton conjectures on the representation theory of the
Monster simple group. They are now frequently called Borcherds algebras.
In Chapter 21 we have given an account of Borcherds algebras, including the
Preface
xvii
definition and statements of the main results concerning their structure and
representation theory, but detailed proofs are not given. Many of the results
on Borcherds algebras are quite similar to those for Kac-Moody algebras,
but there are examples of Borcherds algebras which are quite different from
Kac-Moody algebras. The best known such example is the Monster Lie
algebra, which we describe in the final section.
We conclude with an appendix containing one section for each of the
algebras of finite and affine types, in which the most important pieces of
information about the algebra concerned are collected.
I would like to express my thanks to Roger Astley of Cambridge University
Press for his encouragement to complete the half finished manuscript of this
book. This was eventually achieved after I had reached the status of Emeritus
Professor, and therefore had more time to devote to it. I would also like to
thank my colleague Bruce Westbury for the sustained interest he has shown
in this work.

1
Basic concepts
1.1 Elementary properties of Lie algebras
A Lie algebra is a vector space L over a field k on which a multiplication
L
(x, y) -+ [xy]
is defined satisfying the following axioms:
(i)	(x, y) [xy] is linear in x and in y;
(ii)	[xx] = 0 for all x e L;
(iii)	[[xy]z] + [[yz]x] + [[zx]y]=O for all x, y, z e L.
Axiom (iii) is called the Jacobi identity.
Proposition 1.1 [yx] = — [xy] for all x,y eL.
Proof. Since [x + y, x + у] = 0 we have [xx] + [xy] + [yx] + [yy] = 0. It follows
that [xy] + [yx] = 0, that is [yx] = — [xy].	□
Proposition 1.1 asserts that multiplication in a Lie algebra is anticommutative.
Now let II. К be subspaces of a Lie algebra L. Then [HK\ is defined as the
subspace spanned by all products [xy] with x e H and у eK. Each element of
| //A" | is a sum
к1У1] H----h [xryr]
with x; ePfyiE. K.
Proposition 1.2 [HK] = [KH] for all subspaces H, К of L.
1
2
Basic concepts
Proof. Let ле//, у eK. Then [xy] = [—у, x] e [ AV/|. This shows that [7Ж] c
[7C//]. Similarly we have [КН] c | //A' | and so we have equality.	□
Proposition 1.2 asserts that multiplication of subspaces in a Lie algebra is
commutative.
Example 1.3 Let A be an associative algebra over k. Thus we have a map
A
(x, y) xy
satisfying the associative law
(xy)z = x(yz) for all x, y, z e A.
Then A can be made into a Lie algebra by defining the Lie product [xy] by
[xy] = xy - yx
We verify the Lie algebra axioms. Product [xy] is clearly linear in x and
in y. It is also clear that [xx]=0. Finally we check the Jacobi identity.
We have
[[xy]z] = (xy - yx)z - z(xy - yx)
= xyz — yxz — zxy + zyx.
We have similar expressions for [[yz]x] and [[zx]y]. Hence
[ [ vy] z] + [ [yz] x] + [ [zx]y] = xyz - yxz - zxy + zyx + yzx - zyx - xyz
+ xzy + zxy — xzy — yzx + yxz = 0.	□
The Lie algebra obtained from the associative algebra A in this way will be
denoted by [A].
Now let L be a Lie algebra over k. A subset H of L is called a subalgebra
of L if H is a subspace of L and [ ////1 с II. Thus H is itself a Lie algebra
under the same operations as L.
A subset I of L is called an ideal of L if I is a subspace of L and [/L] с I.
We observe that the latter condition is equivalent to [L/] с I. Thus there is no
distinction between left ideals and right ideals in the theory of Lie algebras.
Every ideal is two-sided.
Proposition 1.4 (i) If H, К are subalgebras of L so is HC\K.
(ii)	If H, К are ideals of L so is IP'K.
1.1 Elementary properties of Lie algebras
3
(iii)	If H is an ideal of L and К a subalgebra of L then H + K is a sub-
algebra of L.
(iv)	If H, К are ideals of L then H + K is an ideal of L.
Proof, (i) IKK is a subspace of L and [HПК, HПТС] с \HH\ П [АТС] c
НГ\К. Thus НГ\К is a subalgebra.
(ii)	This time we have [77ПК, L\ c [77L] П [7CL] с H ПК. ThusНПК is an
ideal of L.
(iii)	H + К is a subspace of L. Also [77 + К, H + 7C] c [7777] + [НК] + [KH] +
[KK]cH + K, since [7777] c 77, [НК] cH, |AW| с К. Thus H + K is a
subalgebra.
(iv)	This time we have [77 + K, L\ c [77L] + [7CL] c 77 + K. Thus 77 + К is
an ideal of L.	□
We next introduce the idea of a factor algebra. Let 7 be an ideal of a Lie
algebra L. Then 7 is in particular a subspace of L and so we can form the
factor space L/I whose elements are the cosets I + x for xeL. I + x is the
subset of L consisting of all elements у + x for у El.
Proposition 1.5 Let I be an ideal of L. Then the factor space L/I can be
made into a Lie algebra by defining
[7 + x, I + у] = I + [xy]	forallx,yEL.
Proof. We must first show that this definition is unambiguous, that is if
I + x = I + x' and I + y = I + y' then 7+[xy] = 7 +[x'y'].
Now I + x = I + x' implies that x = x' + for some f e 7. Similarly I + у =
7 + у' implies у = у' + i2 for some i2 e I. Thus
=	-';l
= 7 + [z\y'] + [x'z2] + [z\ z2] + [x'y']
= 7+[x'y']
since [z\y'], [x'y, [y2] all lie in 7. Thus our multiplication is well defined.
We also have
[7 + x, 7 + x] = 7 + [xx] = 7
and the Jacobi identity in L/I clearly follows from the Jacobi identity in L.
4
Basic concepts
Now suppose we have two Lie algebras Lp L2 over A map L2
is called a homomorphism of Lie algebras if 0 is linear and
0[xy] = [0x, 0y] for all x, у e .
The map в:	L2 is called an isomorphism of Lie algebras if в is a
bijective homomorphism of Lie algebras. The Lie algebras Lp L2 are said to
be isomorphic if there exists an isomorphism в:	L2.
Proposition 1.6 Let ():L . ^L2 be a homomorphism of Lie algebras. Then
the image of в is a subalgebra of L2, the kernel of в is an ideal of and
Lj/ker0 is isomorphic to im 0.
Proof im в is a subspace of L2. Moreover for x, у in we have
[0(x), 0(y)] = 0[xy] e im 6.
Hence im в is a subalgebra of L2.
Now ker в is a subspace of . Let x e ker в and у e . Then
0[xy] = [0(x),0(y)] = [o, 0(y)]=o.
Hence [xy] ekerd and so kerd is an ideal of LP
Now let x, у e LP We consider when 0(x) is equal to 0(y). We have
0(x) = 0(y) 4>0(x — y) = 0ox — у eker0
О ker в + x = ker в + у.
This shows that there is a bijective map 0(x) -+ ker 0 + x between im в and
Lj/ker0. We show this bijection is an isomorphism of Lie algebras. It is
clearly linear. Moreover given x, y, z e we have
[0(x), 0(y)] = 0(z)O0[xy] = 0(z)
О ker в + [xy] = ker в + z
О [ker в + x, ker в + у] = ker в + z.
Thus the bijection preserves Lie multiplication, so is an isomorphism of Lie
algebras.	□
Proposition 1.7 Let I be an ideal of L and H a subalgebra of L. Then
(i)	I is an ideal of I+ H.
(ii)	IПН is an ideal of H.
(iii)	(I + 11)/1 is isomorphic to
1.2 Representations and modules
5
Proof. We recall from Proposition 1.4 that lrdl and I + H are subalgebras.
We have [7,I + H] c [ZL] С I, thus I is an ideal of I + H. Also |/ ПH, H\ c
[IH] П [НН] с / П //, thus 1ПН is an ideal of H.
Let в: H(I + H)/1 be defined by 0(x) = I + x. This is clearly a linear
map, and is also evidently a homomorphism of Lie algebras. It is surjective
since each element of (f + H)/I has form I + x for some .it//. Finally its
kernel is the set of xeII for which I + x = I, that is ICtH. Thus (J + H)/I is
isomorphic to НЦ1 Г\ II) by Proposition 1.6.	□
1.2 Representations and modules
Let Mn(k) be the associative algebra of all и x и matrices over the field k and
let [Mn (A) | be the corresponding Lie algebra. This is often called the general
linear Lie algebra of degree n over k and we write
Qln(k) = [Mn(k)].
We have dim gfn(^) = и2.
A representation of a Lie algebra L over k is a homomorphism of Lie
algebras
p:L^gfn(L)
for some n, and p is called a representation of degree n. Two representations
p, p' of degree n are called equivalent if there exists a non-singular n x n
matrix T such that
p'(x) = T‘p(x)T forallx^L.
A left L-module is a vector space V over k together with a multiplication
Lx V V
(x, v) —> XV
satisfying the axioms:
(i) (x, v) —> xv is linear in x and in v;
(ii) [xy]u = x(yu) — y(xv) for all x, у e L and v e V.
Suppose V is a finite dimensional L-module. Let ex,... , en be a basis of
V. Let
xej = l^Pij(x)ei
6
Basic concepts
with p,7Cv) e k and let p(x) = (p,;(v)). Then p is a representation of L. For
we have
|vv|e; = Xve;)
=x HPkj(y)ek “У UPkj^ek
\ к	/	\ к
= HPkj(y)xek ~HPkj(x)yek
к	к
= Y,Pkj(y) IEp«^)li -Ep^W (Ep»(y)ei
к	\ i	/к	\ i
= E X,(Pik(x)Pkj(y) -Pik(y)Pkj(x)) e;
i \ к	/
= E (рСФСу) - р(у)р W)y ei-
Thus p|vv| =p(v)p(v) — p(v)p(v) = |р(л), p(v)| and p is a representation
of L.
Suppose now we take a second basis /x,... , fn of V. Let p' be the repre-
sentation of L obtained from this basis. Then p' is equivalent to p. For there
exists a non-singular n x n matrix T such that
i
Thus we have
xfj = E Tkjxek = E Tkj (ЕРлСФ/) =E [EPiktxyTkj] ec
On the other hand
x// = Ep^WA=Ep^W (Er»ed =E (EWjW) ei-
к	к	\ i / i \ к	/
It follows that p(x)T = Tp'(x), that is p'(x) = T 1p(x)T for all x e L. Hence
the representation p' is equivalent to p.	□
Example 1.8 L is itself a left L-module.
The left action of L on L is defined as x • у = [xy]. Then we have
[ky]z] = [x[yz]] - [y[xz]]
1.3 Abelian, nilpotent and soluble Lie algebras
7
which is a consequence of the Jacobi identity. This shows that L is a left
L-module. This is called the adjoint module. We define adx:L^ L by
ad x • у = [xy]	for x, у e L.
Then we have
ad[xy] = ad x ad у — ad у ad x.	□
Now let V be a left L-module, U be a subspace of V and H a subspace
of L. We define HU to be the subspace of V spanned by all elements of the
form xu for x e H, ue U.
A submodule of V is a subspace U of V such that LU C U. In particular
V is a submodule of V and the zero subspace 0 = {0} is a submodule of V.
A proper submodule of V is a submodule distinct from V and O.
An L-module V is called irreducible if it has no proper submodules. V is
called completely reducible if it is a direct sum of irreducible submodules.
V is called indecomposable if V cannot be written as a direct sum of two
proper submodules. Of course every irreducible L-module is indecomposable,
but the converse need not be true.
We may also define right L-modules, but we shall mainly work with left
L-modules, and L-modules will be assumed to be left L-modules unless
otherwise stated.
1.3 Abelian, nilpotent and soluble Lie algebras
A Lie algebra L is abelian if [LL] = O. Thus [xy] = 0 for all x, у e L when
L is abelian.
Given any Lie algebra L we define the powers of L by
LX=L, L"+1 = [L"L] forn>l.
Thus L is abelian if and only if L2 = О.
Proposition 1.9 L" is an ideal of L. Also
L = L1z>L1z>L3z>--- .
Proof. We first observe that if I, J are ideals of L then | /./] is also an ideal
of L. For let x e I, у e J, z e L. Then
[Mz] = Ы] - [y[xz]] e [77].
8
Basic concepts
It follows that L" is an ideal of L for each n > 0. Thus we have
L"+1 = [L"L]cL".	□
A Lie algebra L is called nilpotent if Ln = О for some n > 1. Thus every
abelian Lie algebra is nilpotent. It is clear that every subalgebra and every
factor algebra of a nilpotent Lie algebra are nilpotent.
We now consider a different kind of powers of L. We define
= L, L("+1) = [L("fL(n)] forw>0.
Proposition 1.10 is an ideal of L. Also
l = l(0)dL(1)dL(2)d--- .
Proof. is an ideal of L since the product of two ideals is an ideal. Also
L("+r> = [Lw,LW]cL(").	□
A Lie algebra L is called soluble if = О for some n > 0.
Proposition 1.11 (a) [LmLn]cL1"^ for all m,n>l. (b) if"1 cl1" for all
n>0. (c) Every nilpotent Lie algebra is soluble.
Proof, (a). We use induction on n. The result is clear if n= 1. Suppose it is
true for n = r. Then
[LmLr+1] = \Lm\LrL\\ = \\LrL\Lm]
C [[LLm]Lr] + [[LmLr]L] by the Jacobi identity
C [Lm+1Lr] + [[LmLr]L]
C Lm+r+1 by inductive hypothesis.
Thus the result holds for n = r+ 1, so for all n.
(b). We again use induction on n. The result is clear if n= 1. Suppose it is
true for n = r. Then
L(r+i) = I I c |	| c /
by (a). Thus the result holds for n = r+ 1, so for all n.
(c). Suppose L is nilpotent. Then L2" = О for n sufficiently large. Hence
£(n) _ q (b) anj so £ is Soiuble.	□
1.3 Abelian, nilpotent and soluble Lie algebras
9
It is clear that every subalgebra and every factor algebra of a soluble Lie
algebra are soluble.
Proposition 1.12 Suppose I is an ideal of L and both I and L/I are soluble.
Then L is soluble.
Proof. Since L/I is soluble we have (L/7)^ = О for some n. This implies
L^ G I. Since I is soluble we have I1'-™1 = О for some m. Hence
jfn+m) _ (jfn)ym)	_ q
and so L is soluble.	□
Proposition 1.13 Every finite dimensional Lie algebra L contains a unique
maximal soluble ideal R. Also L/R contains no non-zero soluble ideal.
Proof. Let I, J be soluble ideals of L. Then 1 + J is also an ideal of L and
(I + J)/I is isomorphic to J/(JП J) by Proposition 1.7. Now J is soluble,
thus J/(IП J) is soluble and so (I + J//I is soluble. Since I is soluble we see
that I + J is soluble by Proposition 1.12. Thus the sum of two soluble ideals
of L is a soluble ideal. It follows that L has a unique maximal soluble
ideal R.
If //R is a soluble ideal of L/R then I is a soluble ideal of L by Proposi-
tion 1.12. Hence I = R and I/R = O.	□
The ideal R is called the soluble radical of L. A Lie algebra L is called
semisimple if R = O. Thus L is semisimple if and only if L has no non-zero
soluble ideal.
L is called simple if L has no proper ideal, that is no ideal other than L
and O.
Suppose L is a Lie algebra of dimension 1 over k. Then L has a basis {x}
with 1 element. Since [xx] = 0 we have L2 = 0. Thus L is abelian. We see that
any two 1-dimensional Lie algebras over k are isomorphic. Of course any such
Lie algebra is simple, because L has no proper subspaces. The 1-dimensional
Lie algebra is called the trivial simple Lie algebra. A non-trivial simple Lie
algebra is a simple Lie algebra L with dim L > 1.
Proposition 1.14 Each non-trivial simple Lie algebra is semisimple.
Proof. Suppose L is simple but not semisimple. Then the soluble radical R
satisfies RfO. Since R is an ideal of L this implies R = L. Thus L is soluble.
10
Basic concepts
Hence = О for some n > 0. This implies that L since = L would
imply =L for all n. Now is an ideal of L, hence = О since L is
simple. Thus [LL] = O. But then every subspace of L is an ideal of L. Since
L is simple L has no proper subspaces, so dim L = 1. Thus the only simple
Lie algebra which is not semisimple is the trivial simple Lie algebra. □
2
Representations of soluble and nilpotent
Lie algebras
2.1 Representations of soluble Lie algebras
We shall now and subsequently take the base field k to be the field C of
complex numbers. We shall also assume until further notice that L is a finite
dimensional Lie algebra over C, although at a later stage we shall also consider
infinite dimensional Lie algebras.
We first consider 1-dimensional representations of a Lie algebra L.
A 1-dimensional representation is a linear map p-.L^-C such that p[xy] =
[p(x),p(y)] for all x. у e L.
Lemma 2.1 A linear map p:L^C is a 1-dimensional representation of L
if and only if p vanishes on L2.
Proof Suppose p is a representation. Then for x, у e L we have
Pby] = [p(*)> p(y)] =p{x)p{y) -p{y)p{x) = 0.
Hence p vanishes on L2.
Conversely suppose that p vanishes on L2. Then
p[xy] = 0 = [p(x),p(y)]
and so p is a representation of L.	□
We shall now prove a theorem of Lie which shows that any irreducible
representation of a soluble Lie algebra is 1-dimensional.
Theorem 2.2 (Lie’s theorem). Let L be a soluble Lie algebra and V be a
finite dimensional irreducible L-module. Then dim V = 1.
11
12
Representations of soluble and nilpotent Lie algebras
Proof. Since L is soluble we have L2 f L. Let I be a subspace of L such that
/ D L2 and dim I = dim L — 1. Then I is an ideal of L since
[ZL]c[LL]=L2cL
Thus I is an ideal of L of codimension 1.
We shall prove Lie’s theorem by induction on dim L. Suppose dimL = 1
and V be an irreducible L-module. Let L = Cx and v be an eigenvector of
x in V. Then Cv is an L-submodule of V. Since V is irreducible we have
V = Cv and dim V = 1.
Now suppose dimL > 1 and V is an irreducible L-module. We may regard
V as an /-module. Then V contains an irreducible /-submodule W and we
may assume dim W = 1 by induction. Let w be a non-zero vector in W. Then
yw = X(y)w for all у el
where A is the 1-dimensional representation of I given by W. Let
U = {ue V ; yu = X(y)u for all у el}.
Then we have
OfWcUcV.
We shall show that U is an L-submodule of V. Let ueU, xeL. Then
y(xu) = x(yu) - [ху]и = Л(у)хи - Л([ху])и
since [xy] e I. We shall show A([xy]) = 0. Once we know this we have xueU
and so U is an L-submodule. Since V is irreducible we have U = V. Hence
yv = A(y)v for all v e V, у e I.
Since dim I = dimL — 1 we can write L = I ®Cx, a direct sum of subspaces.
Let v be an eigenvector for x on V. Then Cv is an L-submodule of V, being
invariant under the action of both I and x. Since V is irreducible we have
V = Cv and so dim V = 1.
In order to complete the proof we must show that A([xy]) = 0 for all x e L,
у el An fact it is sufficient to prove this for the element x chosen above such
that L = I® Cx.
Let и be any non-zero element of U. We write
v0 = «, Vj=xm, v2 = x(xm), ...
We have v0, vb v2,... e V and so there exists p > 0 such that v0, vb ... , vp
are linearly independent and vp+1 is a linear combination of these. Consider
the subspace (v0, vr,... ,v ) of V spanned by these vectors. This subspace
2.1 Representations of soluble Lie algebras
13
is invariant under the action of x. We consider the effect on this subspace of
elements у El. We have
yv0 = yu = X(y)u = X(y')v0.
We shall show
yv, = A(y)v; + a linear combination of v0,... , u;_P
This is true for i = 0. Assuming it for we have
= y(xVi_f) = x(yvi_1) -
= x(A(y)v;_1 + a linear combination of v0,... , u;_2)
— (a linear combination of v0,... , v^f)
= A(y)u; + a linear combination of v0,... , v;_P
Thus the subspace (v0, vr,... , v ) is invariant under the action of у for all
у el, as well as being invariant under x. Hence it is an L-submodule of V.
Since V is irreducible we have
V= Uj,... ,vp).
Now [xy] e 1 and we see from the above description of the action of 1 that
tracey[xy] = (p+l)A([xy]).
Thus we have (p + l)A([xy]) = tracev|xv| = traceyxy — tracevyv = 0, since
traceyxy = traceyyx. Hence A([xy]) = 0 and the proof is complete. □
Corollary 2.3 Let L be soluble and V be a finite dimensional L-module.
Then a basis can be chosen for V with respect to which we obtain a matrix
representation p of L of the form
(* \
0 *	*
p(x)= 0	forallxeL.
*
• -00*^
Thus the matrices representing elements of L are all of triangular form.
14
Representations of soluble and nilpotent Lie algebras
Corollary 2.4 Let L be a soluble Lie algebra with dim L = n. Then L has a
chain of ideals
O = I0Gl1G---Gln_1Gln = L
with dim /.. = r.
Proof We apply Theorem 2.2 to the adjoint L-module L. The submodules of
L are the ideals of L. By taking a maximal chain of submodules we obtain
ideals of L with the required property.	□
2.2 Representations of nilpotent Lie algebras
When L is a nilpotent Lie algebra we can obtain even stronger results about
its representations. Moreover these results on representations of nilpotent Lie
algebras play a crucial role in the understanding of semisimple Lie algebras,
which we shall deal with subsequently. We begin by recalling results from
linear algebra related to the Jordan canonical form. Any n x n matrix over C
is similar to a diagonal sum of Jordan block matrixes of form
/А 1	\
А 1	0
• 1
О	Л 1
\	4
In a similar way any linear transformation в: V V on a finite dimensional
vector space V over C gives rise to a decomposition of V as in the following
proposition.
Proposition 2.5 Let 0:V V be a linear map with characteristic polynomial
where Ab ... , Ar are the distinct eigenvalues of в and тъ ... ,mr are their
multiplicities. Let Vt be the set of all v e V annihilated by some power of
0 — A;l. Then we have
v=v1®v2®---®vr.
Moreover dim V; = m;, 0(V;) c V; and the characteristic polynomial of в on
V; is (t-Xf)”1-.
2.2 Representations of nilpotent Lie algebras
15
Proof. Although this is a standard result from linear algebra we shall prove it
in view of its importance for the theory of Lie algebras.
We begin by showing that V; is equal to W; = {ue V ; (0 — A;lpu = 0}. It
is clear that Wt C V;. So let v e V;. Then
(0 — A;1)P = O for some A.
We may choose N >mt. Also
П(0-А;1Гг = О
7=1
for, by the Cayley-Hamilton theorem, в satisfies its own characteristic equa-
tion. Now the polynomials
(t-Xf, П(г-Л?Г>
7=1
have highest common factor (t —A;p- Thus there exist polynomials
p(f), qft) e C[t] such that
(t - A;p = /,(t)(t - A,)v + <?(t) П (t - X^ •
7=1
Hence
(0 - A; l)m- и = />(0)(0 - А;1)"и+ ?(0) П (0 - Aj ip V = 0.
7=1
ThusveW; and V; = W;.
We next show that V = V1 ® • • • ® Vr. Let /;(t) = (t — Axp ... (t — A;_1)'"i-1
(t — A;+1p+1... (t — Xf)"1'. Then the polynomials (t), • • • , fr(t) have high-
est common factor 1. Thus there exist polynomials pff),... , pr(f) e C[t]
with E;/;(tp(f) = 1-
Let veV. Then v = ^fifi(ff)pi(ff)v. Let vt= ffff)pfff)v. Then
(0-A;1Pu; = x(0)(a(0)h) = O
by the Cayley-Hamilton theorem. Thus vte V;. Hence и=1р------hur with
vt e V;, and so V = Vr 4-h Vr.
In order to show the sum is direct we must prove
v;n(Li + --- + Pi + Pi + --- + vr) = c»-
16
Representations of soluble and nilpotent Lie algebras
Now the polynomials (t — A;)m‘ and fff) have highest common factor 1, thus
there exist p(f), q(f) eC[t] with
Let v e V; П (Vx 4--h V;_j + V;+14----h Vr). Since v e V; we have
(0 —A;l)m,u = 0.
Since v e V14----h Vi_1 + V;+14----h Vr we have
/;(0)u = O.
Hence u = /7(0)(0 —A;l)m‘T> + <?(0)/;(0)u = O. Thus we have shown that
V= Vj©---®Vr.
We next observe that в acts on each V;. For let v e V;. Then
(0 - A; l)m'6v = 6(6 - A; l)m'v = 0(0) = 0,
thus 6vE Vj.
We next show that the only eigenvalue of 0:V;^V; is A;. Suppose if
possible that Xj is an eigenvalue for some j i and let v e V; be an eigenvector
for Xj. Then (0 —A;l)m,u = 0 and (0 — A^l)v = 0. But the polynomials
(t —A;)m‘ and t — Xj have highest common factor 1 so there exist p(t), q(t) e
C[t] with
Ht)(t-A;r + (?(t)(t-Ap = l.
Hence u = /?(0)(0 —A;l)m,u+<?(0)(0 —Ajl)u = O, a contradiction. So all
eigenvalues of 0,: V; Vt are equal to A;. It follows that dim V; < mt since
mt is the multiplicity of eigenvalue A; on V. But
dim V = dim V) 4----hdim Vr = т14-----h mr.
It follows that dim Vi = mi. Finally the characteristic polynomial of 0 on V; is
(t —A;)m-.	□
The subspace V; is called the generalised eigenspace of V with eigen-
value A;. Thus the ordinary eigenspace of A; lies in the generalised eigenspace.
It is not in general true that V is the direct sum of its eigenspaces with respect
to its different eigenvalues, but Proposition 2.5 shows that this result is true
if the eigenspaces are replaced by the generalised eigenspaces.
The relevance of the decomposition into generalised eigenspaces for the
representations of nilpotent Lie algebras is shown by the following theorem.
2.2 Representations of nilpotent Lie algebras
17
Theorem 2.6 Let L be a nilpotent Lie algebra and V be an L-module. Let
у eL and p(y): V V be the map v yv. Then the generalised eigenspaces
Vi of V associated with p(y) are all submodules of V.
Before proving this theorem we need a preliminary result.
Proposition 2.7 Let L be a Lie algebra and V be an L-module. Let v e V,
x,y eL and а, /ЗеС. Then
(pCv)-(a + /3)l )"w>=]T	((ady - /31fx) (fp(y) - «!)"+).
i=o \1/
Proof. We use induction on n. The result is clear when n = 0. We assume it
for n = r. We write
xt = (ad у — /31fx e L.
Then we have
(p(y) - (a + /3)l)r+1 xv = (p(y) - (a + /3)1) £ (Л p(xt)(p(y) - al)r~lv.
i=o \1/
Now
(p(y) - (a + /3)l)p(x;) = p([yx;]) + p(+)p(y) - (a + /3)p(x;)
= P ((ad у - /31)+) + p(+)(p(y) - al)
= p(++i)+р(+)(р(у) - “О-
Hence
(p(y)-(a + /3)l)r+1xu
= E (P(++i) (Р(У) - «I / ' ’ ’ + E P,- ) P(xi) (Р(У) ~ al)'1' v
i=0 \1/	i=0 \1/
r+1 /	\	r+1 / \
= E ( ; \ ) p(+)(p(y) - a I)''1 ')’ + E ( • ) p(+)(p(y) - a I )'1 +
;=o v V	i=o \V
(I r \	I r \	\
interpreting I ) = 0 and I ^1=01
r+1 /r_i_ i\
= E( ; ) ((ady-/31)!x) ((pCv)-al)'1 T).
i=o \ 1	'
This completes the induction.	□
18
Representations of soluble and nilpotent Lie algebras
Proof of Theorem 2.6. Let v e V;, x, у e L. Then
(p(y) - 1)" xv = £ (П\ ((ady)'x) ((p(y)-A;l)"~7u)
j=o \J/
by Proposition 2.7 with a = A;, /3 = 0. Since veVt, (p(y) — A(l)"7r = 0 if
n — j is sufficiently large. Since L is nilpotent (ady)7x = 0 if j is sufficiently
large. Thus (p(y) — А(1)"лт = 0 if n is sufficiently large. Hence xve Vt and
so V; is a submodule of V.	□
Corollary 2.8 Let L be a nilpotent Lie algebra and V a finite dimensional
indecomposable L-module. Then a basis can be chosen for V with respect to
which we obtain a matrix representation p of L of the form
^A(x)
p(x) =
\
for all xeL.
Proof. We can choose a basis as in Corollary 2.3 with respect to which each
p(x) is triangular. The generalised eigenspaces of V with respect to p(x) are
all submodules of V by Theorem 2.6 and V is their direct sum. Since V is
indecomposable only one of the generalised eigenspaces is non-zero. Thus
all the eigenvalues of p(x) on V are equal. Let this eigenvalue be А (л). Then
the diagonal entries of the triangular matrix p(x) are all equal to А (л). □
We observe that the map x^ А (л ) is a 1-dimensional representation of L,
as it arises from a 1-dimensional submodule of V.
We have seen from Proposition 2.5 and Theorem 2.6 how to obtain a direct
decomposition of V into submodules for any element y&L. We may use this
result to obtain a direct decomposition of V into submodules which does not
depend on the choice of any particular element of L.
Theorem 2.9 Let L be a nilpotent Lie algebra and V a finite dimensional
L-module. For any 1-dimensional representation A ofL we define l';= {i-1 V’:
for each x e L there exists N(x) such that (p(x) — A(x)l)'v^u = 0}. Then
V = ®Vs
Л
and each VA is a submodule of L.
2.2 Representations of nilpotent Lie algebras
19
Proof. We first express V as a direct sum of indecomposable L-modules. Each
of these defines a 1-dimensional representation A of L as in Corollary 2.8.
Let WA be the direct sum of all indecomposable components giving rise to A.
Then we have
V = ®WA.
л
We shall show that WA= VA and so that WA is independent of the decom-
position chosen into indecomposable components. It is clear that WA c VA
by Corollary 2.8. Suppose if possible that WA^VA. Then there exists
v e	with v # °- We write =	with w^eW^, where
the set 5 is finite. Since ir, e there exists such that (p(A) —
p.(A)l)A'	= 0. Hence
П (p(v)-p(x)l)A''‘u = 0.
However, we also have (p(x) — A(.v) 1)лА> = 0.
We recall from Lemma 2.1 that the 1-dimensional representations of L are
in bijective correspondence with linear maps	The vector space
Lf!? over C cannot be expressed as the union of finitely many proper
subspaces. Lor each pceS the set of x satisfying A (A) = p. (A) is a proper
subspace. Thus there exists .it/. such that A(A)^p.(A) for all pceS. Thus
the polynomials
are coprime. Thus there exist polynomials a(t), b(t) e C[t] such that
a(t) П (/ — p (Av))V/' + b(f) (/ — A(.v))v' = 1.
jj.eS
Hence
ФМ) П (p(v)-p(x)l)A''‘u + fo(p(x))(p(x)-A(x)l)A'iu=u.
The left-hand side of this expression is zero, as we have seen above. Thus
v = 0, a contradiction. Hence VA = WA, V = фА VA and each VA is a submodule
of V.	□
A 1-dimensional representation A of L is called a weight of V if VA 0, and
VA is called the weight space of A. The decomposition V = фА VA is called
the weight space decomposition of V. It follows from Corollary 2.8 that a
20
Representations of soluble and nilpotent Lie algebras
basis can be chosen for VA with respect to which the matrix representation of
L on VA has form
^A(x)
p(x) =
\
for each .re/..
We shall make frequent use of the weight space decomposition in subsequent
chapters.
We next prove a theorem of Engel which gives a useful characterisation of
nilpotent Lie algebras in terms of the adjoint representation.
Theorem 2.10 (Engel’s theorem). A Lie algebra L is nilpotent if and only if
ad .v:	L is nilpotent for each xeL.
Proof. Suppose L is nilpotent. Then Ln=O for some n. Let y&L. Then we
have
ad.v-ve/.2,	(ad.v)2-v e/.3,
and so (ad x)!li y = 0 for each у e L. Thus (ad .v)" 1 = 0 and so ad x is a nilpo-
tent linear map.
Now suppose conversely that ad x is a nilpotent linear map for each x e L.
We wish to show L is nilpotent. We suppose if possible that this is false and
let H be a maximal nilpotent subalgebra of L. Thus H is nilpotent but any
subalgebra properly containing H is not nilpotent. We may regard L as an
//-module. Then H is an //-submodule of L and we can find an //-submodule
M of L containing H such that M/H is an irreducible //-module. We have
dim(A//Z/) = 1 by Theorem 2.2.
Moreover the 1-dimensional representation of H afforded by M/H is the zero
representation, as otherwise adv would fail to be nilpotent for some xeII.
Hence we have [НМ] с H. Now there exists x e M such that
M = H®Cx.
We have
[MM] c [HH] + [Hr] c H.
Thus M is a subalgebra of L and H is an ideal of M.
2.2 Representations of nilpotent Lie algebras
21
We shall show that for each positive integer i there exists a positive integer
e(z) such that
ме(;) c Hl
This is true for i = 1 since M2 C H. We prove it by induction on i. Assume
that Me(r> c Hr. Then
Me(r)+1 = [Me(r),H + Cx] c Hr+1 + |ЛГ1Л1, .v|.
Hence Me«+1 c7/r+1+advMe«.
We shall show that
Me(r)+j c Hr+1 + (ad xy . Me(r)
for each positive integer j. This is true for j = 1. Assuming it inductively for
j we have
Me(r)+7+1 c [Яг+1 + (adx)J.Me(r), M]
C Hr+1 + [(adx)jMe(r),H + Cx]
C Z/'1 + (ad.v)7 'ЛГ'1'''1
since Hr+1 is an ideal of M and (adcHr. Thus we have shown
M<r)+j c Hr+1 + xy Me(r) for ац j
Now we know that (ad.v)7 = 0 when j is sufficiently large. For such j we
have
Thus we define e(r+ 1) = e(r) + j and then cH'+1 as required.
Now H is nilpotent so Hl = О for i sufficiently large. For such i we have
Me^ = 0. Thus M is nilpotent. But this contradicts the maximality of H.
Thus our initial assumption was incorrect and so L must be nilpotent. □
Corollary 2.11 A Lie algebra L is nilpotent if and only if L has a basis with
respect to which the adjoint representation of L has form
/0
0
p(x) =
for all xeL.
0
0/
22
Representations of soluble and nilpotent Lie algebras
Proof. Suppose L is nilpotent. Then L has a series of ideals
L D L1 D L3 D • • • D Lr = О for some r.
We refine this series by choosing a sequence of subspaces between consec-
utive terms, each of codimension 1 in its predecessor. Such subspaces are
automatically ideals of L since if □ Li+1 we have
[7L]c[L;L]=L;+1 С I.
Thus we have a chain of ideals
= О
with dim/,. =/: and cIk_v By choosing a basis of L adapted to this
chain of ideals the map adx:L^ L is represented by a matrix p(x) of zero-
triangular form (i.e. triangular with zeros on the diagonal).
Conversely if L has a basis with respect to which adv is represented by
a zero-triangular matrix p(x) for all .it/., we have p(x) nilpotent and so
adv is nilpotent. Thus L must be a nilpotent Lie algebra by Engel’s theorem
(Theorem 2.10).	□
3
Cartan subalgebras
3.1 Existence of Cartan subalgebras
Let H be a subalgebra of a Lie algebra L. Let
N(H) = {л- e L; [/ix] e II for all h e H}.
N(H) is called the normaliser of II.
Lemma 3.1 (i) N(H) is a subalgebra of L.
(ii) II is an ideal of N(II).
(iii) N(II) is the largest subalgebra of L containing H as an ideal.
Proof (i) Let x, у e N(II'). Then
[/i[xy]] = [[y/i]x] + [[/ix]y] e H.
Hence [xy] e N(H) and N(II) is a subalgebra.
(ii) This is clear from the definition of N(H).
(iii) If II is an ideal of M then [HM]cH soMc N(II).	□
Definition A subalgebra H of L is called a Cartan subalgebra if H is
nilpotent and II = N(II). Cartan subalgebras play a very important role in
the theory of semisimple Lie algebras. Our aim in this section is to show that
L contains a Cartan subalgebra.
Let us take an element x e L and consider the linear map ad x : L^L. Let
LOx be the generalised eigenspace of adx with eigenvalue 0. Thus LOx =
fcl', there exists n such that (ad x)"y = 0}, and Lo x will be called the null
component of L with respect to x.
An element x e L is called regular if dim Lo x is as small as possible. The
Lie algebra L certainly contains regular elements.
23
24
Cartan subalgebras
Theorem 3.2 Let x be a regular element of L. Then the null component Lo^
is a Cartan subalgebra of L.
Proof Let H = LOx. We must show that H is a subalgebra of L, that II is
nilpotent, and that II = N(II).
We first show that H is a subalgebra. Let y,ztH. We must show that
[yz] e H. By Proposition 2.7 we have
(ad.v)"| vz| =	f”') [(ad.v)'v, (adл)" 'z].
i=o \l/
(We take V = L, a = /3 = 0 in Proposition 2.7 to obtain this.) Since у e H we
have
(adx)!y = 0 if i is sufficiently large.
Since z £ H
(ad xf / = 0 if n — i is sufficiently large.
Hence (ad x)”[yz] = 0 if n is sufficiently large. Thus |vz|e/7 and H is a
subalgebra of L.
We next show that H is nilpotent. To do this we shall prove that all the
matrices in the adjoint representation of H are nilpotent and use Engel’s
theorem (Theorem 2.10). Let dim 77 = / and bx,... , bt be a basis for 77. Let
y = A1A>1 + -- - + A;A>;e77	Ai,... ,A^gC.
Consider the linear map ad у : L^L. We have ad у : II II since 77 is a
subalgebra and we obtain an induced map ad у : L/H	L/H.
Let xft) be the characteristic polynomial of ad у on L, Xi(0 be its charac-
teristic polynomial on 77 and y2L) be its characteristic polynomial on L/H.
Then we have
L(0 = Li(0L2(0-
Since xft) = det(/1 — ady) and у depends linearly on Ap ... , A; we see that
the coefficients of xif) are polynomial functions of Ap... , A;. The same
applies to Xi(0 and Аг(0- Let
T2(0 = ^o + ^if + ^2f2_l----
where d0, cf,d2,  are polynomial functions of Ab ... , A;. We claim that d0
is not the zero polynomial. For in the special case when у = x we know that
3.2 Derivations and automorphisms
25
all eigenvalues of ad у on L/H are non-zero, so y2(0 has non-zero constant
term. Let
Xl(f) = f,"(Co + Clf+C2f2“l-)
where c0, cb c2,... are polynomial functions of Ab ... , A; and c0 is not the
zero polynomial. We have
m</ = degxi(t).
We then have
Xfi) = tm(codo + terms involving positive powers of t).
Now codo is not the zero polynomial so we can choose Ap... , A; e C to
make codo non-zero. For such an element у e H we have
dimL0 = m.
Since x is regular and dimL0 = / we have m>l. Since we also know m < I
we have m = l. Now Xi(0 has degree / and is divisible by tl, hence
Li(0 = f'-
It follows by the Cayley-Hamilton theorem that (ady); : H H is zero.
Hence by Engel’s theorem we deduce that H is nilpotent.
Finally we show that H = N(H). It is certainly true that HcN(H). So let
z e N(H). Then [xz] e H. Thus
(adx)"[xz] =0 for some n.
But then (adx)"+1z = 0 and so z&H. Thus H = N(H) and we have shown
that H is a Cartan subalgebra of L.	□
3.2 Derivations and automorphisms
A derivation of a Lie algebra L is a linear map 8 :	L such that
<5[xy] = [<5x, y] + [x, <5y] for all x, ye L.
Lemma 3.3 Let xeL. Then ad x is a derivation of L.
Proof, ad x[yz] = [ad x • y, z] + [y, ad x • z] by the Jacobi identity.	□
An automorphism of L is an isomorphism в : L^L. The automorphisms
of L form a group Aut L under composition.
26
Cartan subalgebras
Proposition 3.4 Let 8 be a nilpotent derivation of L. Then exp S is an
automorphism of L.
Proof Since й is nilpotent we have 8n = 0 for some n. Then we have
n— 1 Or
exp <5 =	—
r!
r=0 1 •
The map exp <5 : L L is clearly linear. Let x, у e L. Then
<5[xy] = [<5x, y] + [x, Sy]
8r[xy] = E (;) [<5'w,	'y]
i=0 V/
as is easily seen by induction on r. Hence
exp 8  [xy] = E E "7 (•) У '>’] = E E Y7
r>0i=0r' V/	i>Oj>Ol'J-
Thus exp 8: L^L is a homomorphism. Similarly exp (—5) is a homo-
morphism and we have exp <5 exp (—S) = l. Thus exp S :	L is an
automorphism.	□
The subgroup of Aut L generated by all automorphisms exp ad x for all x e L
with adx nilpotent is called the group of inner automorphisms InnL. Every
element of Inn L has form
exp ad Xj • exp ad x2.exp ad xr
where xx,... , xr e L and ad xx,... , ad xr are all nilpotent.
Lemma 3.5 Inn L is a normal subgroup of Aut L.
Proof Let 0 e AutL. It is sufficient to show that 0(exp adx)0 1 e InnL for
all x e L with ad x nilpotent. Now we have
d(adx)d 'у = в [x, d ' v] = [0x, y] = (ad 0x~)-y
for all y&L. Hence
0(adx)0 '=addx.
3.3 Ideas from algebraic geometry
27
It follows that
0(exp adx)d 1 =exp ad(tfv) elnnL.
Thus InnL is normal in AutL.	□
Two subalgebras M1,M2 of L are called conjugate in L if there exists ве
InnL such that в(М1) = М2.
We wish to show that any two Cartan subalgebras of L are conjugate in L.
However, we first need some concepts from algebraic geometry.
3.3 Ideas from algebraic geometry
Let H be a nilpotent subalgebra of a Lie algebra L and regard L as an
//-module. Then we obtain a decomposition
^ = ф-^л
as in Theorem 2.9, where
LA = {x e L; for each h e H there exists n such that (ad h — X(h) l)"x = 0}.
Now H lies in Lo by Corollary 2.11. We shall suppose that the nilpotent
subalgebra H satisfies the condition H = L0. Then there exist 1-dimensional
representations Ap ... , Ar of H with Ax ^0,... , Ar 0 and
/, = //ф/.Л| ф---ф/.Л;.
Given x e L we then have
x = x0 + xx + • —h xr
with x0 e H and x; e LA for i = 1,... , r. We claim that ad x; : L L is nilpo-
tent when i 0.
To see this let p. : HC be a weight of the H-module L and let
Then by Proposition 2.7 we have
(ad/i —/z(/i)l — A;(/i)l)" [x;y] = J2 ( П ) [(ad/i — A;(/i)iyx;,
j=o \J/
(ad/i-,u(/i)l)"-7y].
Because x;eLA. then (ad/i —A;(/i)l)-'x; = 0 if j is sufficiently large. Since
y^b,j then (ad/z—/x(/z)l)"7v = O if n — j is sufficiently large. Thus
(ad/i —/z(/i)l — A;(/i)l)" [x;y] = 0
28
Cartan subalgebras
if n is sufficiently large, and so [x;y]	Thus we have
adx;-LMcLAi+A[.
Since A; 0 and there are only finitely many /х : H C for which
L^O we see that (ad .v,);V = 0 if N is sufficiently large. Thus ad xt is
nilpotent.
We deduce that exp ad xt e Aut L for z^O. We now define a map
f : L L by
/(x) = exp ad • exp ad x2.....exp ad xr • x0.
We shall discuss some properties of this function f. We choose a basis {btj}
of L for 0 < i < r where for fixed z the elements btj form a basis of LA. with
respect to which the elements of H are represented by triangular matrices, as
in Corollary 2.3. Here Ao = 0.
Lemma 3.6 f : L —> L is a polynomial function. Thus
where each is a polynomial in the Xkl.
Proof. Each map ad xt : L L is linear. Also we have
since ad xt is nilpotent. Thus exp ad xt : L L is a polynomial function.
The given map f is a composition of the linear map x x0 with polynomial
functions exp ad xt for i > 0, so is a polynomial function.	□
We write ptj = fij(^kl) where ftj is a polynomial. We define the Jacobian
matrix
Л/) = (^/али)
and the Jacobian determinant det J(f) of f. det./(/') is an element of the
polynomial ring С[ЛЫ].
Proposition 3.7 det J(f) is not the zero polynomial.
Proof. We shall show det J(/) is not the zero polynomial by showing that it
is non-zero when evaluated at a carefully chosen element of H. So let h eH
and consider (аД/аЛы)А.
3.3 Ideas from algebraic geometry
29
First suppose k^G. Then
(df/dXkl)h = lim---------------------
= lim (eXP ad tbkl>)h ~ h
t^o	t
_ цт h + t[bkl, 7г] 4-----h
= [bki, h] = -\hbkl\
= — Xk(h)bkl + a linear combination of bkl,, bkl_x
Next suppose k = Q. Then
\ г f(h+tboi)~f(h)
W/dX0l)h = lim-----------------
h + tb0l-h
= lim-----------= b0l.
t^o t
and so (det J(/))A = ±П[=1 A;(/i)d‘ where <7; = dimLA,.
Now the linear maps A; : H C for i = 1,... , v are all non-zero. Thus
we can find an element h e H with A;(/i) 0 for i = 1,... , v. For such an
element h we have (det J(ff)h ф 0. Hence det J(/) is not the zero polynomial.
□
Proposition 3.8 The polynomial functions ftj are algebraically independent.
30
Cartan subalgebras
Proof. Suppose if possible that there is a non-zero polynomial F(xy) e C[xy]
such that = 0. We choose such a polynomial F whose total degree in
the variables xy is as small as possible. Then
and so
dF dftj
-----— = ().
dftj dXkl
Let v be the vector (5F/5/y). Then
vJ(/) = (0,...,0).
Since det J(/) is non-zero this implies that v = (0,... ,0), that is
dF/dfij = Q for each/y.
Now dF/dx^ is a polynomial in C[xy] of smaller total degree than F. By the
choice of F dF/dx^ must be the zero polynomial. Hence F does not involve
the variable x;;. Since this is true for all x;; F must be a constant. Since
F(Jfj = Q this constant must be zero. Thus F is the zero polynomial and we
have a contradiction.	□
Let B = C[/y] be the polynomial ring in the ftj and A = C[Ay] the poly-
nomial ring in the Ay. We have a homomorphism в : B^> A uniquely deter-
mined by
0(/y) = /y(AJeA.
Proposition 3.9 The homomorphism в : В A is injective.
Proof. Suppose FeB satisfies 0(F) = 0. Then F(/y) = 0, regarded as a
function of the Xkl. Since the /y are algebraically independent this implies
that F = 0. Thus в is injective.	□
Thus we may regard В as a subring of A. A and В are integral domains
with a common identity element and A is finitely generated over B. We next
prove a general result which applies to this situation.
Proposition 3.10 Let A and В be integral domains such that В с A, A, В
have a common identity element 1, and A is finitely generated over B. Let
p be a non-zero element of A. Then there exists a non-zero element q of В
such that any homomorphism ф : B^C with di(q)fi) can be extended to a
homomorphism ф : A C with Ф(р) 0.
3.3 Ideas from algebraic geometry
31
Proof We may assume that A is generated over В by a single element f.
For then by iterating the process we can prove the result when A is finitely
generated over B. Thus we assume that A = B[£] for some f e A.
Suppose first that f is transcendental over B. Given a non-zero element
p = p(if) e A we choose q eB to be one of the non-zero coefficients of p(if).
Suppose we are given a homomorphism ф : B^> C with <b(q) 0. We write
ф(Ь) = b e C. By applying ф to the coefficients of p(if) we obtain p(if) e C[£].
The element p(if) is not the zero polynomial since ф(с[) ^0. We can find an
element /3 e C with p(/3) 7^ 0. We now define a homomorphism ф : A -+ C by
ФШ)=№-
ф is well defined since f is transcendental over B, and ф is a homomorphism,
being a composite of the homomorphisms
А = В[£]^С[Д^С	(3.1)
g{^)	-^g(f) -^g(/3)	(3.2)
ф clearly extends ф. Finally we have Ф(р) = р([3) ф 0.
Next suppose that f is algebraic over B. Then we can find /(t) e B[t] of
minimal degree such that /(£) = 0. We write
/(t) = foot"+fo1t"-1 + ---+fe„ wo.
Now let g(t) be any polynomial in B[t] satisfying g(f) = 0. We divide g(t) by
/(t) using the Euclidean algorithm. We are working over an integral domain В
rather than over a field. However, provided we multiply g(t) by a sufficiently
high power of the leading coefficient b0 of /(t) we can carry out the Euclidean
process over B. We thus obtain
4>40 = «(0/(0 + 40
where w(t), u(t) e B[t] and deg ;’(/) < deg f(f). Thus
<) = 44C)-«(W)=o.
Since degu(t) < deg f(t) this implies that v(t) = 0. Hence
440 = «(0/(0-
Let p be the given non-zero element of A. The element p is algebraic over
В since A is generated over В by the single algebraic element if Thus there
exists a polynomial h(f) e B[t] with non-zero constant term hm such that
li(p) = 0. We define the element q e В by q = bohm. Thus q f 0. We assume
we are given a homomorphism ф : B^> C with (l>lq) f 0. Then ф(Ь0) f 0 and
32
Cartan subalgebras
ф(И.т) 0. We write ф(Ь) = b e C. The polynomial /(t) e B[t] gives rise to a
polynomial /(t) e C[t]. We choose an element /3 e C with /(/3) = 0. We note
that
^(/3) = й(/3)/(/3) = О,
hence g(/3) = 0 since b0 = ф(Ь0) Ф 0.
We now define a homomorphism
ф : A^C
by = A'(/3). We note that the map ф is well defined, since we have
shown that g(£) = 0 implies g(/3) = 0. The map ф is a homomorphism since
the maps
B[t] -+ C[t] -+ C
are homomorphisms. The definition of ф shows that ф extends ф. Finally we
show ф(р)^£0. Since h(p) = 0 we have 1г(ф(р>У) = 0. However, the constant
term of hft) is which is non-zero. Since hft) has non-zero constant
term and Ь(ф(р)) = 0 we must have Ф(р) 0.	□
We now apply this result to our earlier situation. Let d = dimL and
f : Cd^Cd
be the polynomial function
(Лу)^(/у(Лы)).
We write V = Cd and for each polynomial p e C[xy] we write
Vp = {veV; p(»)^0}.
Corollary 3.11 For each non-zero polynomial p e C[xy] there exists a non-
zero polynomial q e C[xy] such that f(V ) D Vq.
Proof. We apply Proposition 3.10 to the integral domains Bed discussed
earlier. Thus A is the polynomial ring C[Ay] and В is the polynomial ring
C[/y]. We choose a non-zero polynomial p e A. Then there exists a non-zero
polynomial qE В such that any homomorphism ф : B^C with (l>lq) 0 can
be extended to a homomorphism ф : A^ C with Ф(р) ^0. This means that
given any ve V we have v = f(w) for some weV. Hence V <zf(V ) as
required.	□
3.4 Conjugacy of Cartan subalgebras
33
3.4 Conjugacy of Cartan subalgebras
We showed in Theorem 3.2 that the null component LOx of a regular ele-
ment x eL is a Cartan subalgebra of L. We shall now show conversely that
any Cartan subalgebra is the null component of some regular element. We
shall then prove that, given two regular elements, their null components are
conjugate in L.
Proposition 3.12 Let H be a Cartan subalgebra of L. Then there exists a
regular element xeL such that H = LOx.
Proof Since II is nilpotent we may regard L as an //-module and decompose
L into weight spaces with respect to H as in Theorem 2.9. H lies in the zero
weight space Lo by Corollary 2.11. Since H = N(II) we can show that H = L0.
For if	the //-module Lo/H will have a 1-dimensional submodule
M/H on which H acts with weight 0. Hence [HM\ ell and so M <z,\'(H).
This contradicts II = N(II). Thus we have H = L0. Let
L = H®LXi®---®Lx? Лр.-.Д^О
be the weight space decomposition of L with respect to H. Let xeL and
x = x0 + xx + • —h xr
with xoeH and x; e Lk. for i 0. Then we can define a polynomial function
f : L^> L as in Section 3.3 with
/(x) = exp ad xx • exp ad x2.exp ad xr • x0.
We define p : L C by
/?(х) = A1(x0)A2(x0) • • • Ar(x0).
Then p is a polynomial function on L. p is not the zero polynomial since
we can find xoeH for which each A;(xo)^O for i=l,... , r. Hence by
Corollary 3.11 there exists a non-zero polynomial function q : L^ C such
that/(Lp) D L?.
We next consider the set R of regular elements of L. Let у eL and
Т(у) = det(/1 - ady) = /" + м! (У)/" 1 + • • • + pn (y)
be the characteristic polynomial of ad у on L. Then  , P-,, are poly-
nomial functions on L. There exists a unique integer k such that p.tl к is
not the zero polynomial but pn_k+i,... , pn are identically zero. The gener-
alised eigenspace of ad у with eigenvalue 0 has dimension к if p.. , k (y) 0
34
Cartan subalgebras
and dimension greater than k if lxn_k(y) = Q- Thus у is regular if and only if
Now there exists у El. such that yELqC\R. For we may choose у with
# 0- Since L c f{L } we can find xcf. such that f(xj = y. Thus
we have
exp ad xx • exp ad x2 • • • • exp ad xr-x0 = y.
Hence x0, у are conjugate elements of L. Since у is regular, x0 must also be
regular. Sincex^Lp we have
M*o)M*o) • • • Ar(*o) # 0.
Now xoeH and H is nilpotent, hence /.,,,.,□// by Corollary 2.11. On the
other hand Lo cannot be larger than H since
Ai(*o)#°, •••
Hence H = Lo,Xo where x0 is regular.	□
Theorem 3.13 Any two Cartan subalgebras of L are conjugate.
Proof. Let II. H' be Cartan subalgebras of L. We regard L as an
//-module and decompose L into weight spaces with respect to H. We have
seen in the proof of Proposition 3.12 that H = L0. Let the weight space
decomposition be
L = H®LXi®---®Lx? Ap ... ,Ar#0.
For each re / we have
x = x0 + xx 4— • + xr
with x0 e H and x; e LA for i 0.
Now for each xoe// we have LOxod// and for some xoe// we have
Lo = H since H is a Cartan subalgebra. An element x0 e H is regular if and
only if Lo =H. This is equivalent to the condition
M*o)M*o) • • • Ar(*o) # 0.
We now consider the polynomial function f :	L defined by
/(x) = exp ad xx • exp ad x2.exp ad xr • x0.
Let p : L -e C be the function given by
/?(x) = Aj (x0)A2(x0) • • • Ar(x0)
3.4 Conjugacy of Cartan subalgebras
35
where p is a polynomial function on L which is not identically zero, since
p(x) is non-zero when x0 is a regular element of H. By Corollary 3.11 there
exists a non-zero polynomial function q :	C such that f(Lpj D L .
We now start with the second Cartan subalgebra 1Г. We can define a corre-
sponding function f :	L and a corresponding function p' : L —> C. There
exists a non-zero polynomial function q' : L^C such that f'(Lp,j □ L^.
Now LqПLq, = {xeL ; (qq')(x)^0}. Thus Lq^L^ is non-empty. We
choose zeL П£^. Thus z G/(Lp)n/'(Ly). Thus there exists xeL with
z = /(x) and /?(x)^0. Similarly there exists x' eL with z = /'(x') and
p' (x) 0. Thus
z = exp ad • exp ad x2...exp ad xr  x0
and so z is conjugate to x0. Since p(x)^£Q x0 is regular. Similarly z is
conjugate to xqi and xqi is regular. Thus we have found regular elements x0 e H
and Хд e H' such that x0, x't are conjugate in L.
Now we have H = LOxo and ir = I.;i q since xo,x'o are regular. Thus an
inner automorphism of L which transforms x0 to x'o will transform H to H'.
Hence H, H' are conjugate in L.	□
The dimension of the Cartan subalgebras of L will be called the rank of L.
4
The Cartan decomposition
4.1 Some properties of root spaces
Let L be a Lie algebra and H be a Cartan subalgebra of L. We regard L as
an //-module. Since H is nilpotent we have a weight space decomposition
^ = ф^л
л
as in Theorem 2.9, where
L^ = {xeL ; for each heH there exists n such that (ad/? — Л(/г) 1)"л=0}.
Proposition 4.1 L0 = H.
Proof. The algebra H is contained in Lo by Corollary 2.11. Suppose if
possible that HfLQ. Then Lo/H is an //-module, and this module contains
a 1-dimensional submodule M/H on which H acts with weight 0. Hence
[НМ] G H and so M c N(H). This implies H N(H), a contradiction. □
The 1-dimensional representations A of H such that A^O and	are
called the roots of L with respect to H. The set of roots of L with respect to
H will be denoted by Ф. Thus we have
\аеФ /
This decomposition is called the Cartan decomposition of L with respect
to H. La is called the root space of a.
Proposition 4.2 Let A, /z be 1-dimensional representations of H. Then
[^Л’ ^x] C^A+/z-
36
4.1 Some properties of root spaces
37
Proof. Let у eLA, zgL^. We show that [yz]GLA+/z. Let xeH. Then by
Proposition 2.7 we have
" fn\
(adv —A(x)l — m(x)1)"[k] = 52 ( • ) [(adv —A(x)l)!y, (adx — /х(л)1)" 'z].
Since yEl.A (adv — A(x)l)!y = 0 if i is sufficiently large. Since zgLm
(adv — /х(л)1)" 'z = 0 if n — i is sufficiently large. Hence
(adv — А (л) I — ^(х)1)Л[уг] =0
if n is sufficiently large. This shows that [yz] gLa+/z.	□
Corollary 4.3 Let a, /3 e Ф be roots of L with respect to H. Then
[La,Lp]cLa+p	if а + /ЗеФ
[La,Lp]cH	if 13=-a
=0 if a + [3f() and а + [3£Ф.
Proof. This follows from Proposition 4.2 and the fact that Lo = H. □
Proposition 4.4 Let a e Ф and consider the subspace [LaL_a] of H. Given
any /3 e Ф there exists a number r G Q, depending on a and /3, such that
(3=ra on \LaL_a],
Proof. If —a is not a weight of L with respect to H then L_a = О and there
is nothing to prove. Thus we assume —a is a weight. Then — (»еФ since
a#0.
We consider the functions ia + /3 : H C where i G Z. Since Ф is finite
there exist p, q e Z with p > 0, q > 0 such that
-pa + /3, ... ,/3,... ,qa + j3
are all in Ф but — (p+ +)a + /3, (<?+l)a + /3 are not in Ф. If either
— (p++)a + /3 = 0 or (<?+l)a + /3 = 0 the result is obvious. Thus we assume
— (p+ l)a + /3^0, (q+ l)a + /3^0. Thus — (p+ V)a + /3, (q+ l)a + /3 are
not weights of L with respect to H.
Let M be the subspace of L given by
M = L_pa+/3®    ®Lqa+/3.
38
The Cartan decomposition
Let у &La, zeL_a. Let x = [yz] £ [LaL_a]. Then we have
ady(M)cM	by Proposition 4.2, since L^ +^a+p = О
adz(/W)c/W	by Proposition 4.2, since L_^ +^a+p = O.
Thus
ad x(M~) = (ad у ad z — ad z ad y)M с M.
We calculate the trace trwad.v. Since xeH each weight space Lia+p is
invariant under ad x. Thus
ч
trwad.v = E trLra+/jadx.
i=-F
Now adv acts on the weight space Lia+p by means of a matrix of form
'(ia + l3)x	*
0	(ia + /3)x;
Thus trL,a+fi adv =dimLta+y3.(za + /3)(x). Thus
<7
trM adx= 52 dimLta+y3(/a(x)+ /3(x))
i=-p
(<7	\	/ <7	\
52 idimLia+fs j a(x)+	I 52 dim	j /3(x).
i=-p	/	\i=-p	/
On the other hand we have
trM ad.v = trw(ady adz-adz ady)
= trM(ad у ad z) - trM(ad z ad y) = 0.
Hence
/ ч	\	/ ч	\
I E	I	“(*)+ I E dim/m,/; I /3(x) = 0.
\i=-p	/	\i=-p	/
Moreover dim Lia+p > 0 for — p <i<q. Hence for x e [LaL_a] we have
„ idim
^x) = -^~-p	/^E(x).
(Е/=-И1т£;«+уз)
Thus /3(x) = ra(x) for some reQ independent of x. Hence /3 = ra on
[LaL_ff].	□
4.2 The Killing form
39
4.2 The Killing form
In order to make further progress in understanding the Cartan decomposition
of L we introduce a bilinear form on L called the Killing form. We define
a map
Ax/.^C
x,y^ (x, y)
given by
(x, y) = tr(adx ady).
We have ad x : L L, ad у : L^- L and ad x ad у :	L, so tr(adxady) e C.
Proposition 4.5 (i) (x, y) is bilinear, i.e. linear in x and y.
(ii)	(x, y) is symmetric, i.e. (y, x) = (x, y).
(iii)	(x, y) is invariant, i.e.
([xy],z) = (x, [yz]) for all x,y,zeL.
Proof, (i) is clear from the definition.
(ii)	follows from the fact that tr AB = tr BA.
(ii	i) ([xy], z) = tr (ad [xy] ad z) = tr ((ad x ad у — ad у ad x) ad z)
= tr(adx ad у adz)— tr(ady adx adz)
= tr(adx ad у adz)— tr(adx adz ady)
= tr(adx (adу adz — adz ady)) = tr(adx ad[yz]) = (x, [yz]).
□
Proposition 4.6 Let I be an ideal of L and x,y el. Then
(x, y)i=(x, y)L.
Thus the Killing form of L restricted to I is the Killing form of I.
Proof. We choose a basis of I and extend it to give a basis of L. With respect
to this basis ad x : L L is represented by a matrix of form
О J
40
The Cartan decomposition
since x e I, and similarly ad у : L L is represented by a matrix of form
^1
О О J
Thus ad x ad у : L L is represented by the matrix
\ О О )
Hence trL(adx ady) = tr AjBj = trz(adx ady) and so {x, y)L = (x, y);	□
For any subspace M of L we define M by
,M={.t tA ; (x, y)=0 for all у e/W}.
M is also a subspace of L.
Lemma 4.7 If I is an ideal of L then I is also an ideal of L.
Proof Let x e l\ у eL. We must show that [xy] e 7х. So let z G I. Then
([xy],z) = (x, [yz])=0
since [yz] GI and ,i e/ . Thus [xy] eI and I is an ideal of L.	□
We see in particular that Л is an ideal of L. The Killing form of L is said
to be non-degenerate if Л = 0. This is equivalent to the condition that if
(x, y) = 0 for all у e L then x = 0.
The Killing form of L is identically zero if /.=/.. This means that
(x, y) = 0 for all x, у e L.
We now prove a deeper result on the Killing form which will be very
useful subsequently.
Proposition 4.8 Let L be a Lie algebra such that l.f() and L2 = L. Let H
be a Cartan subalgebra of L. Then there exists xeH such that (x, x) 0.
Proof We consider the Cartan decomposition of L with respect to H. Let
this be L = ®LK. Then we have
L2 = [LL\ =



4.2 The Killing form
41
Now we have [LaLm] cLa+/z by Proposition 4.2. Now LK+fl = O if A + /z is
not a weight. Thus each non-zero product lies in some weight space
Lv. We consider the zero weight space Lo. Since L2 = L we have
^o = Z2 [^_A]
A
summed over all weights A such that —A is also a weight. Now I. . = II by
Proposition 4.1, thus we have
a
summed over all roots a e Ф such that —a is also a root.
Now L is not nilpotent since L2 = L. H is nilpotent and soHfL. So there
is at least one root /3 e Ф. /3 is a 1-dimensional representation of H and so
vanishes on [7/7/] since
/3[ху] =/3(x)/3(y) —/3(y)/3(x) = 0 x,yeH.
But /3 does not vanish on H since /3^0. So using the above decomposition
of H we see that there is some root a e Ф such that — a e Ф and /3 does not
vanish on [LaL_a],
We choose x e [LaL_a] such that /3(x) ^0. Then we have
(x, x) = tr(ad x ad x) = EdimLA(A(x))2
A
since ad x is represented on LA by a matrix of form
^A(x)	*
\ О A(x)/
Now by Proposition 4.4 there exists rA a e Q such that A(x) = rA ffa(x). Thus
we have
X) = I EdimLArA,« I “W2-
\ A	/
Now /3(х) = rp ,,a(x) and /3(x)^0. Thus a(x)^0 and rpa^0. It follows
that (x, x)	□
We shall now obtain some important consequences of this result.
42
The Cartan decomposition
Theorem 4.	9 If the Killing form of L is identically zero then L is soluble.
Proof We use induction on the dimension of L. If dimL=l then L is
soluble. So suppose dimL > 1. By Proposition 4.8 we have Lf^L2. L2 is an
ideal of L so the Killing form of L2 is the restriction of the Killing form of
L, by Proposition 4.6. Thus the Killing form of L2 is identically zero. By
induction L2 is soluble. Since L/L2 is soluble it follows that L is soluble, by
Proposition 1.12.	□
Theorem 4.1	0 The Killing form of L is non-degenerate if and only if L is
semisimple.
Proof. Suppose first that the Killing form of L is degenerate. Then /. f O.
Now /. is an ideal of L by Lemma 4.7. Thus the Killing form of /. is the
restriction of that of L by Proposition 4.6. Thus the Killing form of /. is
identically zero. This implies that /. is soluble, by Theorem 4.9. Thus L has
a non-zero soluble ideal, so L is not semisimple.
Now suppose conversely that L is not semisimple. Then the soluble radical
R of L is non-zero. We consider the chain
R D A(1) D A(2) d • • • d K1: '' D K1' = О
where as usual Rb+^ = [TjW/jWj. The subspaces R^ are all ideals of L since
the product of two ideals is an ideal. Let I = Then I is a non-zero ideal
of L such that I2 = О.
We choose a basis of I and extend it to a basis of L. Let x e I and yEL.
With respect to this basis ad x is represented by a matrix of form
/О A\
W °)
since I2 = О and I is an ideal of L, ad у is represented by a matrix of form
(B, B2\
\o bJ
and ad x ad у is represented by the matrix
/ О ABf\
W 0 )
Hence {x, y) = triad .v ady) = 0. Since this holds for all xel and у El. we
have I G I. . Thus I. f() and so the Killing form of L is degenerate. □
4.2 The Killing form
43
We now define the direct sum of Lie algebras L2. Lr ® L2 is the vector
space of all pairs (xb x2) with xx eLj, x2 eL2 under the Lie multiplication
given by
[(*1, *2) (?!, y2)] = ([x^], [x2y2]) .
In this direct sum we define = { (xj, 0) ; xx e Lx} and I2 = {(0, x2) ; x2e£2}.
Then f and I2 are ideals of фL2 such that f ПI2 = О and I1+/2 = L1$L2.
Moreover f is isomorphic to and I2 is isomorphic to L2.
Conversely let L be a Lie algebra containing two ideals f,I2 such that
fp\I2 = O and f+I2 = L. Then the Lie algebra IX®I2 is isomorphic to L
under the isomorphism
в : f®I2^L
(Xl, x2)^Xj+x2.
For в is certainly an isomorphism of vector spaces. But 0 also preserves Lie
multiplication. To see this we first observe that
[7172]c71n72=o.
Thus
[0(хрх2), 0(У1,у2)] = [Xj+x2, y1+y2] = [x1y1] + [x2y2]
= 0 ([^1У1], feyzD = 0 [(*i, *2) , Oh, y2)] •
Thus if a Lie algebra has two complementary ideals f, I2 the Lie algebra is
isomorphic to f ®I2.
We may in a similar way consider direct sums LX®L2®    ®Ln of more
than two Lie algebras.
Theorem 4.1	1 A Lie algebra L is semisimple if and only if L is isomorphic
to a direct sum of non-trivial simple Lie algebras.
Proof Suppose L is semisimple. If L is simple then L must be non-trivial
since the trivial simple Lie algebra is not semisimple. Thus we suppose L is
not simple. Let I be a minimal non-zero ideal of L. Then If О and I^fL.
Consider the subspace I of L; I is also an ideal of L by Lemma 4.7. Now
the Killing form of L is non-degenerate by Theorem 4.10. Thus an element
x e L lies in 7 х if and only if the coordinates of x with respect to a basis of L
satisfy dim I homogeneous linear equations which are linearly independent.
It follows that
dim 7 х = dim L — dim 7.
44
The Cartan decomposition
Now consider the subspace I T\I . This is an ideal of L. Thus the Killing form
of IA I1- is the restriction of the Killing form of L, by Proposition 4.6. Hence
I Г\ I1- is soluble, by Theorem 4.9. Since L is semisimple we have I Ct I1- = O.
Thus
dim (I + / ) = dim I + dim I - dim (l A 7х)
= dim I + dim / = dim L.
Hence I + Il = L. Thus L is the direct sum of its ideals I and I . Hence L is
isomorphic to the Lie algebra I .
We shall now show that I is a simple Lie algebra. Let J be an ideal of I.
Then we have
[Л.] G [Л] + [Лх] G [Л] G J
since [Лх] С [Лх] с IA I1- = O. Thus J is an ideal of L contained in I. Since
I is a minimal ideal of L we have J =0 or J = 1. Thus I is simple.
We show next that I is semisimple. Let J be a soluble ideal of I . Then
[JL] с [Л] + [Лх] с [Лх] c J
since |.//| c [7х/] СIА 7х = О. Thus J is an ideal of L. Since L is semisimple
and J is soluble we have J =0. Thus / is semisimple.
Now we know dim 7х < dimL. By induction we may assume / is a direct
sum of simple non-trivial Lie algebras. Since L = I®I1- and I is simple and
non-trivial, L is also a direct sum of simple non-trivial Lie algebras.
Conversely suppose that
L =	ф • • • ф Lr
where each L; is a simple non-trivial Lie algebra. Each L; is semisimple so
has non-degenerate Killing form by Theorem 4.10. Now each L; is an ideal
of L. Moreover if xt e L;, Xj e Lj and i ф j then (x;, x^ = 0. For
ad X; ad Xj • у e L; A Lj = О for all у e L
thus (x;, Xj'j = tr(adx; adXj) = 0.
Now let x = xx 4-----hxr e Lx with x; e L;. Let y^Lf. Then we have
= л)=0-
Since this holds for all y; e L; we have x; = 0. This holds for all i, hence x = 0.
Thus Lx = О and the Killing form of L is non-degenerate. This implies that
L is semisimple by Theorem 4.10.	□
4.3 The Cartan decomposition of a semisimple Lie algebra
45
4.3 The Cartan decomposition of a semisimple Lie algebra
When L is semisimple we can say much more about its Cartan decomposition
than in the general case. We shall now investigate this Cartan decomposition
in detail.
Let L be semisimple, H be a Cartan subalgebra of L, and L = ф LA be the
Cartan decomposition of L with respect to II. We recall from Proposition 4.1
that Lo = II.
Proposition 4.12 LA and I.,j are orthogonal with respect to the Killing form,
provided p —A.
Proof Let x e LK, у e L^. We assume A + p 0 and must show that (x, y) = 0.
Now for any weight space Lv we have
ad x ad у Lv c LA+ +v by Proposition 4.2.
We choose a basis of L adapted to the Cartan decomposition. With respect
to such a basis ad x ad у will be represented by a block matrix of form
/0	\
0	*
*	0
\	0/
since A + p + v v. Hence we have
(x, y) =tr(ad x ady) = 0.	□
Proposition 4.13 If a is a root of L with respect to H then —a is also a
root.
Proof. We recall that a is a root if a 0 and La O. Suppose if possible that
— a is not a root. Since —a^O we have L_a = O. By Proposition 4.12 we
see that La is orthogonal to all LA, hence La G I. . But since L is semisimple
we have /. = О by Theorem 4.10. Thus La = 0, which contradicts the fact
that a is a root.	□
Proposition 4.14 The Killing form of L remains поп-de generate on restric-
tion to H. Thus if x Ell satisfies (x, y) = 0 for all у E II then x = 0.
46
The Cartan decomposition
Proof. Let x e H and suppose {x, у) = 0 for allytH. We also have (x, у) = 0
for all y&La where a 0, by Proposition 4.12. Thus (x, y) = 0 for all у e L
and so x e I. . Since L is semisimple 7. = O, hence x = 0 as required. □
Note that the Killing form of L restricted to II does not coincide with the
Killing form of H. The latter is degenerate since II is not semisimple.
Theorem 4.15 [7/7/] = O. Thus the Cartan subalgebras of a semisimple Lie
algebra are abelian.
Proof. Let x e [7/77] and y&H. Then we have
(x,y) = tr (adx ady) = J2dimLA A(x)A(y)
Л
since adx ad у is represented on LA by a matrix of form
^A(x)A(y)	*
О
However, A is a 1-dimensional representation of H and xe[7/7/], hence
A(x) = 0. Thus (x, y) = 0 for all у eH. This implies x = 0 by Proposition 4.14.
Thus [7/7/] = 0.	□
Let H* =Hom(7/, C) be the dual space of H. This is the vector space of
all linear maps from H to C. We have dim//* = dim77.
We define a map II II using the Killing form of L. Given hcd! we
define h* e H* by
h* (x) = fh, x) for all x e H.
Lemma 4.16 The map h^ h* is an isomorphism of vector spaces between
H and H*.
Proof. The map is certainly linear. Suppose h e H lies in the kernel. Then
{h, x)=0 for all xe/7. This implies h = 0 by Proposition 4.14. Thus the
kernel is O. Hence the image must be the whole of H*, since dim 77* = dim H.
Hence our map is bijective.	□
Now we have a finite subset Фс//*, the set of roots of L with respect
to H. For each a e Ф there is a unique element h'a e H such that
a(x) = <X,x)
for all x e 77.
4.3 The Cartan decomposition of a semisimple Lie algebra
47
(The notation ha might seem more natural, but this will be reserved for the
coroot of a, to be discussed in Chapter 7.)
Proposition 4.17 The vectors h'a for a e Ф span H.
Proof Suppose if possible that the h'a lie in a proper subspace of H. Then
there exists an element x eH with xf() and (h'a, x) =0 for all a e Ф. Thus
a(x) = 0 for all a e Ф. Let у eH. Then we have
(x, y) = tr (ad x ad y) = dim LA A(x)A(y) = 0
л
since A(x) = 0 for all weights A. Thus (x, y)=0 for all yeH. This implies
x = 0 by Proposition 4.14, a contradiction.	□
Proposition 4.18 h'a e [LaL_a\ for all a e Ф.
Proof La is an H-module. Since all irreducible //-modules are 1-dimensional
La contains a 1-dimensional //-submodule Ce„. We have [xea] = a(x)eff for
all x e H.
Let yet. Then [eay] e [LaL_a\ eH. We shall show that [eay] =
(ea,y) h'a. In order to prove this we define
z=Ky]-(ea,y) h'aeH.
Let x e H. Then
(x, z) = (x, [e„y]) - (eff, y) (x, h'a}
=	, y) - (eff, y) a(x)
= «(*) К, У) ~ К, У) a(x) = 0.
Thus (x, z) = 0 for all x eH, and it follows that z = 0. Hence
К?] = У> h'a for ad У-1-
Now we can choose yeL_a such that {ea, y) ^0. Otherwise ea would be
orthogonal to L_a, so orthogonal to the whole of L by Proposition 4.12. Then
ea e I. . But Lx = 0 since L is semisimple. Thus ea = 0, a contradiction. Thus
we can find у e L_a with (ea, y) 0. Then
h'a = , 1 X [e«y] e lLaL-a\ 	□
\e«, У/
48
The Cartan decomposition
Proposition 4.19 {h'a, h'f) 0 for all a e Ф.
Proof. We suppose that {h'a, h'f) =0 for some a e Ф and obtain a contra-
diction. Let /3 be any element of Ф. By Proposition 4.4 there is a number
rp a e Q such that f}=rpaa when restricted to [LaL_a], Since h'a e [LaL_a]
by Proposition 4.18 we obtain
/3 (Ю = (Ю
that is (h'p, h'a] = rp a (h'a, h'a} = 0.
This holds for all /3 e Ф. But by Proposition 4.17 the elements h'p for /3 e Ф
span H. Thus we have (x, h'f) =0 for all xeH. This implies that h'a = Q by
Proposition 4.14. This in turn implies that a = 0, which contradicts a e Ф.
□
Having obtained a number of results on the Cartan decomposition of a
semisimple Lie algebra, each depending on previous results, we are now able
to obtain one of the most important properties of the Cartan decomposition.
Theorem 4.20 dim La = 1 for all a e Ф.
Proof We choose a 1-dimensional //-submodule Cea of La as in Propo-
sition 4.18 and, as in the proof of that proposition, we can find an element
with [eae_a] = h'a.
We consider the subspace M of L given by
M = Cea ф Ch'a ф L_a ф Т_2а Ф • • •
There are only finitely many summands of M since Ф is finite and there are
only finitely many non-negative integers r with L_ra O.
We observe that ad eaM G M. For
k/'J = 0
кЛ1 = -«К) ea
КуЖ.уЖ for all y е/. ,,,
by the proof of Proposition 4.18, and
ade„•/. iYb. с/.	for all r> 2,
by Proposition 4.2.
4.3 The Cartan decomposition of a semisimple Lie algebra
49
Similarly we can show that ad e_aM с M. For we have
Kaea] = —h'a
[е-Л] = “(Ч) e-a
and ad e_aL_ra c L for all r > 1.
Now h'a = [eae_a] and so
ad h'a = ad ea ad e_a - ad e_a ad ea.
Hence ad h'aM G M. We shall calculate the trace of ad h'a on M in two
different ways. On the one hand we have
trM (ad /i'J = a (h'a) + dimL_a (-a	+ dimL_2a (-2a (/i'J) + •••
= a (/i'J (i - dim I. ,, - 2 dim L_la-).
On the other hand we have
trM (ad h'a) = trM (ad ea ad e_a - ad e_a ad ej = 0.
Thus
a (h'f) (1 - dim L_a - 2 dim L_la---) = 0.
Now a (/i'ff) = (h’a, by Proposition 4.19. Thus
1 — dim /. ,b — 2dim /.	----= 0.
This implies that dimL_a = 1 and dim/. jYb, = 0 for all r>2. Now аеФ if
and only if —a e Ф, by Proposition 4.13. Thus dimLa = 1 for all а e Ф. □
Note that although all the root spaces La are 1-dimensional the space
H = L0 need not be 1-dimensional.
Proposition 4.21 If a e Ф and га e Ф where r e Z then r = 1 or — 1.
Proof This follows from the proof of Theorem 4.20, where we showed that,
for all a e Ф, — га f(l> for all r > 2. This, together with the fact that га e Ф
if and only if —га e Ф, gives the required result.	□
We shall now obtain some further properties of the set Ф of roots. Let
a, /3 G Ф be such that and /3 f —a. Then /3 cannot be an integer multiple
of a, by Proposition 4.21. There exist integers p>G, q>0 such that the
elements
—pa + P,... , — a+f), P, a +P,    , qa +P
50
The Cartan decomposition
all lie in Ф, but — (p+ l)a + /3 and (<?+ l)a + /3 do not lie in Ф. The set of
roots
-pa + f},... ,qa + (3
is called the а-chain of roots through /3. Let M be the subspace of L defined by
Л/ = L_pa+p Ф • • • Ф Lqa+p.
Then we have ad eaM G M. This follows from the fact that ad eaLra+p G
L(r+i)a+/3 and ^(?+i)«+y3 = 0 since (q+ 1)а + /З^Ф and (q+ 1)а + /3#0. Sim-
ilarly we see that ad e_aM G M.
We assume that [eae_a] = h'a, as in the proof of Theorem 4.20. Then we
have
ad h'a = ad ea ad e_a - ad e_a ad ea
and so ad h'a,MG M. We calculate the trace of ad h'a on M in two different
ways. We have
trM (ad /i'J = (ra + /3) (/i'J
r=-p
since dimLra+yS = 1. We also have
trM (ad /i'J = trM (ad ea ad e_J - trM (ad e_a ad e J = 0.
Thus
£ (ra +/3)O'J = 0,
r=-p
that is
( ---2----------2---) a(flo) + (P+(l+1)l3(ha) = Q-
Since p + q+1^0 we obtain
^^-(h'a,h'a)+{h'a,h^ = Q,
that is
since (h'a, h'a) ^0 by Proposition 4.19. Thus we have proved the following
result.
4.3 The Cartan decomposition of a semisimple Lie algebra
51
Proposition 4.22 Let a, /3 be roots such that [3fa and [3f—u. Let
-pa + /3,... ,(3,... ,qa + /3
be the a-chain of roots through /3. Then
2кл[
(IQ IQ
= p-q.
□
This result has some useful corollaries. The first gives a strengthening of
the result of Proposition 4.21.
Proposition 4.23 If a e Ф and fa e Ф where f e C, then f = 1 or — 1.
Proof. Suppose if possible that f #±1. We put /3 = fa and apply Proposi-
tion 4.22. This gives
2^ = 2	Pl=p-q.
(IQ IQ
Hence If e Z. If f e Z then f = ±1 by Proposition 4.21. Hence f ^Z. Then
the a-chain of roots through /3 is
д (p~Q (p+q\
\ 2 J н \ 2 /	\ 2 J
Now p, q are not both 0 since /3^0. So all the roots in the а-chain are odd
multiples of |a. Since the first and the last are negatives of one another
and consecutive roots differ by a it is clear that | a lies in the chain. Hence
|аеФ. Since аеФ we have a contradiction to Proposition 4.21. Hence f
must be 1 or — 1.	□
Thus the only roots which are scalar multiples of a root a are a and —a.
Proposition 4.24 (IQ h'^ eQfor all а, /3 e Ф.
Proof. We know from the outset that (1Q h'^ e C. Now we have
(lQ h'e}
2------— e Z by Proposition 4.22.
(iQ Ю
(Q
Г
(h
e Q. It will therefore be sufficient to show that (IQ IQ e Q.
Now we have
(Q h'a} = tr (ad h'a ad IQ = £ (/3 (iQf = £ (iQ htf.
/ЗеФ	/ЗФ
52
The Cartan decomposition
If follows that
i
<h’a,h’a) ^\{h'a,h'a})
Hence (h'a, h'a) e Q and the result is proved.
□
4.4 The Lie algebra §I„(C)
We shall now illustrate the general results about the Cartan decomposition of
a semisimple Lie algebra by considering in detail the Lie algebra §I„(C). The
special linear Lie algebra §I„(C) is the Lie algebra of all и x и matrices of
trace 0 under Lie multiplication [AB] = AB — BA. §I„(C) is a subalgebra of
0l„(C) = [<(C)]. We have
dimg(„(C) = и2, dim§(n(C) = w2 —1.
We shall assume n > 2. Then §(Л(С) has a basis
Ец—E22, E22 — E33,	Ел_1л_1 — Елл, Ey i^j
where the Ey are elementary matrices.
Theorem 4.25 §(„(C) is a simple Lie algebra.
Proof. We have д(л(С) = §(л(С)фС7л. Now every ideal of §(Л(С) is an
ideal of gI„(C). For [7, §1л(С)]с/ implies [7, gIn(C)]c7 since [x, /„|=0
for all x e I. It will therefore be sufficient to show that the only non-zero ideal
of gI„(C) contained in §(Л(С) is equal to §(Л(С).
Let I be a non-zero ideal of gI„(C) contained in §(Л(С). Let xel with
xf 0. Then
x = E xpqEpq with xpq e C.
Not all xpq are zero.
Suppose first that there exist i^f j with xiq ^0. Then
[Eii, E Xp4Epq] = E XiqEiq ~ E XpiEpi E E
4	P
Also
| [Ea, v|, E..\ = x^E^ + x.E. : I.
4.4 The Lie algebra §l„(C)
53
Hence
[Eu ~Ejp xijEij+xjiEji]=2xijEij-2xjiEjieE
Thus 4Х;;Е;; £ I. SLUX X;; 0 W6 liaVC E;; £ I.
Now suppose that xiq = 0 for all i j. Then x = xppEpp. Since XPP = 0
and not all x„„ = 0 the x„„ are not all equal. Suppose xp ^xfl. Then
[х, E ,\ (x x.f E !
and so E;i e I.
lJ
Thus in either case there exist i j with Eiqel. Let q i, j. Then
[Eij’ Ejq]=E4 eE
Thus Eiq £ I for all q ф i. Now let p^i, q. Then
[Ep;,E;?]=Ep?eE
Hence Epq £ I for all pf^q. Also
lEРЧ ’ E4p ] = Epp ~EqqEl fOT a11 P # <?•
But the Epp—Eqq for p^fq and the Epq for p^fq generate §1Л(С). Thus
Z = §(n(C) and §(n(C) is simple.	□
We next determine a Cartan subalgebra of §I„(C). We write L = M„(C).
Proposition 4.26 Let H be the set of diagonal matrices in L. Then dim H =
n — 1 and H is a Cartan subalgebra of L.
Proof. The vector space of diagonal n x n matrices of trace 0 clearly has
dimension и — 1. It is a subalgebra H of E with [HH] = (). Thus H is nilpotent.
To show H is a Cartan subalgebra we must show H = N(EE).
Let X-ijEtj lie in N(H). Suppose if possible that Ay ^0 for some i^j.
We have
£^Ett,£AyEy e//
к	i,j
for all	The coefficient of Eiq in this matrix is (/x,—/xjA/;.
Thus if we choose (z, j) such that i j and Ay 0 and choose ^к р-кЕкк £ H
with	we obtain a contradiction. Hence Ay=0 for all i^j. Thus
N(H) = H and H is a Cartan subalgebra of L.	□
We next obtain the Cartan decomposition of L with respect to H.
54
The Cartan decomposition
Proposition 4.27 Let H be the subalgebra of diagonal matrices in L. Then
the Cartan decomposition of L with respect to H is
L = H®''£CEij.
Proof This is certainly a decomposition of L into a direct sum of sub-
spaces. To show it is a Cartan decomposition it is sufficient to verify that
the 1-dimensional subspaces CEy for i^j are //-submodules of L. Now we
have
n
k=l

and so CE;i is indeed an //-submodule.
lJ
□
We next obtain the roots of L with respect to H.
Proposition 4.28 The roots of L with respect to H are the functions H C
given by
Proof. This follows from the Cartan decomposition given in Proposition 4.27.
□
We next calculate the value of the Killing form (x, y) when x,yeH.
Proposition 4.29 Let x = fff=1XiEii,y = fff=1pbiEii lie in H. Then (x,y) =
2n tr(xy).
Proof. We have
(x, y)=tr(adx ady) =
ij
since ad x ad у E,, = (A; — A J (/x, — p.f Etj for i j, and ad x ad у H = O.
4.4 The Lie algebra §l„(C)
55
Hence
(x, y) =E(A/- AJ (л-Му)
ij
i,j	i,j	i,j	i,j
= 7.n tr(xy)- (A; j (EmJ - (EAj) (EM;
\ i / \ j /	\ j / \ i
= 2n tr(xy), since У^А, = У^/х,=О.
We may use this knowledge of the Killing form of L restricted to H to
determine the elements h'a e H corresponding to the roots a e Ф.
Proposition 4.30 Let e Ф satisfy
aU
О
\
Then h' = — (E:: — ЕЛ.
Proof. Let x =	M:E1:1: G H. Then we have
/ 1 /	\ \	( 1 / s A
у (En - Ejj) ’ x = 2n tr — (E;; - Ej } x
\2n	I	\2n	/
= A; — Aj = CEy(x), by Proposition 4.29.
However, h'a.. eH is uniquely determined by the condition [h'a , x^= а^(х~)
for all x e H. Hence h' =—(E,-:—ЕЛ.	□
5
The root system and the Weyl group
5.1 Positive systems and fundamental systems of roots
As before, let L be a semisimple Lie algebra and H be a Cartan subalgebra.
Let Ф be the set of roots of L with respect to H. We know by Proposition 4.17
that the elements h'a, a e Ф, span H. Thus we can find a subset which forms
a basis of H. Let h' ,..., h' form a basis of H.
Proposition 5.1 Let аеФ. Then h'a = ^f\=iP‘ih'a. where each /z; lies in Q.
Proof. We know that h'a = ^f!i=i lLh'a for uniquely determined elements
/X; e C. Let (h'a , Ь'а^ = ^. Then e Q by Proposition 4.24. We consider the
system of equations:
= -----------------------------------------------
^a2} = ^1^12 + M2^22 + ’	Ь
(h'a’ h'ai} =	+	---
This is a system of / equations in / variables ... , /х;. Now det (£y) f()
since the Killing form on L is non-degenerate on restriction to II. by Propo-
sition 4.14. Thus we may solve this system of equations for , /x; by
Cramer’s rule. Since (h'a, h'a}e<Q) and all £y eQ we deduce that /z; eQ for
i=l,...,l.	'	□
We denote by HQ the set of all elements of form J2;=1 ILh'a. for /z; e Q and
/Л the set of all such elements with /j. elR. Proposition 5.1 shows that II
and /Л are independent of the choice of basis h'a,. Also II is the set of all
56
5.1 Positive systems and fundamental systems of roots
57
rational linear combinations of the h'a, аеФ, and HK is the set of all real
linear combinations of such elements.
We show next that the Killing form of L behaves in a favourable manner
when restricted to HK.
Proposition 5.2 Let x e Нж. Then (x, x) e IR and (x, x) >0. If (x, x) = 0 then
x = 0.
Proof Let x = J2;=1 ILh'a . Then we have
Now A (h'a.) =	h'ae Q by Proposition 4.24. Thus we have (x, x)eIR,
and also (x, x) > 0.
Suppose that (x, x) =0. Then we have J2;/z;A (h'a ) =0 for all A e Ф. In
particular JT fj^aj (h'a^ =0 for j = l,... ,This gives JT /j^ l^h'^,,h'aJ = 0,
that is 22iMi£y=O- Since the matrix is non-singular we deduce that
/X; = 0 for all i. Thus x = 0.	□
This proposition shows that the Killing form restricted to HK is a map
HK x IL IR which is a symmetric positive definite bilinear form. The vec-
tor space HK endowed with this positive definite form is a Euclidean space.
This Euclidean space contains all vectors h'a for a e Ф.
We recall from Lemma 4.16 that we have an isomorphism h^ h* from H
to H* given by /i*(x) = (h, x). We define H^_ to be the image of IL under
this isomorphism. H^_ is the real subspace of II spanned by Ф. We may also
define a symmetric positive definite bilinear form on by

58
The root system and the Weyl group
Thus //; becomes a Euclidean space containing all the roots a G Ф. We shall
investigate the configuration formed by the roots in the Euclidean space Hg.
We shall, for the time being, write V = H^.
A total ordering on V is a relation < on V satisfying the following axioms.
(i)	A < p and p < v implies A < v.
(ii)	For each pair of elements A, p e V just one of the conditions A < p,
A = p, p < A holds.
(iii)	If A < p then A + v < p + v.
(iv)	If A < p and £ G Ii with i> 0 then i'A < £p, and if £ < 0 then £p. <
Every real vector space has such total orderings. If vr,... , are a basis of V
and A = J) A;u;, p = J)ptvt with A p then we may define A < p if the first
non-zero coefficient pt — A, is positive. This gives us a total ordering on V.
A positive system Ф+ с Ф is the set of all roots a e Ф satisfying 0 < a
for some total ordering on V. Given such a positive system Ф+ we define
the fundamental system П с Ф+ as follows: a e П if and only if a e Ф+
and a cannot be expressed as the sum of two elements of Ф+. Ф is the
corresponding set of negative roots.
Proposition 5.3 Every root in Ф+ is a sum of roots in П.
Proof. Let аеФ+. Then either « G 11 or a = /3 + y where /3, у G Ф+ and
/3 < a, у < a. We continue this process, which must eventually terminate since
Ф+ is finite. Thus a is a sum of elements of П.	□
Proposition 5.4 Let a, /3 G П with a^X/3. Then (a, /3) < 0.
Proof We first observe that a — /3 0 Ф. For if a — /3 G Ф we would have either
a — /3 G Ф+ or /3 — a G Ф+. If a — /3 G Ф+ then a = (a — /3) + /3 which contra-
dicts a G П. If /3 — a G Ф+ then /3 = (/3 — a) + a which contradicts /3 G П. Hence
a — /3 0 Ф. We now consider the «-chain of roots through /3. This has form
+ ,qa + /3
since — a + /3 0 Ф. By Proposition 4.22 we deduce
However, (h'a, h'a) > 0, hence h'^ < 0. It follows that (a, /3) < 0.	□
Thus any two distinct roots in the fundamental system П are inclined at an
obtuse angle.
5.2 The Weyl group
59
Our next result shows the importance of the concept of a fundamental
system of roots.
Theorem 5.5 A fundamental system П forms a basis of V = Lff
Proof We first show that П spans V. We know by Proposition 4.17 that
Ф spans V. Since a e Ф if and only if —a еФ we see that Ф+ spans V. By
Proposition 5.3 we deduce that П spans V.
We show now that the set П is linearly independent. Suppose this were
false. Then there would exist a non-trivial linear combination of the roots
а;еП equal to zero. We take all the terms with positive coefficient to one
side of this relation. Thus we have
of + • • • + Pi, °f = Ph ah + • • • +
where p.,:,... , pt pj ... , p^>0 and at ,... , at , aj ,... , otj are distinct
elements of П. We write
v = piaiA------\-pi a< =pjajt-\------
' ll ll	* lr lr ' JI JI	' Js Js
Then we have(u, v) = (pt a; 4-----h pt at , р^а^-\---h Pj a,). We deduce
(v, v) <0 by Proposition 5.4. Since the form is positive definite this implies
that v = 0. However, 0 < v since we have 0 < at for all at e П and pt > 0.
This gives a contradiction. Thus П is linearly independent.	□
We see in particular that |П| = / = dim/7. Thus the number of roots in a
fundamental system is equal to the rank of the Lie algebra L.
Corollary 5.6 Let П be a fundamental system of roots. Then each a e Ф can
be expressed in the form a = 'ffniai where (itll. n;eZ and either nt>0
for all i or nt<0 for all i.
Proof. The roots a e Ф+ have all nt > 0 and the roots a e Ф have all n; < 0.
□
5.2 The Weyl group
Inside the root system Ф a positive system Ф+ can be chosen in many different
ways. However, we shall show that any two positive systems in Ф can be
transformed into one another by an element of a certain finite group W which
acts on Ф.
60
The root system and the Weyl group
For each a e Ф we define a linear map sa : V V by
(a, x)
sa(x) = x — 2------a for all x eV.
(a, a)
As before, V = H^. This map sa satisfies
^(“) = -a
sa(x) = x if («,%)= 0.
There is a unique linear map satisfying these conditions - the reflection in
the hyperplane of V orthogonal to a. Thus sa is this reflection.
The group W of all non-singular linear maps on V generated by the sa for
all a e Ф is called the Weyl group. This group plays an important role in the
Lie theory. It is a group of isometries of V, that is we have
(wx, wy) = (x, y) for all x, у e V.
Proposition 5.7 W permutes the roots. Thus if a e Ф and w e IV then w(a) e
Ф.
Proof. It is sufficient to show that л-а(/3)еФ for all а, /ЗеФ since the
elements sa generate W. If /3 = a or —a this is clear. Thus suppose /3^±«.
Let the «-chain of roots through /3 be
-pa + /3, ... ,/3,... ,qa + j3.
Then we have
/ j. о	or \
•?«(/?) = /?-2;----a = f3-(p-q)a
(a, a)
by Proposition 4.22. Now /3 — (p — q)a is one of the roots in the «-chain
through /3. Thus sa (/3) e Ф.
In fact we observe that sa inverts the above «-chain of roots. In particular
we have
sa (qa + /3) = -pa + /3,	sa (-pa + (3) = qa + (3.	□
Proposition 5.8 The Weyl group W is finite.
Proof. W permutes Ф and Ф is finite. If two elements of W induce the same
permutation of Ф they must be equal, since Ф spans V. Since there are only
finitely many permutations of Ф, W must be finite.	□
5.2 The Weyl group
61
Now suppose that Ф+ is a positive system in Ф and that П is the corre-
sponding fundamental system.
Lemma 5.9 Let a e П. If /Зе Ф+ and [3fa then sa(J3) e Ф+.
Proof. We can express /3 in the form
/3 = 22и;а; а;еП, w;eZ, nt>0
i
by Corollary 5.6. Since [ifa there must be some и;^0 with a^a. We
then consider
sa(J3) = /3-2-,---
(a, a)
and express this as a linear combination of the elements of П. The coefficient
of a; in sa(]3) remains n;. Since nt > 0 we deduce from Corollary 5.6 that
^(/3)еФ+.	□
Theorem 5.10 Let Ф^, Ф2 be two positive systems in Ф. Then there exists
w e W such that w (Ф/) = Ф2.
Proof. Let m= |Ф^ ПФ, |. We shall use induction on m. If m = 0 we have
Ф, = Ф2 and so w = 1 has the required property. Thus we may assume m > 0.
Let Пх be the fundamental system in Ф/. We cannot have II, с Ф2 as this
would imply Ф/ с Ф^, contrary to m > 0. Thus there exists a e Пх П Ф2.
We consider sa (Ф^)- This is also a positive system in Ф. By Lemma 5.9
sa (Ф^) contains all roots in Ф/ except a, together with —a. Thus we have
(Ф+)пФ7|=ш-1.
By induction there exists w' e W such that w'sa (Ф^) = Ф^. Let w = w'sa.
Then w (cl*, ) = Ф2 as required.	□
Corollary 5.11 Let П1,П2 be two fundamental systems in Ф. Then there
exists w e IT such that w (П1) = П2.
Proof. Let Ф^, Ф2 be positive systems containing П1, П2 respectively. Let
Ф2 = ш(Ф^). Then w(II1) is a fundamental system contained in Ф^, so
ш(П1) = П2.	□
Proposition 5.12 Let П be a fundamental system in Ф. Then for each a e Ф
there exist at e П and w e W with a = w fctf).
62
The root system and the Weyl group
Proof. Let Ф+ be the positive system with fundamental system П. First
suppose a e Ф+. Then we have
a = ^2niai ell, w;eZ, и;>0
i
by Corollary 5.6. We define the height of a by
ht о- =	/7
i
We shall argue by induction on ht a. If ht a = 1 then a = ai for some i and
аеП. The result is obvious in this case. Thus suppose hta> 1. Then we
have и; > 0 for at least two values of i by Proposition 4.21. Now
(a, a) ='^2,ni{a, at).
i
Since (a, a)>0 and each и;>0 there exist а;еП with (a, aq)>0. Let
sfa) = (3. Then /3 e Ф and
/3 = а-2---------at.
{at, au
Since (a,;,a)>0 we see that ht/3<hta. On the other hand /3 e Ф+ since
only one coefficient nt is changed in passing from a to /3, thus at least one
coefficient remains positive in /3. By Corollary 5.6 this is sufficient to show
that /3 e Ф+. By induction there exist a. e П and w' e W such that /3 = w' (ctj').
Then
a = 5-;(/3) = 5-;w' (aJ
as required.
Finally we suppose that и£ф . Then a = sa(—a) and —аеФ+. Thus
—a = w' (a;) for some w' e W, ate П. Hence a = sa w' (a;) as required. □
Thus each root is the image of some fundamental root under an element of
the Weyl group.
We show next that W is generated by the reflections corresponding to roots
in a given fundamental system.
Theorem 5.13 Let П= {oq,... , oq} be a fundamental system in Ф. Then the
corresponding fundamental reflections sai, ... , sai generate W.
Proof. Let Wo be the subgroup of W generated by sa ,... , sa . Since the
sa generate W for all a e Ф it is sufficient to show that each sa lies in Wo.
We may assume аеФ+ since sa = s_a. Now the proof of Proposition 5.12
5.2 The Weyl group
63
shows that a = w(a;) for some а;еП and some weW0. We consider the
element wxn/w! e Wo. We have
uw w-1(ai) = wsa. (a;) = w (—a;) = —a.
We shall also show wsa.w^1(x) = x if (a, x)=0. For (a, x)=0 implies
(и’ 1 (tt), и’ 1 (л;)) =0, that is {at, w-1(x)) =0. This gives sa.w~1 (л) =
ir '(л), i.e.	(v) = x. Thus wsaw^1 is the reflection in the hyper-
plane orthogonal to a, that is wsaw^1 = sa. This shows that sa e Wo. Hence
Wo = W.	'	□
We now wish to obtain further information about the way in which the
Weyl group W is generated by a set of its fundamental reflections. As before
we let П= {dip ... , U[} be a fundamental system of roots and consider the
corresponding set of fundamental reflections. For simplicity we write
s2 = sa2, ...,	s^s^.
Then each element of W can be expressed as a product of elements st. (We
do not need to introduce inverses since s.! = st.) For each w e W we define
/(w) to be the minimal value of m such that w can be expressed as a product
of m fundamental reflections st. l(w) is called the length of w. It is clear
that /(1) = 0 and / (s,) = 1. An expression of w as a product of fundamental
reflections st with /(w) terms is called a reduced expression for w.
We shall relate /(w) to another integer n(w). We recall that each element
weW permutes the elements of Ф. We define n(w) to be the number of
roots a e Ф+ for which w(a) e Ф . Thus n(w) is the number of positive roots
made negative by w. We aim to show that /(w) = n(w).
Proposition 5.14 n(w) < l(w) for all w e W.
Proof. We shall first compare n(w) with n (wsf. We recall from Lemma 5.9
that transforms a; to —ai and all positive roots other than a; to positive
roots. It follows that
n (wSj) = n(w) ± 1.
In order to determine the sign we consider the effect of w and wst on at. If
w («,) e Ф+ then w transforms at to a positive root and wst transforms at to
a negative root. Hence n (ws^) = n(w) + 1. On the other hand if w (a;) e Ф
then we get the reverse situation and n (wxf = n(w) — 1.
64
The root system and the Weyl group
Now let us take a reduced expression
w = st st ... st r=l(w).
Then we have
w(w) < n (sti ... sir i) + 1 < n (sh ... sir_2)+2< • • • < r.
Thus n(w) < l(w) as required.	□
In order to prove the converse result l(w) <n(w) we shall first prove a
result called the deletion condition which is important in its own right.
Theorem 5.15 Let w = si{... be any expression of w eW as a product of
fundamental reflections. Suppose n(w) < r. Then there exist integers j, k with
1 < j <k <r such that
W = S . . . ?; . . . ?; ... S:
ll lj lk lr
where " denotes omission.
Proof. We recall from the proof of Proposition 5.14 that, for all weW,
n (wsj) = n(w) ± 1. Consider the given expression
w = st ... st .
Since n(w) < r there exists k with 1 < k < r such that
П (S; . . . S; ) =n (S; . . . S: ) — 1 .
This implies st .. .st (a; ) G Ф as in the proof of Proposition 5.14. Since
o'; G Ф+ there exists i with 1 < i < k such that
S; . . . S; («, )сФ
lj+l lk-l X lk/
S; S; . . .S;
lj lj+l lk-l X lk/
By Lemma 5.9 st. transforms only one positive root into a negative root,
namely a;,. Thus we have
S; ...S;	=
lj+l lk-l X lk/ lj
It follows that the reflections S: , S: associated with the roots a,- , a,- are
lk 4	lk 4
related by
5.3 Generators and relations for the Weyl group
65
This implies
Sj.Sj. , ...S: , =S;. , . . .S;, , S;, .
Ij lj+i	4-1 lj+i	lk_Y
Thus we have
S- ... $ =$, ... Sf Sf ... S; S; ... S;
ll lr ll lj-l lj+l lk-l lk+l lr
and so w = st ...	. st ... st as required.	□
Corollary 5.16 w(w) = /(w).
Proof. We know from Proposition 5.14 that n(w) < l(w). Suppose if possible
that n(w) < l(w). Let w = sii...si be a reduced expression, thus r = /(w).
Since n(w) <r we may apply Theorem 5.15 to show that w is a product of
r — 2 fundamental reflections. This contradicts the definition of l(w).	□
Thus the length of w is equal to the number of positive roots made nega-
tive by w.
Proposition 5.17 (a) The maximal length of any element ofW is |Ф+|.
(b)	W has a unique element w0 with I (w0) = |Ф+|.
(с)	»О(Ф+) = Ф-
(d)	»o=1-
Proof. Since /(w) = n(w) we have /(w) < |Ф+|. For each fundamental system
П c Ф, -П is also a fundamental system, coming from the opposite total
ordering. Thus by Corollary 5.11 there exists w0 e W with w0(II) = —П.
Hence 1Л’,(Ф) = Ф and n (w0) = | Ф+|. Thus /(w0)= |Ф+| also and w0 is
an element of W of maximal length.
Now let w'o e W also have l(w'o) =|Ф+|. Then n (w'0) = |Ф+| and so
;<(Ф ) = Ф . Let w=(w'0)~lw0. Then ш(Ф+) = Ф+ and so n(w) = 0.
Hence /(w) = 0 and so w=l. Thus w'0 = w0 and the element w0 of maximal
length is unique.
Finally we have Wq (ф+) = ф+ and so n (wq)=0. Hence l(wfy = 0 and
Wq = 1.	□
5.3 Generators and relations for the Weyl group
In this section we shall give a description of the Weyl group W by means of
generators and relations. Let the order of the element stSj e W be mtj when
i^j-
66
The root system and the Weyl group
Theorem 5.18 W is isomorphic to the abstract group given by generators
and relations:
(Si,--- ,St-, ^=1, (^)my = l forz'^j)-
A group defined by generators and relations of this form is called a Coxeter
group. Thus the theorem asserts that the Weyl group is a Coxeter group.
Proof. Since W is generated by Sp... , s, and the relations s2 = l and
(siSj)m‘1 = 1 hold in W it is sufficient to show that every relation
S: ... S: = 1
И lr
in W is a consequence of the defining relations. Now each st is a reflection,
thus det st = — 1. Hence det ... st ) = (— l)r. If st ... st = 1 we deduce that
r must be even. Let r = 2q. We shall show that
is a consequence of the defining relations, by induction on q. If q = 1 the
relation is s, s, =1, hence s, = v 1 =s; . Our relation is thus s2 = 1, which is
one of the defining relations.
We may therefore assume inductively that all relations in W of length less
than 2q are consequences of the defining relations.
Now the given relation can be written
S; ... S; S; = S; ... S; .
4	lq lq+i l2q	lq+2
Thus l\s, ...St St )< q +1. Hence, by the deletion condition Theorem 5.15,
\ ll lq lq+l ) i-	y J
we have
Si ... Si =Si ... !• ... !• ... Si
ll lq+l ll lj lk lq+i
for certain j, k with 1 <j<k<q+l. Now unless j = 1 and k = q + 1 this is
a consequence of a relation with fewer than 2q terms. It can therefore be
deduced from the defining relations. The relation
S: ... 5; . . . 5; ... S: = S; ... S;
ll lj lk lq+i l2q lq+2
has 2q — 2 terms, so is also a consequence of the defining relations. Thus the
given relation
Sj ... Sj =S: ... Sj
4	lq+i l2q	lq+2
will be a consequence of the defining relations, unless we have j = 1 and
k = q+ 1.
5.3 Generators and relations for the Weyl group
67
We may therefore assume that j = 1 and к = q + 1. Thus we have
that is
We now write the original relation
Sj ... Sj = 1
!1	l2q
in the alternative form
Sj ... Sj S; = 1 .
l2 l2q ll
In exactly the same way this relation will be a consequence of the defining
relations unless
Sj ... Sj = Sj ... Sj .
!2	!«+l	!3 lq+2
If this relation is a consequence of the defining relations then the relation
Sj ... Sj Sj = 1
z2 l2q И
will also be a consequence of the defining relations, by the above argument,
and we are done.
Now st ... st =st ... st is equivalent to
l2 lq+l z3 lq+2 ~
Sj Sj Sj ... Sj Sj Sj Sj ... Sj = 1
г3 l2 z3 lq lq+l lq+2 lq+l l4
and this will be a consequence of the defining relations unless
Sj Sj Si ... Si = Si Si ... Si Si .
z3 l2 1$ lq 12 13 lq lq+l
We may therefore assume this to be true. But we also have
Si Si Si ... Si = Si Si ... Si Si
ll г2 13 lq г2 13 lq lq+l
and so st = si3. Hence the given relation
Sj ... Sj = 1
ll l2q
will be a consequence of the defining relations unless st = .
However, the given relation can be written in the equivalent forms
Sj ... Sj Sj = 1
z2 l2q ll
Sj ... Sj S; S; = 1
Z3	l2q ll 12
68
The root system and the Weyl group
and so on. Thus this relation will be a consequence of the defining relations
unless we have
Si = Si = Si = . . . = Si .
l2 l4 l6	l2q
Thus we may assume that the given relation has form
that is Si V= 1. Now the order of л л is , hence divides q.
Thus the relation 9 = 1 is a consequence of the defining relation
(st st ) 'l'2 = 1. This completes the proof.	□
This remarkable proof, due to R. Steinberg, shows that the Weyl group W
is a finite Coxeter group.
6
The Cartan matrix and the Dynkin diagram
6.1 The Cartan matrix
We shall now investigate in more detail the geometry of the system of roots
Ф in the vector space V = t/L We recall from Proposition 5.2 that V is a
Euclidean space with respect to the scalar product (,). The roots Ф span V
but are not linearly independent. Any fundamental system П с Ф forms a
basis of V.
We first consider the possible angles between pairs of roots a, /3 e Ф and
the relative lengths of the roots a, /3. The angles will be taken to satisfy
0 < в < тг.
Proposition 6.1 Let a, /3 e Ф be such that /3 ±a. Then-.
(i) the angle between a, /3 is one of -тт/6, tt/4, -tt/3, tt/2, 2tt/3, 3tt/4, 5tt/6
(ii) if a, /3 are inclined at -tt/3 or 2tt/3 then a, /3 have the same length
(iii) if a, /3 are inclined at it/4 or Зтт/4 then the ratio of their lengths is ^2
(iv) if a, /3 are inclined at тт/6 or 5 тт/6 then the ratio of their lengths is ^/3.
Proof Let в be the angle between a, /3. Then we have
(a, /3) = |a| \/3\cos в
where |a| = ^/{a, a). Hence
(a,/3)2	(a,f3) ((3,a)
COS Г7 = -------------- =----------- -----
(a, a) (13,13)	(a, a) (/3,13)
and so
4 cos 0 = 2--2----
M (O)
69
70
The Cartan matrix and the Dynkin diagram
Now we recall from Proposition 4.22 that 2^Tl and 2^-^- are integers. Hence
4cos2 в e Z. Since 0 <4cos2 в <4 and /3^ ± a we have 4cos2 в e {0, 1,2, 3}.
We consider in each case the possible factorisations of 4 cos2 в into the product
of two integers.
First suppose 4 cos2 0 = 0. Then в = -тт/2.
, 1 1
Next suppose 4cos 0=1. Then cos 0= - or —hence 0 = tt/3 or 2 tt/3.
The possible factorisations of 4 cos2 0 are
1 = 1-1 or 1 = -1--1.
In either case we have
(a,/3)	(/3,a)
{a, a)	(13,13)
and so (a, a) = (/3, /3) and a, /3 have the same length.
Next suppose 4cos20 = 2. Then cos0= 1/^/2 or — 1/^/2, thus 0 = tt/4 or
Зтт/4. The possible factorisations of 4 cos2 0 are
2=1-2 or 2=—1-—2.
In either case, by choosing a, /3 in a suitable order, we have
(fra) Ja,fr
that is (a, a) = 2(/3, (3) and |a| = fr2|/3|. Thus the ratio of the lengths of a, /3
is ^/2.
Finally suppose that 4cos2 0 = 3. Then cos 0 = fr3/2 or —fr3/2, so 0 = tt/6
or 5tt/6. The possible factorisations of 4 cos2 0 are
3 = 1-3 or 3 = -l--3
In either case, by choosing a, /3 in a suitable order, we have
(fra) (a,/3)
((3,(3)	’ (a, a)’
that is (a, a) = 3(13, (3) and |a| = fr3|fr. Thus the ratio of the lengths of a, /3
is ^/3.
This completes the proof. We do not obtain any information about the
relative lengths of a, /3 in the case when 0 = тт/2.	□
Corollary 6.2 Let П be a fundamental system of roots and let a, /3 e П with
/3=fa. Then the angle between a, /3 is one of -тт/2, 2тт/3, Зтт/4, 5тт/6.
6.1 The Cartan matrix
71
Proof. This follows from Proposition 6.1 together with the fact, proved
in Proposition 5.4, that the angle в between two distinct fundamental roots
satisfies тт/2 < в < тт.	□
Let П = {ар ... , U[} be a fundamental system. We incorporate the infor-
mation about the angles between the at and their relative lengths in the form
of a matrix. We define A;; by
lJ J
Thus Atj e Z. The / x / matrix A= (Ay) is called the Cartan matrix.
Proposition 6.3 The Cartan matrix A has the following properties.
(i)	Au=2for all i.
(ii)	Aye{0,-l,-2,-3}ifi#j.
(iii)	If Ay = —2 or —3 then Ajt = — 1.
(iv)	Ay = 0 if and only if A^ = 0.
Proof. Properties (i), (iv) are obvious and (ii), (iii) follow from the proof of
Proposition 6.1.	□
If we number the fundamental roots in П in a different way we may
well get a different Cartan matrix A. However, apart from this ambiguity of
numbering, the Cartan matrix A is uniquely determined by the semisimple
Lie algebra L.
Proposition 6.4 The Cartan matrix of L depends only on the numbering of
the fundamental roots. It is independent of the choice of Cartan subalgebra
H and fundamental system П.
Proof. The independence of the choice of Cartan subalgebra follows from
the conjugacy of Cartan subalgebras, proved in Theorem 3.13.
Let П' be a second fundamental system. By Corollary 5.11 there exists
w e W with w (П) = П'. Let w (a;) = al. Since w is an isometry of V we have
ctj) {of a’f)
Thus the Cartan matrices defined by П and П' with respect to these labellings
are the same.	□
72
The Cartan matrix and the Dynkin diagram
The only possible 1x1 Cartan matrix is (2). We also see that any 2x2
Cartan matrix must be one of the following:
'2 0\ /2 -1\	/ 2 -1\	/ 2	—2\	/ 2 -1\	/ 2 -3\
0 2/ \-l 2 J	\-2 2 J	\-l	2 J	\-3 2 J	\-l 2 J
The pair
2 -1\	/ 2 -2\
—2 2 )	\-l 2 )
are obtained from one another by reversing the labelling 1, 2, and so are the
pair
2
—3
-1
2
2
-1
-3\
2 )
6.2 The Dynkin diagram
In order to determine the possible / x / Cartan matrices for larger values of
/ it is useful to introduce a graph called the Dynkin diagram. The Dynkin
diagram is determined by the Cartan matrix. It is a graph with vertices labelled
1,... , If i^j the vertices i, j are joined by n^ edges, where
па = АИАЛ-
We see from Proposition 6.4 that the Dynkin diagram is uniquely determined
by the semisimple Lie algebra L.
The Dynkin diagrams of the Cartan matrices of degrees 1 and 2 are as
follows.
Cartan matrix Dynkin diagram
6.2 The Dynkin diagram
1Ъ
Proposition 6.5 n^ e {0,1,2, 3} for all i^j.
Proof. This follows from Proposition 6.3 and the fact that ntj =А^А^.	□
Thus the number of edges joining any two distinct vertices of the Dynkin
diagram is either 0, 1, 2 or 3.
Now the Dynkin diagram need not be a connected graph. However, if it is
disconnected it will split into connected components. If we number the vertices
so that those in each connected component are numbered consecutively, the
Cartan matrix will split into blocks of the form
/ *	0	0	0 \
0	*	0	0
0	0	*	0
\°	0	0	* /
with one diagonal block for each connected component. This diagonal block
will be the Cartan matrix for the given connected component. The set П =
{oq,... , a/} will be partitioned into subsets in a corresponding way, such
that roots in different subsets are mutually orthogonal.
Now the set of graphs which can occur as Dynkin diagrams of semisimple
Lie algebras turns out to be quite restricted. In order to determine the possible
Dynkin diagrams it is useful to introduce a quadratic form Qfxj,... , x;)
which is defined in terms of the Dynkin diagram. We define
i	i
Q(*!>••• ,x;) = 2j2^- Vnijxixj-
t=i	i,j=i
We illustrate this definition in the cases /=1,2.
Dynkin diagram Quadratic form
о	2xj
о о	2xj + 2x|
о----о	2xj — 2xx x2 + 2x|
о	о	2xj — 2л/2х1 x2 + 2x|
<>	<>	2x^ ~ 2V3xix2 + 2x|
74
The Cartan matrix and the Dynkin diagram
Proposition 6.6 The quadratic form Q (xx,... , x;) is positive definite.
Proof We have, for iffy,
/	(“/’“J	I \
hence — fyr, = 2----— since (a,., a;) <0. For i= i we have —!-— = 2.
klkl	klkl
Thus the quadratic form may be written
= 2(y,y) where у = 'Ет-f-
kl
Thus Q (xb ... , x;) > 0. Moreover if Q (xb ... , x;) = 0 then y = 0. Since
aq,... , al are linearly independent this implies that x; = 0 for all i. Thus the
quadratic form is positive definite.	□
Now the connected components of the Dynkin diagram of any semisimple
Lie algebra satisfy the following conditions:
(A)	The graph is connected.
(B)	Any pair of distinct vertices are joined by 0, 1, 2 or 3 edges.
(C)	The corresponding quadratic form Q (xx,... , x;) is positive definite.
We shall approach the problem of finding the possible Dynkin diagrams by
determining all graphs satisfying conditions (А), (В), (C). Having determined
all such graphs we shall consider subsequently which ones occur as Dynkin
diagrams.
6.3 Classification of Dynkin diagrams
The main result which we shall obtain in this section is as follows.
Theorem 6.7 The graphs satisfying conditions (А), (В), (C) shown in Section
6.2 are just those in the following list.
6.3 Classification of Dynkin diagrams
75
Proof. We shall show first that the graphs on this list satisfy conditions
(А), (В), (C). It is obvious that they satisfy (A) and (B). We shall therefore
concentrate on condition (C).
We recall from linear algebra that a quadratic form ^a^x^j is positive
definite if and only if all the leading minors of its symmetric matrix (ay)
have positive determinant. This condition is
lanI > о.
all a12
a21 a22
0,... , det (ay) > 0.
Given a graph Г with / vertices on the list in Theorem 6.7 we shall show
that Q (xj,... , x;) is positive definite by induction on /. If / = 1 then Г =
and Q(x1) = 2xJ is positive definite. If / = 2 then Г is A2,B2 or G2. The
symmetric matrix representing Q (xx, x2) is then
2 -1\
-1 2 )
^2
2 -V2\
-V2 2 )
2 -V3\
-73 2 )
G2
In these cases the leading minors have positive determinant.
Now assume />3. Then inspection of the list of graphs in Theorem 6.7
shows that Г contains at least one vertex which is joined to just one other
vertex of Г, and joined to it by a single edge. Let such a vertex be labelled /,
and let the vertex it is joined to be labelled / — 1. We write Г = Г;, and the graph
obtained from Г; by removing the vertex / by Г;_х, and the graph obtained from
r;_j by removing the vertex / — 1 by Г;_2. Let det Г; be the determinant of the
symmetric matrix representing the quadratic form Q(xp ... , x;) associated
76
The Cartan matrix and the Dynkin diagram
to Г;. We observe from the list of graphs that Г;-1 and Г;_2 also he in the list.
Moreover we have
det Г; =
2
0 ... 0 -1
0
-1
2
= 2 det r;_j — det Г;_2
by expanding the determinant by its last row. This gives us an inductive way
of calculating detT;. In particular we have
detAj=2, detA2 = 3, det A; = 2det A;_x — detA;_2.
Thus det At = I + 1.
detAj=2, detB2 = 2, detB3=2, det Л’,. =2detB;_1 — detB;_2.
Thus det = 2.
detA3=4, detZ>4=4, detZ>5=4, detZ>; = 2detZ>;_1 — detZ>;_2.
Thus det Dl = 4.
det E6 = 2 det D5 — det A4 = 3
det Е-/ = 2 det D6 — det A5 = 2
det Es = 2 det D7 — det A6 = 1
det F4 = 2 det B3 — det A2 = 1.
Thus we have shown that det Г; > 0 for all Г;
Now the leading minors of the symmetric matrix associated to Г; are the
symmetric matrices associated to certain subgraphs of Г;. The numbering
can be chosen so that all these subgraphs are connected. However, the list
of graphs has the property that any connected subgraph of a graph on the
list is also on the list. Thus the determinant of every leading minor of the
given symmetric matrix is positive. Hence the quadratic form Q (x3,... , x;)
associated to Г; is positive definite.
Thus we have shown that the graphs on our list satisfy conditions (A),
(В), (C). We wish to prove the converse, i.e. that any graph satisfying con-
ditions (А), (В), (C) is on our list. Before being able to prove this we shall
need some lemmas.
6.3 Classification of Dynkin diagrams
77
Lemma 6.8 For each of the graphs on the following list the corresponding
quadratic form Q (xx,... , x;) has determinant 0.
Proof. First consider the graphs Г = At. Each row of the symmetric matrix of
the given quadratic form contains one entry 2, two entries —1, and remaining
entries 0. Thus the sum of the columns is zero and det A; = 0.
In all the other graphs Г on the list we can find a vertex / joined to just one
other vertex I — 1. Moreover / is joined to / — 1 by a single edge or a double
edge. If there is a single edge we may use the formula
det Г; = 2 det Г;_х — det Г;_2
as before. If there is a double edge we obtain instead
det Г; = 2 det Г;-1 — 2 det Г;_2.
We may use these formulae to calculate all the determinants inductively.
det If = 2 det A3 — 2 (det Ax)2 = 0
det 71/= 2det/); — 2det/); । = 0 for/>4
det C2 = 2 det B2 — 2 det A x = 0
det C; = 2 det — 2detB;_1 =0 for/>3
det Ь4 = 2 det D4 — (det AJ3 = 0
78
The Cartan matrix and the Dynkin diagram
det /), = 2 det Dt — det Dt_2  det	= 0 for I > 5
det E6 = 2 det E6 — det A5 = 0
det F7 = 2 det E7 — det D6 = 0
det F8 = 2 det Es — det E7 = 0
det F4 = 2 det F4 — det If = 0
det G2 = 2 det G2 — det Ar = 0.
Lemma 6.9 Let Г be a graph satisfying conditions (А), (В), (C) and Г' be
a connected graph obtained from Г by omitting vertices or decreasing the
number of edges between vertices or both. Then Г' satisfies conditions (A),
(В), (C) also.
Proof. Г' clearly satisfies (A) and (B). We must show it satisfies (C). Let
Q (%j,... , x;) be the quadratic form of Г and Q' (xp ... , xm) be the quadratic
form of Г', where m<l. We have
i	i
Q(xp ... ,x;) = 22>-- s/nijxixj
i=i	i,j=i
чЧ
m	m
,xm) = '2^xj- s/n’ifyiXj
;=i	ij=i
чЧ
where м2 < for i, j e {1,... , m}. Suppose if possible that Q' is not positive
definite. Then there exist y1,... , ym G IR, not all zero, with Q' (yp ... , ym) < 0.
Consider Q (lyj,... , |ym|, 0,... ,0). We have
6.3 Classification of Dynkin diagrams
79
Hence Q(|yi|,... , |ym|,0,... ,0)<0 but (Ы,... , |ym|,0,... ,0) is not the
zero vector. This contradicts the fact that Q (xx,... , x;) is positive definite.
Hence Q' (xb ... , xm) must be positive definite also.	□
Having Lemmas 6.8 and 6.9 at our disposal we are now able to complete
the proof of Theorem 6.7.
Let Г be a graph satisfying conditions (А), (В), (C). Then, by Lemmas 6.8
and 6.9, Г can have no subgraph of type Ah Bh Ch Dh E6, E7, Es, F\ or G2.
(By a subgraph of Г we mean a graph obtainable from Г by removing vertices,
or removing edges, or both.) We shall use this information to show that Г
must be one of the graphs on the list in Theorem 6.7.
In the first place we see that Г contains no cycles, otherwise it would
contain a subgraph of type A; for some / > 2.
Suppose that Г contains a triple edge. Then Г must be the graph G2,
otherwise Г would contain a subgraph G2.
Thus we may assume that Г contains no triple edge. Suppose Г contains
a double edge. Then Г cannot have more than one double edge, otherwise
it would contain a subgraph C; for some / > 2. Now Г cannot contain a
branch point in addition to a double edge, as otherwise it would contain a
subgraph B[ for some />3. Thus Г is a chain containing just one double
edge. If the double edge occurs at one end of the chain then Г = for some
I >2. If not then we must have Г = F4, since otherwise Г would contain a
subgraph F4.
Thus we may assume that Г contains no double or triple edges. If Г contains
no branch point then Г = A; for some I > 1. Thus we suppose that Г contains
at least one branch point. Now Г cannot contain more than one branch point,
as otherwise it would contain a subgraph Dl for some Z>5. Thus Г contains
exactly one branch point. There must be exactly three branches emerging
from this branch point, since otherwise Г would contain a subgraph Z)4. Let
the number of vertices on the three branches be f, l2, l3 with f > Z2 > l3. Then
the total number of vertices of Г is I = f + Z2 +13 + 1.
Now we must have l3 = 1, as otherwise we have Z; >2 for i = 1,2, 3 and
Г contains a subgraph F6. If Z2= 1 then Г = Dl for some Z >4. Thus we may
assume Z2 > 2. In fact we must have Z2 = 2, as otherwise we have f > 3, Z2 > 3
and Г contains a subgraph F7. Thus we may assume l3 = 1, Z2 = 2. We must
have lr <4 since otherwise Г contains a subgraph Es. Thus Г has type F6, F7
or F8.
Thus we have now determined all possibilities for Г, and seen that Г must
be one of the graphs which appear on the list in Theorem 6.7. This completes
the proof.	□
80
The Cartan matrix and the Dynkin diagram
Corollary 6.10 Let Д be the Dynkin diagram of a semisimple Lie algebra.
Then each connected component of Д must be one of the graphs
A„ / > 1 ; Bh l>2-„ Dh l> 4 ; F6 ; E7 ; F8 ; F4 ; G2.
We shall consider later whether all these graphs actually occur as Dynkin
diagrams.
6.4 Classification of Cartan matrices
We recall that the Dynkin diagram is determined by the Cartan matrix by the
property
па = АаАп
However, the Cartan matrix is not always uniquely determined by the Dynkin
diagram. If we know the integers ntj e {0, 1, 2, 3} for all i, j with i^£j we
consider to what extent the are determined. If n^ = 0 then we must have
Atj = 0 and Ajt = 0 since Atj = 0 if and only if Ajt = 0. If ntj = 1 then we must
have A,.; = —1 and A;; = —1 since A;; e Z, A;; e Z, A;; < 0, A;; < 0. However,
if ntj=2 there are two possibilities for the factorisation п^=А^А^. Either
we have 2 = — 1 • — 2 or 2 = —2 • — 1. Thus we have either Ay = — 1, Ац = —2
or A;; = —2, A;,. = — 1. Similarly if и,; = 3 we have either A;; = — 1, A;; = —3
ОГ A;; = —3, A;; = — 1.
lJ	J1
In the connected graphs in Corollary 6.10 the only ones which give rise
to such an ambiguity are Bh l>2 ; F4 and G2. In these graphs we shall
place an arrow on the double or triple edges. The direction of the arrow is
determined as follows. The arrow points from vertex i to vertex j if and only
if l“J>l“Athatis 1ал1>1421-
Thus in the situation
» > »
i j
we have |a;| =^/2|aJ, Ay = —1, AJ: = —2. In the situation
» > »
i j
we have |a;| = ^3|а^|, Ay = —1, А^ = — 3. The arrow may thus be
regarded as an inequality sign on the lengths of the fundamental roots at the
vertices.
6.4 Classification of Cartan matrices
81
The set of possible connected Dynkin diagrams, including arrows, is shown
on the following standard list.
6.11 Standard list of connected Dynkin diagrams
g2
We note that, since the diagrams of types B2, F4, G2 are symmetric, it does
not matter in which direction the arrow is drawn in these cases.
The connected components of the Dynkin diagram of any semisimple Lie
algebra must appear on this standard list.
We next obtain a standard list of corresponding Cartan matrices. We say
that two Cartan matrices (Ay), (Ay) are equivalent if they have the same
degree / and there is a permutation <j of 1,... , / such that
Ay — A^yj^yj.
Equivalent Cartan matrices come from different labellings of the same Dynkin
diagram. For each Dynkin diagram on the standard list 6.11 we choose a
labelling and obtain a corresponding Cartan matrix which is uniquely deter-
mined. These Cartan matrices appear on the following list.
82
The Cartan matrix and the Dynkin diagram
6.12 Standard list of indecomposable Cartan matrices
/2-1
\
-1 2 -1
-1 2 -1
-1 •
• -1
-1 2 -1
-1 2 -2
-1 2/
/2 -1	\
-1 2 -1
-1 2 -1
-1 • •
• -1
-1 2 -1
-1 2 -1 -1
-12 0
-10 2/
6.4 Classification of Cartan matrices
83
^2—1	\
-1 2 -1
-1 2 -1 -1
-1 2
-1	2 -1
\ -1 2/
/ 2 -1	\
-1 2 -1
-1 2 -1
-1 2 -1 -1
-1 2
-1	2 -1
\	-12/
/ 2 -1	\
-1 2 -1
-1 2 -1
-1 2 -1
-1 2 -1 -1
-1 2
-1	2 -1
\	-12/
/2-1	\
-1 2 -1
—2 2 -1
\	-12/
A Cartan matrix is called indecomposable if its Dynkin diagram is con-
nected. Any Cartan matrix will determine a set of indecomposable Cartan
matrices, unique up to equivalence, whose Dynkin diagrams are the connected
components of the Dynkin diagram of the given Cartan matrix.
If A is the Cartan matrix of any semisimple Lie algebra, each indecom-
posable component of A will be equivalent to some Cartan matrix from the
above standard list.
Proposition 6.13 If a semisimple Lie algebra L has a connected Dynkin
diagram then L is simple.
84
The Cartan matrix and the Dynkin diagram
Proof. Let L = H® La be a Cartan decomposition giving rise to the
Dynkin diagram Д. Let I be a non-zero ideal of L. We shall show that I = L,
thus proving that L is simple.
We first aim to prove that I Г\Н f(). Suppose if possible that IC\H=O.
Let ea be a non-zero element of La and choose a non-zero element x e I with
x =/1+22 МЛ
аеФ
such that the number of non-zero p.,, is as small as possible. Since I Г\Н = 0
there exists some Then we have
[R] = 22 Pa [ll'pCa] = E Pa<*(^) ?a-
аеФ	аеФ
Now by Proposition 4.18 we can choose e^eLp and e_peL_p such
that [epe_p] = h'p. Thus [[/ijgx] e_p] = p.aa (h'p) [eae_p] = fip/3 (h'p) h'p +
Е«еФ p.na (h'p) Na_pea_p where [еае_^] = Na^ea_p. Now we have
[[Ег]е Je / since xel and e_p) 0 since fip^O and /3 (/1^) =
(h'p, h'p)^0. Moreover the number of non-zero terms coming from the root
spaces La is less for [[й^х] e_(3] than it was for x. This contradicts the choice
of x. We can therefore deduce that I Г\Н f().
The next step is to show that I^>H. Suppose if possible this is not so. Then
This implies that there exist at eП and xelCtH such that (h'ax)^0. For if
I Г\11 were orthogonal to each h'a. it would be orthogonal to the whole of H
and would therefore be 0. Then we have
[*4] = “AR, = (h'a., x) ea. e I.
Since (h'a., x)^0 we deduce that ea. e I. Thus [ea.e_a] = h'a. e I.
We can therefore divide the a; t II into two classes, those with h'a. eI and
those with h'a. gl. Both classes are non-empty. Furthermore if h'a. el and
h'a ^I then {h'a , h'a^ = 0. This means that vertices i, j are not joined in the
Dynkin diagram Д, so Д is disconnected. This is a contradiction, thus we
deduce that I dH.
Finally we show that I = L. Let a e Ф. Then we have
lh'aea\ = a (h'a) Ca = (h'a’ h'a) e a-
Since h!a el we have [h'aea] e I, and since (h'a, h'a) ==0 we deduce that ea e I.
This is true for all a e Ф and so I = L. Thus L is simple.	□
6.4 Classification of Cartan matrices
85
We next consider what happens when the Dynkin diagram of L is discon-
nected.
We first define an action of the Weyl group on H. The Weyl group was
introduced in Section 5.2 as a group of non-singular linear transformations
on the real vector space This action can be extended by linearity to give
an action of W on II by C-linear transformations. We also define an action
of W on H by h wh where
A(w/i) = (»’ 1 A) h for all h e H, A e H*, w e W.
There is a unique element wh e II satisfying this condition, and
W1 (W2^) = (W1 W2) f°r aU Wl’ w2 e W.
The actions of W on II and II are compatible with the isomorphism H* H
given by A h'k where A(x) = (hfi x) for all x e H. For suppose w(A) = /z
for A, fieH*. Then
{w (/iA) , x) = (hfi w^1 (x)) = A (»’ 1 (x)) = (wA)x
= /x(x) = (fi' x) for all x e H.
Hence w(A) = fi implies w (/iA) = h'.
Since we know that
s (A) = A-2-^-^-а ГогаеФ.АеЯ*
(a, a)
it follows that
sg(x) = x-2^,^/i; forxefl.
Proposition 6.14 Let L be a semisimple Lie algebra whose Dynkin diagram
A splits into connected components , Ar. Then we have
L = L}®-  -® Lr
a direct sum of Lie algebras, where L; is a simple Lie algebra with Dynkin
diagram Д;.
Proof We have Д = AjU Д2 U • • • UAr. Let П; be the subset of П correspond-
ing to the vertices in A;. Then we have
п = п1ип2и - unr.
86
The Cartan matrix and the Dynkin diagram
Moreover we have (a, /3) = 0 if a e П;, /3 e 11; and i j. Let //, be the sub-
space of H spanned by the elements h'a with sell;. Then we have
H = H1®H2®---®Hr
where (h, h’} = 0 if h e Ht, h' e Hj and i j.
Now let sell; and consider the fundamental reflection sa e W. It is clear
that s„ transforms 77, into itself and fixes each vector in 77; for all i Ф i. Thus
we have
s„(//,) = //7	j=l,...,r.
Since the elements sa generate the Weyl group W we deduce that
w(Hj)=Hj	,r weW.
Now for all a e Ф we have h'a = w (h'a) for some at e П and some w e W,
by Proposition 5.12 and the definition of the H'-action on 77. It follows that
each h'a, a e Ф, lies in 77; for some i. Let Ф; be the set of all a e Ф such that
h'a e 77;. Then we have
Ф = Ф1йФ2и---иФг.
We define L; to be the subspace of L spanned by 77; and the ea for all
a e Ф;. We deduce from the Cartan decomposition of L that
L = L1 ® L2 ® • • • ® Lr
a direct sum of vector spaces. In fact we can see that each L; is a subalgebra
of L. It is sufficient to verify that [eae^ eLt if a, /3 e Ф;. If a + /3 e Ф then
we have а + /ЗеФ; since h'a+l3 = h'a +h'^ е/f. If a + /3 = 0 then [eae^j is a
multiple of h'a and so lies in 77;, thus in L;. If a + /3 is non-zero but not a root
then [еае^ =0. In either case we have [еае^ eLt. Thus L; is a subalgebra
of L.
We show next that [L;L^] = O if i^j. Let a e Ф; and /ЗбФ;. Then we
have
\Ке/з] = /3 Ю ep = (h'i3’ h'a} e/J = <>
and similarly [ea/i^]=0. We also have [e,,e.j]=0. For а-Н/З^Ф since
h'a +	‘З008 not in апУ subspace Hk of 77. It follows that = O.
We now know that each L; is an ideal of L, since
[L;L]C£[L;LJC[L;L;]CL;.
j
6.4 Classification of Cartan matrices
87
This implies that
[M-----h xr, у ! 4--1- yr] = [%! у ! ] 4-h [Xryr ]
where xh yt e L;. Hence
L = ф /. 2 ф • • • ф Lr
is a direct sum of Lie algebras.
Now each L; is a semisimple Lie algebra. For let I be a soluble ideal of L;.
Since = О for all j ф i the ideal I is an ideal of L. Since L is semisimple
we have 1=0. Hence L; is semisimple.
We next observe that //, is a Cartan subalgebra of L;. The subalgebra //, is
abelian, hence nilpotent. Let satisfy xeN (Hfi Then [xh] eHt for all
hefi. We also have [x/i] = 0 for all h~If with j i. It follows that [x/i] e H
for all h e H. Since H is a Cartan subalgebra of L we have N(H) = H. Hence
xeH. Thus x-JIcf = [f Thus Hi is a Cartan subalgebra of L;.
We now consider the Cartan decomposition
=	£ <Cea
atty
of L; with respect to //,. We see that Ф; is the root system of L; with respect
to //,, that П; is a fundamental system of roots in Ф;, and that Д; is the
Dynkin diagram of Now Д; is connected. Thus the Lie algebra L; must
be simple, by Proposition 6.13. Thus we have obtained a decomposition of
L as a direct sum of simple Lie algebras L;, whose Dynkin diagrams are the
connected components Д; of Д.	□
Corollary 6.15 A semisimple Lie algebra L has a connected Dynkin diagram
if and only if L is simple.
Proof. This follows from Propositions 6.13 and 6.14	□
7
The existence and uniqueness theorems
We have seen that each non-trivial simple Lie algebra L has a Dynkin diagram
Д which appears on the standard list 6.11 of connected Dynkin diagrams. In
the present chapter we shall consider the converse question. Given a Dynkin
diagram Д on the standard list, is there a simple Lie algebra L with Dynkin
diagram Д? If so, is L uniquely determined up to isomorphism? We shall
show that both the existence and uniqueness properties hold. The proof of
the uniqueness property is somewhat easier, and we shall prove this first. In
order to do so we shall need some properties of the structure constants of the
Lie algebra L.
7.1 Some properties of structure constants
Let L be a simple Lie algebra with Dynkin diagram Д. Let H be a Cartan
subalgebra of L and
аеФ
be the Cartan decomposition of L with respect to H. We know from The-
orem 4.20 that dim La = 1 for each a e Ф. Let ea be a non-zero element of
La. Let П be a fundamental system of roots in Ф. Then the elements h'a. for
at e П form a basis for H. It will be convenient to choose a slightly different
basis consisting of scalar multiples of the h'a . We define ht eH by
We note that at (/z;) = 2. Then {hh z = 1,... , / ; ea, a e Ф} is a basis
of L. By Proposition 4.18 we know that h'a e [LaL_a\ for all аеФ. Thus,
88
7.1 Some properties of structure constants
89
if we have already chosen the ea for all a e Ф+, we may choose the e.
uniquely for a e Ф+ to satisfy the condition
(This relation will then be satisfied for a e Ф also.)
We define ha e H for each a e Ф by
h =^-
“ <h’a,h’ay
The element ha is called the coroot corresponding to the root a. In particular
we have ht = h . We then have
= for all а еФ.
We next consider the product when a+ /3^0. We have /S] = О
if a + /30 and a + /3 Ф. If а + /ЗеФ we have /S] cLa+p. We define
Na p e C by the condition
^a,pea+P'
The numbers Na^ for a, /3, a + /3 e Ф will be called the structure constants
of L. They clearly depend upon the choice of the elements ea e La.
We now consider the multiplication of the basis vectors {hh ea} of L. We
have
[M7]=o
= ea
Ke-a] = ha
[eaep\=Napea+p if a,/3, a +/3 e Ф
[eaep]= 0 if a + /3^0 and a + /3^ Ф.
In order to express [eae_a] as a linear combination of basis elements we may
express h'a as a linear combination of the h'a., at e П, and so also express ha
as a linear combination of the ht.
We shall now derive some relations between the structure constants Na p.
Proposition 7.1 The structure constants Na p satisfy the following relations.
(i)	N/3,a = ~Na,p-
(ii)	W a, 0, У e Ф satisfy a + /3 + y = Q then = Np’y =
(y, y) (a, a} (/3, j3)
90
The existence and uniqueness theorems
(iii)	NapN_a_p = —(p +1)2 where the a-chain of roots through /3 is
-pa + /3, ... ,/3,... ,qa + j3.
(iv)	If a, /3, y, 8 e Ф satisfy a + f3+y + 8 = Q and no pair are negatives of
one another, then
^а,/3^у,3	|	। NygN^g _ Q
(a + /3, a + /3)	(/3 + y,/3 + y) (y + a, y + a)
Proof, (i) This relation is clear.
(ii)	Suppose a + f3 + y = Q. We consider the Jacobi identity
+ [[e/+y] ej + [[eyea]	=0.
This gives
N«,I3 [е«+узе-(«+д] +N/3,y [e-aea] + N%a [е^ер]=0
that is
2N __^+±P__ — ум ha +2N
'MJ	e"<h'..k'.+
Now the roots a, /3 are linearly independent since, if they were not,
a + /3 could not be a root. Thus h'a, h'p are linearly independent and
h'a+p = h'a + h'p. We deduce that
_ N/gy _ Nya
^h'a+p, h'a+pj (h'a, h'a} (hy h'^
that is
_ N/gy _ Nya
(y, y) (a, a) (/3, /ЗУ
(iii)	Now suppose a, /3 e Ф are linearly independent. We consider the Jacobi
identity
[[eae--«] eg] + [[e M;] +.] + [[ff ?a]	= 0.
This gives
2 +N_a/3N_a+i3aei3 + N^aNa+i3_aei3 = 0.
We deduce that
{hy Kb)
\ N-a,pN_a+p + Np Na+p — 0.
{h'„ h')
7.1 Some properties of structure constants
91
Using relations (i) and (ii) this may be written
(/3,/3)	( —a + (3, — a+ff)
a’p -a’-p(a+/3,a+p) Na,-a+pN-a,a-p
(a, a)
(If —a + /3 is not a root N_a^ is interpreted as 0 so the middle term
disappears.) We now consider the «-chain of roots through /3. Let it be
-pa,qa+/3.
Weapplythesameformulatothepairs (a, /3)(a, — a + /3)... (a, —pa+/3)
and obtain
„ jy	N	+	+
“•₽ -“•-₽{« + /3,а + /3)	(/3,13)	(a, a)
(-a + (3,-a + (3)	{-2a + (3,-2a + (3)
Na,-a+pN-a,a-p	Na.-^+pN-a.la-p {_a + ^_a + jS}
_2K-«+/3)
(a, a)
(-pa + (3,-pa + (3)	_2(a,—pa + [3)
a.-pa+p -^-^_(р_1)а + ^_(р_1)а + ^	{a,a}	
(The last equation has only one term on the left since — (р+Г)а+/3 is
not a root.) Adding these equations we obtain
N N	Of 4.1^“’^	nX/’+l)
Na,pN-a,-p2-—a—= 2(p+ 0;-----------; -2—7—•
(a + p, a + p)	{a, a) 2
However, we know from Proposition 4.22 that =p — q. Thus we
have
V-^<a + /3^ + /3>--(p+1)<7-
In order to obtain the required result Na pN_a	= — (p+ I)2 we must
show
(a + /3, a + /3) _ p+ 1
92
The existence and uniqueness theorems
We recall from the proof of Proposition 6.1 that
2W)
(a, a)
(B,B)
= 4 cos2 в
where в is the angle between a, В and hence that 2-^-|- e {0, — 1, —2, —3}
Also from Proposition 4.22 we know that 2^^y =p — q. If we choose
/3 to be the initial root in its «-chain we have p = 0 and hence q<3.
This shows that each «-chain has at most four roots. Thus the possible
positions of /3 in its «-chain are
В	a + B 	О			p = Q	</=1
					
в	a + B	2а + В		р = 0	q = 2
					
— a + B	В	a + B		/7=1	</=1
					
В	a + B	2а + В	За + В	/7 = 0	</ = 3
					
-a + B	В	a + B	2а + В	/7=1	q = 2
					
-2а + В	-a + B	В_	а + В	/7 = 2	</=1
In the first case we have (a + /3, a + /3) = (/3, /3) since sa(/3) = a + /3. In
the remaining cases the first and last roots in the «-chain are long roots
and the remainder are short roots. The relative lengths are given in the
proof of Proposition 6.1. We have
(a + B, « T/3)	.	.
2---------- — i 1 о f 1 3
(B,B)	’2’ ’3’ ’
in the above six cases respectively. Thus in each case we have
(a + B, a + B) _ p+1
H) ~~
and so NapN_a_p = -(p + I)2
(iv)	Now suppose that «, /3, y, 8 еФ satisfy a+B+y+8=0 with no pair
equal and opposite. Consider the Jacobi identity
[[CT/;]	+ [[^7] ea] + [[eyea] e?] =0.
7.2 The uniqueness theorem
93
This gives
^а+Р, у + y^P+y, a + N% aNy+a/3 = 0.
Using relations (ii) this gives
Na.pNy’g + NpyNaS	NyaNpS _ q
(a+ /3, a+/3}	(/3 + y, /3 + y)	(y + a,y + a)
(As usual we interpret Ne as 0 if 0 + ф is not a root.)	□
Proposition 7.1 (iii) has a very useful corollary.
Corollary 7.2 If a, /3, a + (3 еФ then	Thus = La+p.
7.2 The uniqueness theorem
We shall now use the above relations between the structure constants to show
that the Lie algebra L is uniquely determined up to isomorphism. A Dynkin
diagram on the standard list 6.11 is given, and this determines uniquely a
Cartan matrix A=(Ay) on the standard list 6.12. Now the Cartan matrix
determines the set Ф of roots as linear combinations of the fundamental roots
П = {oq,... , a/}. For each root a e Ф has form a = w (aq) for some at e П
and some weW, by Proposition 5.12. Moreover each element weW is a
product of elements sq,... , sl by Theorem 5.13. The actions of sq,... , sq on
the fundamental roots ar,... , al are given in terms of the Cartan matrix by
si (aj) = aj~Aijai-
Thus by applying the fundamental reflections successively to the fundamental
roots we obtain all roots as linear combinations of the fundamental roots.
We next observe that all scalar products (h'a, h'^ for a, /3 e Ф are determined
/ Jrf Jrf \
by the Cartan matrix. By Proposition 4.22 2	h,^ is determined by the
root system, hence by the Cartan matrix as shown above. Then (h'a, h'a) is
determined by the formula
I Hh’a,h’p}\2
(h’a,h’a) ^\(h’a,h’a)J
94
The existence and uniqueness theorems
of Proposition 4.24. Thus (h'a, h1^ is also determined by the Cartan matrix.
Thus we see that if the structure constants Nap are known the multiplication
of basis elements
[hiea] = a(hi) ea
Ke-al = ha
[eaep\=Na,pea+p if а,(3,а + (ЗеФ
[eaeg]=0 if a + and а + /З^Ф
will be completely determined.
We shall show that for certain pairs (a, /3) of roots the structure constants
Nacan be chosen arbitrarily, and that the remaining structure constants are
uniquely determined in terms of these by the relations of Proposition 7.1.
We choose a total ordering on the vector space V = //i as in Section 5.1
giving rise to the positive system Ф+ and fundamental system П of roots. An
ordered pair (a, /3) of roots will be called special if a + /3 e Ф and 0 < a < /3.
The pair (a, /3) will be called extraspecial if (a, /3) is special and if, in
addition, for all special pairs (% 3) such that a + /3 = y + 8 we have a < y.
Lemma 7.3 The structure constants Na^ for extraspecial pairs (a, /3) can
be chosen as arbitrary non-zero elements of C, by appropriate choice of the
elements ea.
Proof We choose the ea for a e Ф+ in the order given by c. Suppose (a, /3)
is an extraspecial pair. Then we have
[е«е/з] =Napea+p
and ea, ep have already been chosen. Moreover there is only one extraspecial
pair with given sum a + /3. Thus ea+(3 can be chosen to give any non-zero
value of Na p.	□
Proposition 7.4 All the structure constants Na ^ are determined by the struc-
ture constants for extraspecial pairs.
Proof We consider the set of all pairs of roots (a, /3) such that a + /3 is a root.
Let (a, /3) be such a pair and let y = — a — /3. Then the following 12 pairs of
roots are of the given type.
(a, /3) (/3, y) (y, a) (/3, a) (y, /3) (a, y)
(-a, -/3) (-/3, —y) (-y, -a) (-/3, -a) (-y, -/3) (-a, -y).
7.2 The uniqueness theorem
95
Since a + /3 + y = 0 either two or one of a,/3, у are positive. Thus either
two of a, /3, у are positive or two of —a, —/3, — у are positive. By choosing
two positive roots from a, /3, у or from —a, —/3, — у and by writing them
in the appropriate order we obtain a special pair. Thus just one of the above
12 pairs of roots is a special pair.
Now the relations in Proposition 7.1 (i), (ii), (iii) enable us to express
AL ,f, Np,y, a and N_a_p in terms of Nap. Thus these relations enable us to
express Ne for all the 12 pairs (0, di) above in terms of Ne for the special
pair (в, </>).
The next stage is to show that the Nap for all special pairs (a, /3) are
determined in terms of the Nap for extraspecial pairs. Suppose (a, /3) is
special but not extraspecial. Then there exists an extraspecial pair (y, 3) such
that a + /3 = y + 8. Thus a + /3+ (—y) + (—3) = 0 and no pair of a, /3, —y, —8
are equal and opposite. By Proposition 7.1 (iv) we have
Na,pN y,s	।	^/3,-y^a,-3	।	^-y,a^l3,-3	_ q
(a + /3, a + /3)	</3 — y,/3 — y> (-y + a, -y + a)
Now the roots a, /3, y, 8 are ordered by
0<y<a<(3<8.
Thus we may use relations (i), (ii), (iii) of Proposition 7.1 to express N_y,_s in
terms of Ny s; Np_y in terms of Nyp_y\Na _s in terms of Na S_a; N_y a in terms
of Ny a_y, and Np_s in terms of Np,s_p. Thus Na p is expressed in terms of
'^y./jy’ a’ ^y.a—y’ ^3,6—
Now (y, 3) is an extraspecial pair and (y, /3 — y), (a, 3 — a), (y, a — y) and
(/3, 3 — /3) are all pairs of positive roots whose sums are roots less than
a + /3 = y + <5 in the given ordering. We may therefore argue by induction on
a + /3, using the given order, that Na p can be expressed in terms of Ne ф for
extraspecial pairs (0, di).	□
We can now state our uniqueness theorem.
Theorem 7.5 Any two simple Lie algebras with the same Cartan matrix are
isomorphic.
Proof. We choose the basis elements {/i;, ea} of such a Lie algebra L such
that Na,p= 1 for all extraspecial pairs of roots (a,/3). We may do this by
Lemma 7.3. The remaining structure constants Na,p are all then uniquely
determined by Proposition 7.4. Thus the formulae expressing a Lie product of
96
The existence and uniqueness theorems
basis elements as a linear combination of basis elements are completely deter-
mined by the Cartan matrix. Thus the Lie algebra L is uniquely determined
up to isomorphism.
7.3 Some generators and relations in a simple
Lie algebra
We now turn to the question of the existence of a simple Lie algebra with
Cartan matrix on the standard list 6.12. A proof of the existence theorem has
been given by J. Tits (IHES Publ. Math. 31 (1966)) along the lines of the
arguments used so far. The details are technically quite complicated, however,
and so we prefer to give a different proof of the existence theorem.
Let L be a simple Lie algebra with Cartan matrix A. Let H be a Cartan
subalgebra of L and
L = H®^La
аеФ
be the Cartan decomposition. As before we consider the elements ht e H
given by
where П = {oq,... , aj is a fundamental system in Ф. As in Section 7.1 we
can choose elements e; eL„ , L eL „ such that [e;fl = ht.
We shall show that the elements ex,... , ehhx,... ,hl,fi,... , /) generate L.
(Of course this is equivalent to saying that ex,... , e;, /),... , generate L,
but it will be useful to include hx,... , ht in the generating set.)
Lemma 7.6 If a e Ф+ and a 011 there exists at e П such that a — aie Ф+.
Thus every positive non-fundamental root is the sum of a fundamental root
with a positive root.
Proof. Suppose if possible that the result is false. Then a — at is not a root
and is non-zero for each i. (We can use Corollary 5.6 to see that a — at cannot
be a negative root.) Consider the oq-chain of roots through a. This has form
a, at + a,... , qat + a.
7.3 Some generators and relations in a simple Lie algebra
97
By Proposition 4.22 we have
2 7-----7 =
\ai’ ai)
This implies that (a;, a)<0. Now аеФ+ has form a = 'ffiniai with all
и; > 0. Thus
(a, a') = '^ni (at, a) < 0.
i
This gives a contradiction, since we know (a, a) > 0.	□
Proposition 7.7 The elements ex,... , e;, fy,... , h^fy,... , generate L.
Proof. Since /ij,... , hl span H it will be sufficient to show that each La for
a e Ф+ lies in the subalgebra generated by , e; and each La for a e Ф
lies in the subalgebra generated by ... ,
Let a e Ф+. If a = at for some i we have La = Ce;. If а fl I we can write
a = ai + /3 for some at e П and some /3 e Ф+ by Lemma 7.6. We then have
=La by Corollary 7.2. Thus we may choose ea = [e;, for some
in Lp. By repeating this process we obtain
e„ = Г Г e, e, 1... e; 1
a LL И г2J J
for some sequence q,... , ik. Thus each La for a e Ф+ lies in the subalge-
bra generated by ... , e;. Similarly each La for a e Ф lies in the sub-
algebra generated by /x,... ,	□
Proposition 7.8 The generators er,... , el,h1,... ,hl,f1,... , of L satisfy
the following relations.
(a)	[My]=0
(b)	[hiej]=Aijej
(c)	[hifj] = -Aijfj
(d)	=
(e)
(f)	[ei [«/••• [eiey]]]=°
(g)	[Ш---ИМ if^j-
Note that in relations (f), (g) there are I — Л,; occurrences of et, f respec-
tively. Since Atj < 0 for i j this number 1 — A^ is a positive integer.
98
The existence and uniqueness theorems
Proof. Relation (a) follows from [HH] =0. For relation (b), we have
Relation (c) is obtained similarly. Relation (d) holds by definition of ft.
Relation (e) holds because e La._a. and at — a, is not a root when i^j,
as follows from Corollary 5.6. In order to prove relation (f) we consider the
a;-chain of roots through ctj. Since —ai + ctj is not a root this chain has form
aj, a + aj,... , qat +aj.
By Proposition 4.22 we have Atj = — q. Thus (1 — AA) ai + aj is n°t a root.
Since the element [e; [e;... lies in b(i_A..)a+a. this element must
be 0. Relation (g) is obtained similarly.	□
7.4 The Lie algebras L(A) and L(A)
Let A be a Cartan matrix on the standard list 6.12. Motivated by Proposi-
tions 7.7 and 7.8 we shall construct a Lie algebra L(A) which will be shown
to be a finite dimensional simple Lie algebra with Cartan matrix A.
Suppose A is an I x I matrix. Let be the free associative algebra over C on
the 3/ generators ex,... , ehhx,... ,hl,fi,... The set of all monomials
in these generators form a basis for Let [§r] be the Lie algebra obtained
from by redefining the multiplication in the usual way and let £ be the
subalgebra of [§r] generated by the elements , eh hx,... ,hl,fi,...
Let J be the ideal of £ generated by the elements
lhihi\
lAe?]
Ы]+АЛ
[eifi] ~
[ejJ for z
forz#J
7.4 The Lie algebras L(A) and L(A)
99
where the number of occurrences of e;, ft respectively in the last two elements
is 1 — A,;.
We define /.(A) = £/./. We shall eventually be able to show that I.(A) is
the Lie algebra we require to prove the existence theorem. This description
of L(A) by generators and relations is due to J. P. Serre.
In order to investigate the Lie algebra L(A) it is convenient to define a second,
larger, Lie algebra L( A). Let J be the ideal of £ generated by the elements
lhihi\
lAe?] ~Aijej
[М]+АУл
[e;/;] - \
[ejJ for i # j.
Let L(A) = £/J. Since J c J we have surjective Lie algebra homomorphisms
£^L(A)^L(A)
We shall investigate the properties of the Lie algebra L(A). This is generated
by the images of the generators of £ under the above homomorphism. These
images will continue to be written ex,... , ehhx,... ,ht, fi, ... , fi- These
elements satisfy the relations
[M>]=o
[^iej]=Aijej
[e;/;] =
[ei//]=° fori# J
Proposition 7.9 Let be the free associative algebra over C with gen-
erators fi,... , fi- Then may be made into an L(A)-module giving a
representation p: L(A) [End 7y | defined by:
pl.Of, =fifil---fi,
p fti • • • A = - (E Aat) A • • • A
V=i /
P (e,) fi, fi, = ~Yl 8at [ X Auh ) fi, fi,
k=l \h=M /
where as usual the symbol fr means that fr is omitted from the product.
100
The existence and uniqueness theorems
Proof. Since the monomials Д ... form a basis for iy the endomor-
phisms p (/;) , p (/i;) , p (e;) are uniquely determined by the above formulae.
Thus there is a unique homomorphism g^End iy mapping e^h^f to
p (e;) , p (Ii,), p (/;) respectively. This induces a Lie algebra homomorphism
[§r] [End ,y | and so, by restriction, a Lie algebra homomorphism £
[End /у |. In order to obtain a homomorphism L(A)—> [End iy | we must
verify the following relations.
(a)	[p(A)p(A)]=0
(b)	[p(hi)p(ej)] = AijP(ej)
(с)	[p (A) p (_/})] = -Ayp(/})
(d)	[p(e;)p(/;)] = p(M
(e)	[P(e/)P (/;)]= 0	for/#./
Relation (a) is trivial since p (/i;) multiplies each basis element of by a
scalar.
To prove relation (b) we have
P (^) P (ej fi, • • • fir = - E Sjik ( £ Ajt J ( - £ A;;J Д ... fit... fir
p(ej)p(.hi)ftl...fir = -EEJ E a,;J (-ЕаП A---A---A
k=l \h=fc+l / \ g=l /
Thus
(p (A) P (eJ - P (eJ P (A)) A • • • fir
= E|-EAJ E ал))д...д...4
\ k=l \h=k+l / /
= Аур(е>)Д...Д.
To prove relation (c) we have
P (M P (A) A • • • fir = - (Ay+E Aiit) Л A • - - A
p (Л) p /у... a = - (E A«t) fjfi,
V=i /
Thus
(p (M P (л) - P (л) P (^))	-AM, fi=-AiJp(fj)fil...fir.
7.4 The Lie algebras L(A) and L(A)
101
We next consider relation (d). We have
р(а)рШЛ •	fir = -\ZAiih]fr--fr \h=l	/ -Щ E Aii}ffr--fr--fr k=l	\h=k+l	/
Р(Л)р(ОА •	••E = -EM E Aul]ffir--fir--fr- k=l	\h=k+l	/
Thus
(p(а)рШ-рШр(«/))fi{ • • • fir = - ( EAnh) fir--fir = P(Mfr--fr
\h=l /
Finally we consider relation (e). Suppose i^j. Then
р(е/)р(Л)Л•••Л =E Аи„]fjfir--fir--fr
k=l \h=k+l /
= р(Л)р(е;)Д ...fir.
Thus all the relations are preserved and we have a homomorphism L(A)—>
[Endg-].	□
We can deduce useful information about L(A) from the existence of this
homomorphism.
Proposition 7.10 The elements Ip,... ,lp of L(A) are linearly independent.
Proof. We show that the elements p (hf),... , p (/i;) of End are linearly
independent. We have
Thus if \P Oi) = 0 we would have JT A; Atj = 0 for all j = 1,... , /. Since
the Cartan matrix A = (Af) is non-singular this implies that A; = 0 for each i.
Hence p (/lj ,... , p (h[) are linearly independent, and so Ip,... , Ip must be
linearly independent also.	□
Let H be the subspace of L(A) spanned by Ip,... , Ip. Then we have
dimH = l. Moreover [HH] = 0, thus H is an abelian subalgebra of I.(A).
We consider the weight spaces of L(A) with respect to H. We are no longer
dealing with a finite dimensional //-module as in Theorem 2.9, but analogous
102
The existence and uniqueness theorems
ideas apply in our situation. Elements of I Ioni(/7, C) will be called weights.
For each weight p-.H C we define the corresponding weight space
b(A)g by
/.|T );j = {.rt/.(/I) : [/ix] =/z(/i)x for all h^H}.
Proposition 7.11 L(A) =	£(А)М. Thus L(A) is the direct sum of its
weight spaces.
Proof. We first show that /.(A) =	A vector which lies in a weight
space will be called a weight vector. We observe that, if x,yeL(A) are
weight vectors of weights A, p respectively, then [xy] is a weight vector of
weight A + p. For we have
[/i[xy]] = [[йх]у] + [х[йу]] = А(й)[ху]+^(й)[ху]
= (A + /z)(/i)[xy] iorh^H.
Now L(A) is generated by elements e;, hh ft. Eet at e Hom(/f, C) be
defined by
ai (hj)=AH-
Then e; is a weight vector of weight at, ft is a weight vector of weight —ai
and ht is a weight vector of weight 0. Thus all Fie products of generators e;,
ht, /; are weight vectors. Since every element of /.(A) is a linear combination
of such products we deduce that
Z(A) = £l(A)m.
We next show that this sum is direct. If this is not so we can find a non-zero
vector xeL(A)M such that x = ^vxv where xveL(A)v and v runs over a
finite set of weights all distinct from p. Since xelfA)^ we have
(ad h — p(h) l)x = 0.
Since x = xv with xv e L(A)V we have
]""[ (ad h — v(h~)l~) x = Q.
Now we can find an element h e H such that p(h) A- v(h) for all such v. For
the elements satisfying p(h) = v(h) for some fixed v lie in a proper subspace
of II. and the finite dimensional vector space II over C cannot be expressed
7.4 The Lie algebras L(A) and L(A)
103
as the union of a finite number of proper subspaces. Thus we choose h e H
such that fi(h) v(h) for all such v. Then the polynomials
in C[t] are coprime. Thus there exist polynomials a(t), b(t) eC[t] with
- rW) = 1 •
If follows that
a(ad h) (ad h — fi(h) l)x+b(ad /i)]~~[(ad h — v(h) l)x = x.
We deduce that x = 0, a contradiction. Thus the sum L(A)M is direct. □
We next obtain information about the kind of weights /z which can occur,
that is for which L(A)M 0. The weights ax,... , al e Hom(77, C) are linearly
independent since the Cartan matrix A is non-singular. Thus any weight has
form MjCEjd-----Hn;a( for и;еС. We shall show that all weights /z which
occur in L(A) have this form with и; e Z and with either и; > 0 for all i or
и; < 0 for all i.
Let
Q = {n,aiA— + nlal \ n;eZ}.
-----; и;>0 for all i}
Q~ = {n1a1-\----; и;<0 for all i}.
Let
L(A)+ = £ L(A)g
ме2+
L(A)- = £ L(A)g.
мей
It follows from Proposition 7.11 that the sum L(A)~ + H + L(A~)+ is direct.
We shall show that in fact
7.(Л) = 7.(Л) Ф//ФШ) .
Let N be the subalgebra of 7.(4) generated by gp... , e; and N the
subalgebra generated by /x,... , Since e; has weight at and has weight
— a; we have Kcl.(A) and i\' <zL(A) . Thus the sum N~ +H + N is
direct.
104
The existence and uniqueness theorems
Proposition 7.12 (f)L(A) = N ®H®N
(ii) N = L(A)+, N~ = L(A)~, H = l(A)0
(iii) Every non-zero weight oflfA) lies in Q+ or in Q~.
Proof. The relations [h,e,] = A-ijej show that [ht, 7/] c N since the generate
N. Thus we have [H, TV] cN. It follows that H + N is a subalgebra of L(A),
since
[H + N, H + TV] с [HH] + [HN] + [TVTV] с H + TV.
Similarly N +H is a subalgebra of L(A). We now consider the subspace
N + H + N. The relations [eifi] = hi and [e;/)] = 0 if i^fj show that
[e:,N ]<zN +H.
For this is true for the generators of /V , and the relation
[e;[xy]] = [[e;x] y] + [x[e;y]]
then shows it is true for all elements of N since /V + H is a subalgebra. It
follows that
[e,, /V +/7 + /V] C/V +/7 + /V
since H + N is a subalgebra. Similarly we have
[ft, N~ + H + TV] C N~ + H + TV
and the relation
[ht, N~ + H + TV] c N~ + H + TV
is clear. It follows that the set of all xeL(A) such that
[%, n-®h+n]gn-®h+n
contains e;, ht, ft. However, the relation
[ky]z] = [[xz]y] + [x[yz]]
for zGN + H + N shows that the set of such x is a subalgebra. This sub-
algebra must be the whole of L(A). Thus N + II+ N is an ideal of L(A).
Since L(A) is generated by e;, l+f it follows that N +H + N = 1.(Л). We
know that this sum is direct, so we have
I.(A) = N ®H®N.
7.5 The existence theorem
105
Since /V с/~(Л) ,/Vc/.(A) and the sum 7.(Л) ф/7ф/.(Л) is direct
we deduce that N =L(A) and N = L(A)+. Since 7.(Л) = 7.(Л) ф/7ф
L(A)+, H c L(A)0, and the weights occurring in L(A) and L(A)+ are all
non-zero, we deduce from Proposition 7.11 that H = L(A)0. Thus all parts of
the proposition have been proved.	□
Proposition 7.13 dimL(A)a. = 1 and dimL(A)_a. = 1
Proof. We know that (фШф Also the element e;eL(A) is non-zero,
since it induces a non-zero endomorphism p(e;) on the L(A)-module iy
considered in Proposition 7.9. Hence dimL(A)a. > 1. On the other hand we
have
L(A)a_cL(A)+ = iV.
Now N is generated by gj,... ,e; so is spanned by monomials in these
elements. All such monomials are weight vectors. The only monomial which
has weight at is e;, since the at are linearly independent. Thus we have
dimL(A)a. = 1. The relation dim 7.(Л) ,b = 1 is obtained similarly. □
7.5 The existence theorem
We now turn to a study of the Lie algebra L(A), in order to show that it
is a finite dimensional simple Lie algebra with Cartan matrix A. From the
definitions of L(A), 7.(Л) we see that I.(A) is isomorphic to 7. (/!)// where I
is the ideal of 7.(Л) generated by the elements
h И--М]]
[Ж-W]]
for all i^j. As usual we have 1 — A^ factors e; or ft.
Proposition 7.14 (i) Let I+ be the ideal of N generated by the elements
[e; [e;... for all i^j. Then I+ is an ideal of L(A).
(ii) Let I be the ideal of N generated by the elements [/; [./,••• [/;/,]]]
for all i^j. Then I is an ideal ofl.(A).
(iii) I = I+®I-.
106
The existence and uniqueness theorems
Proof. We write Xtj = [et [e;... [ед]]] and Уу = [f [f ... [/;Д]]]. Then I+
is the set of all linear combinations of elements
[[ХуЧ]-"Ч]
for all i^j and all kr,... ,kr in {1,... , /}. For such linear combinations
certainly lie in I+, and form an ideal of N.
Now Xtj is a weight vector, being a Lie product of weight vectors e;, ej.
Similarly [[-Д/'/, ] • • • ek ] is a weight vector. It is therefore transformed by
each of hx,... , h[ into a scalar multiple of itself. In order to show that / is
an ideal of L(A) it will therefore be sufficient to show
[Д,[[Хд,]...^]]е1+
for all i, j,kx,... , kr, k. We shall prove this by induction on r, beginning
with r = 0. In the following lemma we shall show that [Д, Xy]=0, thus
beginning the induction. So let r>l and write [[Хуе4 ]...	] = y. We
assume [Ду] GI+ by induction. Then
[ft [уч]] = [[Ау] 4] + [у [A4]] •
If kr Д k then \fkek ] = 0 and so
[fk [уч]] = [[Ау] д]е/-
If kr = k then
[А [уч]] = [[Ау] 4] + [Ay] eA.
This completes the induction. Thus I is an ideal of L(A). Similarly I is an
ideal of L(A). Hence /©/ is an ideal of L(A) containing the elements Xy
and Ytj. Moreover any ideal of L(A) containing the Xy and Уу must contain
I+ and I . Hence /©/=/.	□
In order to complete the proof of Proposition 7.14 we need the following
lemma.
Lemma 7.15 [Д, Xy] = 0 for all i, j, k with i Д j.
Proof. If k<f{i, j} this relation is obvious since [Де;] = 0 and [Де,]=0.
So suppose k = j. Then we have
[A he/]] = [ei = thjei]=Ajiei
[Аккд]]] = к[Акд]]]=о
[A leilei • • • кд]]]] =° forr>2
7.5 The existence theorem
107
by induction on r. Hence [f;, X;i] =0 if 1 — A;i > 2, that is A;i < —1. If A;i = 0
J	1У J lJ J	lJ —	lJ —	lJ
then Afi = 0 and ГC, X;i] =0 in this case also.
J1	1У J lJ J
Finally suppose that k = i. In this case we shall show that, for r > 1,
[fi [<h [e; • • •	= ~r (Atj + r- 1) [et [e;... [e;e,]]].
<—r^—>	< r-1 >
For r = 1 we have
[fi = [[/^] ej] = - lhiej] = ~Aijej-
For r> 1 we use induction. We have
= ~[hi 1е> F • • •	- (r- о (Аа+r-2) 1е> F • • •
<—r-1—>	<—r-1—>
= (- (2r -2+Ay) - (r - 1) (Ay + r - 2)) [et [e;... [e^]]]
<—r-1^
= -r(Ay + r-l)[e;
<—r-1—>
as required. We now put r = 1 — Ay and obtain [/;, Xy] =0.
Corollary 7.16 L(A) = N ®H®N where H is isomorphic to H, N is
isomorphic to N~/I~ and N is isomorphic to N/I+.
Proof. This follows from the facts that L(A) is isomorphic to L(A)/Z, L(A) =
N-&H&N, and 1 = 1+®Г.	□
We shall continue to denote the generators of L(A) by e;, ht, ft. These are
the images of the generators of £ under the natural homomorphism £	Т(Л).
Proposition 7.17 The maps ad e;: L(A) L(A) and ad/'(: L(A) L(A) are
locally nilpotent.
Proof. To show that ade; is locally nilpotent we must show that, for all
x e L(A), there exists и(х) such that (ad e^A x = 0. Now if ad e; acts locally
nilpotently on x and у it also acts locally nilpotently on | av| . For
(ad e;)" | a v] =	[(ad e;)r x, (ad e,f r у]
r=0 \ Г /
(ad e;)r x will be 0 if r is sufficiently large and (ad е;)" r у will be 0 if n — r
is sufficiently large. Thus (ad e;)" |.vv| will be 0 if и is sufficiently large.
108
The existence and uniqueness theorems
It follows that the set of elements of L{A) on which ade; acts locally
nilpotently is a subalgebra. However, we have
ad e; • e; = 0
(ad	= 0 if i j
(ad e;)2 hj = 0 for all j
(ade;)3/; = 0
ade;-/y = 0 if
Thus this subalgebra contains all the generators ej, hj,fj of L{A), so is the
whole of L{A).
We see similarly that ad/; is locally nilpotent on I.(A).	□
Now the proof of Proposition 3.4 shows that if 3 : L^-L is a locally
nilpotent derivation of a Lie algebra L then exp 3 is an automorphism of L.
Thus exp ad e, and exp adare automorphisms of 1ЛЛ). We define d, e
Aut L{A) by
d, = exp ad e; • exp ad (—/;) • exp ad e;.
Proposition 7.18 (i) d, (//) = H
(ii)	d, (/i) = (/i) for all heH where st : H H is the linear map given by
Si (hi)=hj- Ajihi-
Proof. We have
exp ad e; • hj = (1 + ad e;) hj = hj— А^е{
exp ad (—f)  exp ad e; • hj = exp ad (—/;) • {hj — Ajt ef
= h-ad/; + ^^17-A7;e;)
=	hj — Аце{ — Ajifi — Ац^ + Ajifi = hf — АЛц—A^
J J1 1 JlJ 1 J1 1 JlJ 1 J J1 1 J1 1
exp ad e; • exp ad (—/;) • exp ad efhj = exp ad e; {hj — Aj^ — Aj^
=	(1 + ade;) {hj - A Jr - A;;e;) =	- A;;e; - A;;e; + 2A;;e;
\	I/ \ J	JI I	JI 1/	J	JI I	JI I	JI I	JI I
=	hj — Ajihi.	□
J J1 1
7.5 The existence theorem
109
Now the action of on H is precisely that of the fundamental reflection
= s,t defined in Section 6.4. We recall that
{h'a ’ h}
sfh) = h-2-^—'-h'	for heH
= л—(л«,. л)л<-
In particular
.. / \ к.л)
si (hj) = hj- K’ hj} h^hj- 2--------rht = hj- A^.
{h'a^h'ai]
Thus Proposition 7.18 shows that the automorphism в. of L(A) induces the
fundamental reflection s. on H.
We now consider the decomposition of I.(A) into weight spaces with
respect to H. This time the weights are elements of I lorn (7/, C). For each
weight	we define the weight space L(A)M by
= t/.(/1) : [/ix] =fx(h)x for all h e H}.
Proposition 7.19 L(A) = L(A)M.
Proof. The algebra I.(A) is the sum of its weight spaces, since its generators
e;, h^ f are weight vectors. Moreover the sum of weight spaces is direct, just
as in the proof of Proposition 7.11.	□
It also follows from Proposition 7.12 and Corollary 7.16 that L(A) =
N~ ®H ®N where all weights coming from N are in Q and all weights
coming from /V are in Q . We also have H = L(A)0.
Proposition 7.20 dimL(A)a. = 1 and dim Л(Л) ,b = 1.
Proof. By Proposition 7.13 we certainly have dim L(A)a. < 1. However, the
ideal I+ of N such that N/I+ = N has the property that I+ is a sum of weight
spaces, and all weights occurring in I+ are sums of ar,... , involving at
least two terms. This is clear from the proof of Proposition 7.14. Thus at is
not a weight of I+. Hence
dim =dim L(A)a = 1.
One shows similarly that dim Л(Л) =1.	□
Proposition 7.21 The automorphism d, of I.(A) transforms T(Afj to L(A) .
Hence dimL(A)/z = dimL(A)S /Z.
по
The existence and uniqueness theorems
Proof. Let	Then [/iv] =fjc(h)x for all heH. We apply the auto-
morphism d,. This fixes H by Proposition 7.18. We have
[Ofi, 0;x] = /z(/i)0;x.
Hence
[h, 0;x] = /X (ff^h) 0tx = /j. (s^h) dtx=	0tx,
again by Proposition 7.18. Thus we have 0,x e L(A)S/X. Hence
et (L(A)jGL(AV
Replacing d, by Of1, p. by s^g. and recalling that sj = 1 we also obtain
Of (L(A)J cL(<.
Hence 0t (L(A)J dL(A)w and we have 0; (L(A)g) =L(A)Si/x.	□
We now define W to be the group of non-singular linear transformations
of H* = Hom {H, C) generated by , Sj and define Ф to be the set of
elements w(a;) for weW and ie{l,... ,/}. Then Ф is the root system
determined by the given Cartan matrix A and W is its Weyl group.
Proposition 7.22 dim L{A)a = 1 for all a e Ф.
Proof We have a = w(a;) for some i and some weW. Since W is gen-
erated by Sp ... , sh w is a product of such elements. Thus it follows from
Proposition 7.21 that
dim/.(.4),b =dim Л(Л),ь = I.	□
We aim to show that the Lie algebra I.(A) is finite dimensional. As a step
in this direction we shall show that the Weyl group W is finite. W is iso-
morphic to the group of non-singular linear transformations of /Л generated
by , Sj where /Л = IR/ij -I-------hЮ;. We have dim /7, = /. We do not
have the scalar product on /Л available from the Killing form, so we define
a scalar product directly from the Cartan matrix A.
Proposition 7.23 The Cartan matrix can be factorised as A = DB where
D is diagonal and В is symmetric. D is the diagonal matrix with entries
dr,... , dj e {1,2,3} defined as follows.
If the Dynkin diagram has only single edges then all dt = l.
If the Dynkin diagram has a double edge then d, = \ if at is a long root
and dt = 2 if at is a short root.
7.5 The existence theorem
111
If the Dynkin diagram has a triple edge then dt= \ if at is a long root and
dt = 3 if at is a short root.
Proof. This may be checked from the standard list 6.12 of Cartan matrices.
□
For example in type G2 we have
\-3 2/ \0 зД-l | J'
We now define a bilinear form on IP by (/i;, h^ = dfijB^. This form is
symmetric since В is a symmetric matrix.
Proposition 7.24 This scalar product is positive definite.
Proof. We have n;; = A;;A;; = dfififi, thus — Jn^ = JdjJdfifi. The matrix
J	lJ lJ J1 1 J lJ	v lJ v 1 v J lJ
of our scalar product is
(2 \
-yJnij \
 I of the quadratic
’	2 /
form Q (xb ... , xf) of Proposition 6.6, which is positive definite. Thus DBD
is also positive definite.
□
Proposition 7.25 Our scalar product on HK is invariant under W.
Proof. We first observe that
{ht, x) = djCtfx) for all x e Нж.
For l[hi, hj} = didjBij = diAji = diai(hj). It is sufficient to show that
{stx, sty) = (x, y) for all x,yeHK. We note that s!(x) = x — afix)hi since
st (fii/) = hj —Ajfhf. Hence
(з.х, sty) = (x — afx)hi, у - afyfiif)
= (x, y'f—afix) (h^y) — afiy) (ht, x) + afipfafy) (ht, hf)
= (x,y) — dt at (x)a (y) — dt at (x)a (y) + 2dt at (x) at (у) = (x, у). □
112
The existence and uniqueness theorems
Thus the Weyl group W acts as a group of isometries on the Euclidean
space Z7R.
We define certain subsets of Z7R as follows:
Z7; = {x g77r; (hir x) =0}
/// = {x eZ7r; (ht, x) > 0}
77f = {xeZ7r; (hir x) < 0}
C = 771+n---n77;+.
C is called the fundamental chamber.
Let Wtj be the subgroup of W generated by st,Sj, where i^j. stSj has finite
order mtj given in terms of the Cartan matrix by 2cos(ir/?My) = Thus
Wtj is a finite dihedral group.
Lemma 7.26 Let w e Wtj with i ф j. Then either w (Hf П ) с //, or
w (Hf П) c Hr and I (stw) = I (w) — 1.
Proof. Let U be the 2-dimensional subspace of Z7R spanned by hh hj and
U be the orthogonal subspace. Then Z7R = U ® U and the elements of W4
act trivially on U . It is therefore sufficient to prove the result in U. Let
Г = U HHf nHj~. We obtain a configuration of chambers in U as shown in
Figure 7.1.
Figure 7.1 Configuration of chambers in U
7.5 The existence theorem
113
The chambers Г, ^(Г), у^;(Г),... , SjSt... (Г) lie in II. and the cham-
bers
^(Г), ^(Г), ^^(Г),... , s^j... (Г)
lie in II. The elements sh stSj,, stSj... of all satisfy / (s;w) = /(w) —
1. Thus for each w e Wtj we have either w (Hf П II.) c II, or w (II, П Hj) c
II. and / (stw) = l(w) — 1.	□
Proposition 7.27 (a) Let w e W. Then either w(C) G II or w(C) G Hr and
I (stw~) = l(w) — 1.
(b) Let w e IT and i^j. Then there exists w'such that w(C) G
w' (H+ QH+) and l(w) = l(w') +1 (iT 1 w).
Note. Part (a) is the result we shall need. To prove it we must also prove part
(b) at the same time.
Proof. We prove both statements together by induction on /(w). If /(w) = 0
then w=l and (a), (b) are true. So suppose /(w)>0. Then w = SjW' with
/ (w') = /(w) — 1 for some j. We prove (a).
First suppose j = i. By induction w'(C) G Hf or w'(C) G Hr and /	=
l(w') — 1. But I (siw’') = I (w') + 1, so w'(C)g7/;+. Then w(C)G//f and
/ (,J;w) = /(w) — 1.
Now suppose j^i. By induction there exists w" e with w'(C)g
w" (Я+ПЯ+) and /(w') = /(w") + / (w"-1»'). Thus w(C) Gs,w" (Н+ПН+).
By Lemma 7.26 we have either SjW" (Hf M/t) GHf or Sjiv" (Hf П G
II. and /	= / (sjiv"^ — 1. In the first case w(C) G If. In the second
case w(C) G Hr and / (s;w) = /	= / (^siSjW"w"^1 w') < /	+
/ (w"-1 w') = I faw") - 1 +1 (w'^w') < I (w") + l(w"-1w') = I (w') = l(w)-l.
Thus / (stw) = l(w) — 1 and (a) is proved.
We now prove (b). If w(C) GHf CHf then (b) holds with w'= 1. Thus
we may assume without loss of generality that w(C) <fHf. So by (a), which
is now proved under the assumption of the inductive hypothesis, w(C) G Hr
and / (s;w) = /(w) — 1. By induction there exists w' e W4 such that yirfC) G
w' (Hf and / (s;w) = / (w') + / (w'^Sjw). Thus w(C) Gstw' (H
and
/(w) = 1 +1 (stw) = 1 +1 (w') + l (w'^Sjw) > I (SjW') + /	1	> l(w).
Thus we have equality throughout and l(w) = /	+ / ^(.spy’') 1 . Hence
SjW' e Wtj is the required element and (b) is proved.	□
114
The existence and uniqueness theorems
Proposition 7.28 If weW satisfies CCiw(C) is non-empty, then w=l.
Proof Suppose w^l. Then w = SjW' with I (w') = l(w) — 1. By Proposi-
tion 7.27 (a) w'(C) C Hf. Thus w(C) c Hr. Hence
С П w(C) C Hf ПНг = 0.
So if СП w(C) is non-empty, w= 1.	□
Now the Euclidean space /Л has an orthonormal basis, and the isometries
of are represented by orthogonal matrices with respect to this basis. Thus
W С О/ where O; is the group of / x / orthogonal matrices. Ol0Ml, the set
of all / x / matrices over IR.
For any matrix M = (m^ eM1 we define ||M|| = mq and f°r апУ
column vector v= (Ap ... , A;)‘ e IR; we define || v|| = y/'ZL A?.
Lemma 7.29 (a) If M e Cf, v e IR; then ||Mu|| = ||u||.
(b) If M eO^N Elvfthen ||AW|| = ||2V||.
(c) If M ueIR; then ||Mu|| < \\M || || v ||.
Proof. Straightforward.
Proposition 7.30 (a) W is finite.
(b) Ф is finite.
Proof. Since Ф = H'(l I) where П = {cej,... , ce;} it is clear that (a) implies (b).
Thus we show that W is finite. We consider the И'-action on the Euclidean
space HK. We give elements of IP coordinates relative to our orthonormal
basis. Let v = (Ax,... , A;)1 e C. By definition of C there exists r> 0 such that
Br(v) с C where
Br(v) = {л e IR;; ||x — v|| < r}.
Let weW with w^l. Then w(C)nC = 0 by Proposition 7.28. Thus
w(v) C so
|| w(v) — u|| > r.
Hence
||w-l|||M| > ||w(u)-u|| >r
7.5 The existence theorem
115
so IIw— 111 > Put £=1^. Then ||w — 1|| >e for all w^l in W. Now let
w,w'eW have w w'. Then
||w — w'|| = ||w' — 1) || = ||w'-1w— 1|| >£
since w' e Op Thus distinct elements of W are separated by a distance of at
least e. Since Op and hence W, is bounded it follows that W is finite. □
We now return to our Lie algebra L(A). We know that dimL(A)0 = I and
dim L(A)a = 1 for all a e Ф.
If we can prove that L(A)M = О for all peH* with p ^Фи {0} we shall
be able to deduce that I.(A) is finite dimensional.
Proposition 7.31 Suppose pel! satisfies ррЧ and p g Ф. Then
L{A\ = O.
Proof We assume that and L(A)M O. Since dimL(A)/z ^dimL^)^
we see by Proposition 7.12 (iii) that p e Q+ or p e Q . In particular p lies in
the vector space Hg of real linear combinations of ar,... , a;.
Suppose first that p is a multiple of some root a e Ф. Then p = na or — na
with n > 0 and a e Ф+. Now a = w (a;) for some w e W and some а;еП and
we have dim L(A)na = dimL(A) by Proposition 7.21. Hence L(A)na. O.
Now N is generated by elements ex,... , e; and no non-zero Lie product of
these can have weight nai unless n = 1. Thus p = a or —a, that is p e Ф.
Secondly suppose p is not a multiple of a root. Let
p(h) = 0}
Ha = {heHs ; a(h)=0}.
Then is distinct from all the Ha, аеФ. Since Ф is finite we can find
h e such that h<fHa for all a e Ф. It follows that w(/i) gHa for all a e Ф,
since W permutes the Ha.
We claim there exists w e W such that a;(w(/i)) > 0 for all i = 1,... , To
see this we define the height of an element of HK by
ht =
We choose an element w e W such that ht w(/i) is maximum. This is certainly
possible as W is finite. Then
s;(w(/i)) = w(/i) — afwtfifjhp
116
The existence and uniqueness theorems
Since htst(w(h)) <ht w(h) we have a;(w(/i)) >0. However, a;(w(/i)) = 0
would imply w(h)eHa. which is impossible. Thus a;(w(/i))>0 for each i
and so w(/i) e C.
Now we have (w(/z))(w(/i)) = /z(/i) = 0. We write w(/z) = J).=1 m^. Then
we have
i
y'miai(w(hy) = ('i.
1=1
Since a;(w(/i)) > 0 for each i we must have some mt > 0 and some mj < 0 in
the sum. Thus w(/z) g Q+ and w(/z) 0Q . Hence L(A)W( j = O. By Proposi-
tion 7.21 we deduce £(А)М = О, and so L(A)M = О. This gives the required
contradiction.	□
Corollary 7.32 (i) L(A) = H®'Eae<s>L(.A)a-
(ii) dimL(A) = 1+ |Ф|.
Proof. This is evident since I.(A) is the direct sum of its weight spaces. The
0-weight space is H and this has dimension /. The only non-zero weights are
the elements of Ф and the corresponding weight spaces are 1-dimensional.
Thus we have the required formula for the dimension of IfA).	□
We now know that I.(A) is a finite dimensional Lie algebra - indeed it has
the dimension required for a simple Lie algebra with Cartan matrix A. We
shall now be readily able to show that I.(A) has the required properties.
Proposition 7.33 The Lie algebra I.(A) is semisimple.
Proof. Let R be the soluble radical of I.(A) and consider the series
R = A(o) D A(1) D • • • D R'"}' D //'" = О
where R(-l+A =	\\/e write I = R" We suppose if possible that
R±(). Then I is a non-zero abelian ideal of L. Moreover I is invariant under
all automorphisms of L.
Since [HI] c 7 we may regard I as an //-module. We decompose it into its
weight spaces. This weight space decomposition is
аеФ
7.5 The existence theorem
117
For let x e I have x = x0 + Ц, ф xa where ,i , t W and x,. e La. We show x0 e I
and each xa el. There exists heH such that a(/i)^0 and /3(h) a(h) for
all /3 e Ф with (3j^a. Then
ad h П (ad h —/3(/i)l)x=a(/i) ]~[ (a(/i) —/3(/i))
/ЗФ	/ЗФ
and this is an element of I. Hence xa e I. Since this is true for each a e Ф we
also have x0 e I. Hence
/ = (//П/)ф£(/.„П/).
аеФ
We claim that LaC\I =0 for each a e Ф. Otherwise we would have La с I.
Now a = w (a;) for some w e W and some i = 1,... , By Proposition 7.21
we can find an automorphism of L(A) which transforms La to La,. Since I is
invariant under all automorphism we would obtain La. с I. Hence e; e I. But
then [e;/;] = htel and we would have = 2e; e I, contradicting the fact
that I is abelian. Hence La ПI = О for all a e Ф and so I с H. Let x e I. Then
[xe;] = a;(x)e; e I hence a;(x) = 0. Since ax,... , al are linearly independent
we deduce that x = 0. Hence I = O, a contradiction.	□
Proposition 7.34 H is a Cartan subalgebra of L(A).
Proof. Since H is abelian it is sufficient to show that H = N(H). Let x e N(H).
Then x = h' + Ц,ф Xaea for h' eH, eaeLa. Then for all h e H we have
[M = E A«“(/l)e«eR
аеФ
However, we can find h e H such that a(/i) f 0 for all a e Ф. We deduce that
= 0 for all a e Ф, hence xeH.	□
Proposition 7.35 I.(A) is a simple Lie algebra with Cartan matrix A.
Proof. L(A) = H® Л„ фА(Л )„ is the Cartan decomposition of L(A) with
respect to H. Thus Ф is the root system of L(A). The Cartan matrix A’ = (Ay)
of L(A) is determined by the condition
si (aj) = aj~A'ijai-
118
The existence and uniqueness theorems
However, we have
= hj — Ajihi by Proposition 7.18.
Using the facts that hk = ctj (sfif) and cij (hk~) = Akj we deduce that
si (aj) = aj~ Anai-
Hence A' = A and the Cartan matrix of L(A) is A.
Since the Dynkin diagram of A is assumed connected, L(A) must be a
simple Lie algebra, by Proposition 6.13.	□
Thus we have constructed, for each Cartan matrix on the standard list 6.12
a finite dimensional simple Lie algebra L(A) with Cartan matrix A.
Theorem 7.36 The finite dimensional non-trivial simple Lie algebras
over C are
Ai	/> 1
Bi	l>2
Cl	l>3
Di	Z>4
E^yE-/, Es
F.
G2
These Lie algebras are pairwise non-isomorphic.
Proof. For each Cartan matrix on the standard list 6.12 there is a corresponding
finite dimensional simple Lie algebra, which by Theorem 7.5 is determined
up to isomorphism. Simple Lie algebras with different Cartan matrices cannot
be isomorphic since, by Proposition 6.4, the Cartan matrix on the standard
list is uniquely determined by the Lie algebra.	□
The description in Proposition 7.35 of the simple Lie algebras by generators
and relations enables us to choose the root vectors ea in a way which makes
the structure constants Na & very simple.
Theorem 7.37 It is possible to choose the root vectors ea in the simple Lie
algebra L(A) in such a way that Na = ±(/>4-1) where —pa + /3,... , qa + /3
are the a-chain of roots through /3.
7.5 The existence theorem
119
Proof. L(A) is the Lie algebra generated by elements , eh
hr,... , hh fx, ...fi subject to relations
[M>]=o
[hiej]=Aijej
М]=-А^
[eifi] =
[ejy]= ° if i^j
[e;[e;...[e;e,]]]=°	if i^j
[Л[Л---[ЛЛ]]]=о	if
with 1 — Ajj occurrences of e;,/; respectively.
We now define
e'i = -fi, h'i = -hi, f-=-ei.
It is straightforward to check that e., /i., f. satisfy the above relations. Thus
there is a homomorphism <y : L(A)^L(A) satisfying <y(e;) = — ft,
<y (/i;) = — ht, w (/;) = — e;. Since <y2 = 1, <y is an automorphism of I.(A).
Let a be a positive root of I.(A). Then
[hiea] = a(hi)ea
and so
[-h^d^f)] = a(hi)d(ea)
that is
[hi,0(.ea)] = -a(hi)e(ea)
whence в (ea) e L_a. Let в (ea) = Ae_ff. Then Л 0 and we may choose p. e C
with fi2 = — A 1. Then
(.P^a) P
and
2/i'
e-“] = [e“e-“] = l^h~hf)-
We now change our choice of the root vectors ea -J.„. For each positive root
a we take jxea as our root vector and for the corresponding negative root
-a we take /i 'e (( as the root vector. Changing the notation to call these
120
The existence and uniqueness theorems
new root vectors ea, e_a we retain the multiplication formulae of Section 7.1,
except that the structure constants Nap may now be altered. We also now
have ы (ea~) = —e_a. Now
\_eaep\=^a,pea+p
and so
= Naj (~e_a_p) .
This implies N_a_p = —Nap. By Proposition 7.1 (iii) we have NapN_a_p =
— (/?+l)2, where —pa+/3,... ,qa + /3 is the «-chain of roots through /3.
Hence A(;.,. = (/?+I)2 and/V,b.,j = ±(/?+I).	□
This result has important implications in the theory of Chevalley groups
over arbitrary fields. (See, for example, R. W. Carter, Simple Groups of Lie
Type, Wiley Classics Library, 1989.)
The signs in the formula Na p = ±(/»+1) can be chosen in various ways.
By Lemma 7.3 and Proposition 7.4 the signs can be chosen arbitrarily for
extraspecial pairs of roots (a, /3) and are then determined for all other pairs
(a,/3).
8
The simple Lie algebras
Having obtained a classification of the finite dimensional simple Lie algebras
over C we shall in the present chapter investigate them individually in order
to obtain their dimensions and a description of their root systems. In the case
of Lie algebras of type A;, B;, C; or /) we shall also give a description in
terms of Lie algebras of matrices.
The strategy for obtaining this information will be as follows. Given a
Cartan matrix A on the standard list 6.12 we shall describe a symmetric scalar
product {,} on an /-dimensional vector space V over R with basis a1,... , al
such that
We compare this scalar product with the Killing form l[ai, obtained when
oq,... , al are interpreted as a fundamental system of roots in the simple Lie
algebra with Cartan matrix A. We claim there exists a constant к such that
{ai,aj} = K{ai,aj}
In fact we can define к by the equation
Then we have
for all i, j.
«ib
for all i, j
{a;, O';}
and since both scalar products are symmetric we deduce that
121
122
The simple Lie algebras
Putting j = 1 we deduce
{at, = к {a;, a;} for all i
and it follows that
Оу) = к {a;, « J for all i, j.
Thus the symmetric scalar product {,} is the same as the Killing form up
to multiplication by the constant к. In practise it will not be necessary to
determine this constant.
We then consider the fundamental reflections st : V V defined by
si (aj) = aj - AHai-
The maps Sp ... , s, generate the Weyl group W of transformations of V. The
vectors in V of form w (a;) for all w e W and all i will then give the full root
system Ф. We shall then be able to obtain the dimension of the simple Lie
algebra L by the formula
dimL = 1+ |Ф|.
8.1 Lie algebras of type At
It will be convenient to describe the vector space V as a subspace of a larger
vector space V of dimension I + 1.
Let V be a vector space over R with basis /3r,... , /3l+1 and let the sym-
metric scalar product {,} on V be defined by
i,j=i,...,i+i.
We define ar,... , al by
ai=Pi~Pi> =	/З3, , al = fil~ fil+l-
Let V be the subspace of V spanned by ar,... , a;. Then we have dim V = I.
Our scalar product satisfies
{a;,a;} = 2, {a;, a;+1} = —1, {a;,a,y}=0 if|z—j|>1.
Hence
where A = (Ay) is the Cartan matrix of type A;.
8.1 Lie algebras of type At
123
We now consider the action of the fundamental reflections on V. We define
linear maps st : V -+ V by
^Ш=/з;+1
^(/3;+1)=/3;
^(/3;)=/37 j^i,i+l.
Then we have
Si(aj) = aj-Aijai
and so st restricted to V is the zth fundamental reflection.
We consider the group of transformations of V generated by Sp... , s^
Since st acts on V by permuting /3;, /3i+1 and fixing the remaining f3j the
group generated by the st is the group of all permutations of ... , /3;+1.
This group leaves the subspace V invariant and induces on V the Weyl group
W. Thus we have a surjective homomorphism
whose kernel is trivial. Hence the Weyl group of type A; is isomorphic to the
symmetric group Sl+1.
The full root system Ф of type A; is the set of vectors of form w (a;) for
all w e W and all i. This is the set
Ф={/3;-/37-i^j	,1+1}.
Thus we have |Ф| = /(/+ 1) and dimL = 1 + |Ф| = /(/ + 2).
We shall now show that §I;+1(C) is a simple Lie algebra of type A;.
We discussed this Lie algebra in Section 4.4. In particular we know from
Proposition 4.26 that the subalgebra H of diagonal matrices in L = ёГш(С)
is a Cartan subalgebra. Moreover by Proposition 4.27
L = H®^CEij
is the Cartan decomposition of L with respect to H. By Theorem 4.25 L is a
simple Lie algebra. By Proposition 4.28 the roots of L are the functions
124
The simple Lie algebras
and a system of fundamental roots is given by
A
—	^i+1-
^/+1/
We can now determine the Cartan matrix A= (Ay) of L. We recall from
Proposition 4.22 that the at-chain of roots through ctj when i^j has the form
ctj, at + ctj, ... , qai + a;
where q = —Atj. Since we know the roots we can determine the numbers q.
We have q = 1 if i = j — 1 or j + 1 and q = 0 otherwise. Thus the Cartan matrix
A is the same as the Cartan matrix of type A; in the standard list 6.12. Thus
we have proved:
Theorem 8.1 (i) The simple Lie algebra of type A; has dimension /(/ + 2).
(ii) The Lie algebra §(;+1(C) of all (/+ 1) x (/+ 1) matrices of trace 0 is
simple of type A;.
8.2 Lie algebras of type /Л
We recall that the Dynkin diagram of type Dt has form
Let V be a real vector space of dimension / and basis /3X,... , /3;. Let the sym-
metric scalar product {,} be defined by {/3;, /ЗД = 8tj. We define ar,... , al by
ai=Pi~ /32, a2 = ^2~ /З3, ..., o';-! =/3;-! —/3;, al = /3l_1 +/3;.
Then we have
{a;, aj = 2
“/+i} = -l
{“/-2, “/} = -!
for all i
for 1 < i < I — 2
for i, j e {1,... , I — 1} with I i — j I > 1
{a;, a;} = 0 forz^Z — 2,1.
8.2 Lie algebras of type If
125
It follows that
l“i> “/I
2-H—4=Д.;	for all/, j
and hence that the scalar product {,} is a non-zero multiple of the Killing form.
We now consider the fundamental reflections st on V. For 1 < i < I — 1 we
have
Ш=/з;+1
(/3i+1) = /3;
^(/з7)=/з7 for ;#/,/+1.
For i = l we have
si (Pi)= ~Pi-i
^i(Pj)=Pj forj^/ —1,/.
Thus the Weyl group W generated by Sp ... , s, has form
w (/3;) =	w e W
for some permutation <j of 1,... , Let w (/3;) = Then an even number
of the signs £; are equal to —1. Conversely for any permutation <j of 1,... , /
and any set of signs £; with П Ei = 1 there is an element w e W acting as
above. It follows that the order of the Weyl group of type /) is given by
| W| = 2;~1/!.
We now consider the root system Ф. The elements of Ф have form w (a;) for
all w e W and all i. Since w acts on the /3; by a permutation combined with
certain sign changes we obtain
ф={±/3;±/3; ; / #je{l,... ,/}}.
All combinations of signs are possible. Hence |Ф| = 21(1 — 1) and so
dimL =/+|Ф| =/(2/— 1).
We now wish to describe L as a Lie algebra of matrices. We begin with a
lemma which will be useful both for the type being considered and for certain
other types also.
126
The simple Lie algebras
Lemma 8.2 Let M be an и x и matrix over C. Then the set of all и x и
matrices X over C satisfying
XlM + MX=O
forms a Lie algebra under Lie multiplication of matrices.
Proof The set of such matrices X is clearly closed under addition and scalar
multiplication. Let Xx, X2 be matrices satisfying the given condition. Thus
we have
X{M = -MXr, X\M = -MX2.
It follows that
[XrX2fM = (XjX2 - X2Xj)‘ M = Х\Х\М - Х\Х\М
= -х'2мхг+X\MX2=MX2Xr - MXfX2
= -M[XiX2\.
Thus the set of such matrices X forms a Lie algebra.	□
We now consider the special case when M is the 2/ x 2/ matrix
Then a 2/ x 2/ matrix X =
^12 j sat2sf2es X'M + MX=O if and only
•^21 ^22/
if X22 = — and X12, X21 are skew-symmetric. Let L be the Lie algebra of
all such matrices X and H be the set of diagonal matrices in L. The elements
of H have form
/Ai	\
\	-4
Let us number the rows and columns 1,—1,... , —Then we have
L = H®^Ce,
8.2 Lie algebras of type If
127
where
— E i /+£/; forO<z</,
p —	’’ J J1	J
“ I/-.
-Е-.} + Е-}.
that is for each pair i, j with 0 < i < j we have four vectors ea as above.
Moreover each of the 1-dimensional spaces Cea is a H-module, and we have:
(Еч-Е-г-д
[h, E -E.\ = (Ay - A;) (-E_h_]+E]i)
[1г,Е^-Е^] = ^-^ (E^-E^
[h, -E_hJ +Е_^ = (—A; - Ay) (-E_t} + E_],;).
We write [hea] = a(h)ea for all such ea.
Now the argument of Proposition 7.34 shows that H is a Cartan subalgebra
of L. The decomposition
/. = //®£Ce„
a
is then the Cartan decomposition of L with respect to H.
We next verify that L is semisimple. Suppose not. Then L has a non-zero
abelian ideal I. Since [HZ] с I we may regard I as a //-module and consider
the decomposition of I into weight spaces with respect to H. This gives
/=(//П/)ф£(СфП/)
just as in the proof of Proposition 7.33. Suppose if possible that Cea П1^О
for some a. Then we have eael. We then define ha by ha = [eae_a] and
observe that [haea] = 2ea. Then ea,hael and we have a contradiction to the
fact that I is abelian. Hence Се,,П/ = О for all a and so IcH. Let xeZ.
Then [xen.\ = a(x)ea el so a(x) = 0. This holds for all a and so x = 0. Thus
1 = 0, which gives a contradiction. Hence L is semisimple.
128
The simple Lie algebras
We now know that the functions a : H C given above are the roots of L
with respect to H. A system of fundamental roots is given by
a1(/i) = A1-A2
Ocfh) = ^2 — ^3
al-lW —	\
“/W = V1 + A;
since all the other roots are integral combinations of these with coefficients
all non-negative or all non-positive.
We now determine the Cartan matrix of L. Let the a;-chain of roots through
aj for i j be
a j,	at + aj,
qUf + aj.
Then Atj = — q by Proposition 4.22. Since we know the roots we can find
the number q and hence Atj for each i j. This gives us the Cartan matrix
A=(Aij) of type Di on the standard list 6.12.
Finally we note that since this Cartan matrix is indecomposable the Lie
algebra L must be simple by Corollary 6.15. Thus we have proved the
following result:
Theorem 8.3 (i) The simple Lie algebra of type if has dimension 1(21 — 1).
(ii) The Lie algebra of all 21 x 21 matrices X satisfying X‘M + MX = 0
where
is simple of type If when I > 4.
8.3 Lie algebras of type Bt
We recall that the Dynkin diagram of type If has form
1	2
8.3 Lie algebras of type Bt
129
Let V be a real vector space of dimension / with basis Д,... , /3;. Let the scalar
product {,} on V be defined by {/3;, /3;) = 8tj. We define ar,... , al e V by
ai=Pi~Pi> ai = Pi~/З3, , al-l=Pl-l~Pb ai = Pi-
Then we have
{a;,a';} = 2 forl<z</—1
{“/,“/} = !
{a;, a;+1} = — 1 forl<z</—1
{a;,aJ=0 if |z-j| > 1.
It follows that
“J
2-H—~г = Ац	for all z, j
{a;, aj
where А=(Ау) is the Cartan matrix of type B; on the standard list 6.12.
Thus the scalar product {,} is a non-zero multiple of the Killing form.
We now consider the fundamental reflections st on V. We have, for 1 <
z</-l,
^Ш = /3;+1
^(/3i+1) = /3;
jYM+l-
For i = l we have
si (Pi) = Pi i#l-
Thus the Weyl group W generated by Sp ... , consists of elements w of
the form
w (Pi) = ±Pa(i)
for some permutation cr of 1,... ,/. Let w(J3i) = si/3a.[iy Then, given any
permutation cr of 1,... , / and any set of signs £; e {1, — 1} there is an element
w e W such that w (/3;) = for all i. Thus the order of the Weyl group
W of type В/ is
I W| =2'/!.
130
The simple Lie algebras
We now consider the root system Ф. The elements of Ф have form w(a;)
for all w e W and all i. Since w acts on the /3; by means of a permutation
combined with sign changes we obtain
ф={±/з;±/з7
All combinations of signs are possible. Thus we have |Ф| = 2/2 and so
dimL = /+^| = /(2/+l).
We shall now describe L as a Lie algebra of matrices. We use Lemma 8.2
and this time we take the (2/ + 1) x (2/ + 1) matrix M given by
M =
/2
0 ... 0\
О A
\o A o)
Let L be the Lie algebra of all (2/ + 1) x (2/ + 1) matrices X satisfying the
condition
XlM + MX=0
We consider X as a block matrix
1 / /
Then X satisfies XlM + MX = О if and only if X22 = — Х*п, X12 and X21 are
skew-symmetric, Xlo = —2Xq2, X20 = —ZX^ and Xoo = 0.
Let 77 be the set of diagonal matrices in L. The elements of H have form
We number the rows and columns 0, 1,—1,... , —Then we have
L = H®^2Cea
8.3 Lie algebras of type Bt
131
where
ЕУ-Е-Г i
-E-i, -j + Ejt
E: j—Ej i for 0 < i < j
a I /	,
2Ei0 — Eo _t for 0 < i
_~2E_i0 + E0i
Each of the 1-dimensional spaces Ce is an //-module and we have
[^Ey-E^ _;] = (A;-A;) (Е,-Е_ь _;)
[h, -E_,	 /:] (ЛЛ) (-E_h _] + E]i)
[h,Eh ^-E^ _;] = (A; + Ay) (E;, _7-E7, _;)
[h, -E_hJ+E_^ ;] = (-A;-Ay) (-E_;, 7 + E_7, ;)
[й,2Е;о-Ео,_;] = А; (2E;0-E0>_;)
[^> ~^E_q 0 + /,,,] = —A; (—2EL; 0 + ^oi) •
We write [hea] = a(h)ea for all such a.
We now show that H is a Cartan subalgebra of L; L = H®'ffa'Cea is the
Cartan decomposition of L with respect to H, and L is semisimple. These
facts can be proved in exactly the same way as that used in Section 8.2 for
type Di.
We now know that the functions or.H C given above are the roots of L
with respect to H. A system of fundamental roots is given by
“iW = Ai-A2
uslA)= A2 — A3
U'; (A)-A A;
“/W = A;
since all the other roots are integral combinations of these with coefficients
all non-negative or all non-positive.
We can now determine the Cartan integers A^. Let aj, at + aj,... ,	+ aj
be the tt,-chain of roots through aj for i^j. By Proposition 4.22 we have
Atj = —q and so the Cartan matrix A= (Ay) can be determined. This turns
132
The simple Lie algebras
out to be the Cartan matrix of type on the standard list 6.12. Finally we
observe that L must be a simple Lie algebra by Corollary 6.15, since its
Cartan matrix is indecomposable. Thus we have
Theorem 8.4 (i) The simple Lie algebra of type If has dimension 1(21 + 1).
(ii) The Lie algebra of all (2Z+l)x (21 + 1) matrices X satisfying X! M +
MX = О where
/2
M =
0\
A
\o I, o)
is simple of type If when I > 2.
8.4 Lie algebras of type Cz
We recall that the Dynkin diagram of type C; has form
1
2
l-l
I
О
Let V be a real vector space of dimension / with basis /3,,... , /3;. Let the scalar
product {,} on Vbe defined by {/3;, /ЗД = 8tj. We define ar,... , a; e Vby
al — Pl Pl’	a2~ Pl Pl’ •••> al-l~Pl-l Pl’ al~2Pl-
Then we have	{a;,a;} = 2 forl<z</ —1 {a;, a;}=4 {a;, a;+1} = —1 forl<z</-2 =-2 {a;,a,z}=0 for |z — j > 1.
It follows that	{°+ 2-H—j-L=a..	for all z, j {a;, aj
where A=(Ay) is the Cartan matrix of type C; on the standard list 6.12.
Thus the scalar product {,} is a non-zero multiple of the Killing form.
8.4 Lie algebras of type Ct
133
The fundamental reflections ... , sl act on , /3; in exactly the same
manner as in type If, considered in Section 8.3. Thus we have
| W| =2'/!
as in Section 8.3 and each w e W acts on the /3; by means of a permutation
combined with sign changes. Both the permutation and the sign changes can
be chosen arbitrarily. Thus we obtain the root system Ф as the set of all
vectors of form w (a;) for all w e W and all i. Thus
ф={±/з;±/з7 г-#У;±2/з;}.
All combinations of signs are possible. Thus we have |Ф| = 2/2 and
dimL =/+|Ф| =/(2/+1).
We next describe L as a Lie algebra of matrices. Again we use Lemma 8.2.
This time we take the 2/ x 2/ matrix M given by
Let L be the Lie algebra of all 2/ x 2/ matrices satisfying the condition
X'M + MX = O.
Let X = ( 11 ^12 |. Then X lies in L if and only if X?? = —AT and X1?, AT
\ Y	Y /	J 2.2.	11	1Z’ Z1
\л21 л22/
are symmetric.
Let H be the set of diagonal matrices in L. The elements of H have form
/Ai	\
\	“V
We number the rows and columns 1,... , I, —1,... , —I. Then we have
L = H®^2Cea
134
The simple Lie algebras
where
Each of the 1-dimensional spaces Cea is a //-module. We have
1^Еу-Е-г =	(Е,-Е_ь _;)
[h, -E_,	 /:] (ЛЛ) (-E_h _] + E]i)
Мь _7 + E7, _;] = (A; + Ay) (E;,	_;)
[h,E_,j + E^ ;] = (-A;-A7)(E_;,7 + E_7, ;)
[h,Et _i]=2XiEi _t
[h, E_t, ;] = -2A;E_; ;.
We write [hea] = a(h)ea for all such a.
We observe that H is a Cartan subalgebra of L, that L = H®'2^alCea is
the Cartan decomposition of L with respect to H, and that the Lie algebra L
is semisimple, using the same arguments as given in Section 8.2 for type Dt.
The functions a : H C given above are the roots of L with respect to
H. A system of fundamental roots is given by
ai (/i) = Ax — A2
ai(E) = A2 — A3
— A;_i A;
a;(/i) = 2A;
since all the other roots are integral combinations of these with coefficients
all non-negative or all non-positive.
We can now determine the Cartan integers A^. Let aj, at + aj,... , qat + aj
be the tt,-chain of roots through aj, for i^j. Then А^ = — q. The Cartan
matrix A = (Ay) determined in this way turns out to be the Cartan matrix of
8.5 Lie algebras of type G2
135
type Ci on the standard list 6.12. Finally L is a simple Lie algebra, since its
Cartan matrix is indecomposable. Thus we have
Theorem 8.5 (i) The simple Lie algebra of type C[ has dimension 1(21 + 1).
(ii) The Lie algebra of all 21 x 21 matrices X satisfying X! M + MX = О where
M =
OJ
is simple of type Ct when I > 3.
□
The Lie algebras of type A;, If, Ct or Dl are called the simple Lie alge-
bras of classical type. The remaining simple Lie algebras E6, E2, Es, F4,
G2 are called the exceptional simple Lie algebras. We now determine the
dimensions and root systems of the exceptional Lie algebras.
8.5 Lie algebras of type G2
The Dynkin diagram of type G2 is
o—>—c>
1	2
and the corresponding Cartan matrix is
/ 2 -1\
\-3 2/
Let aj, a2 be the fundamental roots in a root system of type G2. Then we
have
•?i(“i) = -“i	^(“i) = “i + 3a2
•^i (“2) = “i + “2	+ (“2) = - “2
and W= {sr, sf). Thus each root in Ф is obtained from ar or a2 by applying
Sp s2 alternately. Now we have
«1^ —— a1 — 3a2^ — 2a1 — 3a2
S1	s2	S1
a1^a1 + 3a2^2a1 + 3a2
s2	S1
a2^a1 + a2^a1+2a2
S1	s2
«2^ ~ a2~^ - “1 - a2~^ - “1 - 2“2
s2	S1	s2
136
The simple Lie algebras
and
s2 (-2“i - 3a2) = -2“i - 3«2
s2 (2aq + 3ce2) = 2cej + 3ce2
sq (tt| + 2ce2) = oq + 1a2
•?i (-“i - 2ai) = -“i - 2a2-
Thus all the vectors in the above sequences are roots, and we do not obtain
new vectors by continuing the sequences further. Hence
Ф= {oq, a2, аг + а2, a1+2a2, oq + 3cE2, 2a,1+3a2, — oq,
-a2, -aq-a2, -ar-2a2, -aq-3a2, -2ar-3a2}.
Thus we have |Ф| = 12 and dimL= 14. Hence we have proved
Theorem 8.6 The simple Lie algebra of type G2 has dimension 14.
Figures 8.1, 8.2 and 8.3 compare the simple root systems of types A2, B2
and G2.
Figure 8.1 Simple root system of type A2
8.5 Lie algebras of type G2
137
Figure 8.2 Simple root system of type B2
Figure 8.3 Simple root system of type G2
138
The simple Lie algebras
8.6 Lie algebras of type F4
The Dynkin diagram of type F4 is
О----O > t)---О
12	3	4
and the corresponding Cartan matrix is
/2—10 0 \
-12-10
0-22-1
\ 0	0-12/
Let V be a real vector space with dim V = 4 and /32, P3, P3 be a basis
of V. Let the scalar product {,} on V be defined by {/3;,	= <5^. We define
«j, a2, a3, a4 e V by
ai= Pi~ Pi	а1 = Р1~Рз	аз=Рз a4= 2 (~Pi ~ Pi ~ Рз~^~ Pi) •
Then we have
{a1,a1} = {a2,a2} = 2
{a3,a3} = {a4, a4} = l
{“п“2} = {“2,“з} = -1
{a3,a4} = -1
{“o“J=0 if |z-j|>l-
It follows that
{a;, “J
2-!——}-Т=Аи	for all//.
Thus the scalar product {,} is a non-zero multiple of the Killing form. We
consider the action of the corresponding fundamental reflections jj, s2, s3, s4.
We have
S1(P1)=P1’ S1(P1)=P1’ 51(Рз)=Рз’ ^1(.Р1)=Р1
S1(P1) = Pl’ S1(.P1)= Рз’ 51(Рз)=Р1’ s1(.Pa)=P4
S3 (Pl) = P1’ $з(Р1)=Р1’ 5з(Рз) = ~Рз’ $з(Р1)=Р4‘
We consider the subgroup (sj, s2, s3) of the Weyl group W generated by Sj,
s2, s3. Elements in this subgroup all fix /34 but act on /Зь /32, /З3 by means of a
permutation combined with sign changes. Thus w (/3;) = for i = 1, 2, 3.
8.6 Lie algebra of type F4
139
Moreover each permutation a and each choice of signs £, arise in this way.
Applying the elements of this subgroup of W to ar, a2, a3, a4 we see that
the vectors
±/3;	l</<3
±/3;±/37 ij^j l<i,j<3
| (±/Зх ±/32 ±/З3 ±/34)
all lie in Ф. We next consider the action of s4. We have
^4(/31) = |(/31-/32-/Зз + /34)
.4(/з2) = 1(-/з1+/з2-/з3+/з4)
.4(/Зз) = Н-/31-/32 + /Зз + /34)
s4(/34) = i (ft+&+&+&)•
Since $1=1 we have s4 (/3X + ft + ft + ft)) = ft. Hence ft e Ф. We also
have s4 ([f + ft) = — /З3 + ft. Hence —/33 + /34 еФ. Thus, applying further
elements of the subgroup s2, s3) we see that the vectors
±/3;	l</<4
±ft±ft ij^j l<z,j<4
i(±ft±ft±ft±ft)
all lie in Ф, where the choice of signs is arbitrary.
We show this set of vectors is the whole of Ф. To do so it is sufficient to
show that the set is invariant under Sp s2, s3, s4. The set is clearly invariant
under jp s2, s3 because of the simple action of these reflections on /З3, ft, /З3,
/34 described above. Thus it is sufficient to show the set is invariant under s4.
Now the action of s4 given above shows that
s4 (±/?;) = | (si+ s2(32 + s3(33 + s4(34)
where £;е{1,—1} and Пя,= 1- Thus s4 transforms vectors ±ft into the
given set, giving as images vectors | with П£; = 1- Since there are
eight such vectors they all appear as vectors s4 (±/3;). Since s4 = 1 we deduce
that all vectors | EiPi with П Ei = 1 are transformed by s4 into the given set.
140
The simple Lie algebras
The formulae for s4 (/3;) also show that, for all i^j, s4 (±/3; ±/3j has form
±/3k ± /3, for certain к 7^ I. Thus s4 transforms vectors ±/3; ±/3j, i^ j, into the
given set.
It remains to show that s4 transforms all vectors | with Пя,=
— 1 into the given set. We may clearly assume e4= 1. There are four such
vectors. One of them is a4 and we have s4 (a4) = — a4. The other three are
all orthogonal to a4 and so are transformed into themselves by s4.
Thus the given set of vectors is invariant under Sp s2, s3, s4 so is the whole
of Ф. Thus we have
Ф= {±/3; l</<4
±/3;±/37 ij^j l<i, j<4
In particular we have |Ф| =48, hence dimL = 52. Thus we have
Theorem 8.7 The simple Lie algebra of type F4 has dimension 52.
We observe that the roots of F4 are of two different lengths. There
are 24 short roots and 24 long roots. The short roots are ±/3; and
| (±^±^±^±^4). The long roots are ±/3;±/3^.
8.7 Lie algebras of types E6, E7, If
We now consider the simple Lie algebra of type E8. Its Dynkin diagram is
1
a
2	3	4	5	7	8
-o------О-------О----------------о-------о
6
Let V be a real vector space with dim V = 8 and with basis /3; i = 1,... , 8.
Let the scalar product {,} on V be defined by {/3;, /3j} = 8tj. We wish to find
a fundamental system of roots of type Es in V. We note that if the vertex 8 is
removed from the Dynkin diagram we obtain a Dynkin diagram of type i)7.
This indicates how the first seven vectors in the fundamental system should
be chosen. The last one is chosen to be linearly independent of the others and
8.7 Lie algebras of types E6, E7, Es
141
to satisfy the appropriate conditions relating to the scalar product. Thus we
define . , as e V by:
“i=/3;-/3;+i	1<*<6
а1 = 06 + /^7
i=l
Then we have
{a;,a,;}=2 forl<z<8
{a;, a;+1} = — 1 forl<i<5
{a5, a7} = —1
{a7, a8} = —1
{a;, ttj = 0 for all other pairs i, j.
It follows that
where A = (Ay) is the Cartan matrix of type Л8 on the standard list.
In order to obtain the remaining roots we consider the action of the funda-
mental reflections ... ,s8. We have
^Ш=/з;+1
s; (/3j) = /3j	forjVh'+l
when 1 < i < 6. Thus the subgroup of the Weyl group W generated by
,$1,... , s6 will give all permutations of /3X,... , /37 and will fix /38. The fun-
damental reflection s7 acts by:
si (&) = —/37
si (Z^7) = ~06
s7(J3i) = l3i /#6,7.
Thus the subgroup of W generated by Sp ... , s7 will act on /3X, ... , /37 by
permutations and sign changes, and will fix /38. Moreover the number of sign
changes will be even, and any permutation of /3X,... , /37 combined with any
even number of sign changes will arise in this way.
142
The simple Lie algebras
It is then clear that the vectors

are all in the root system Ф. We also have
^*8 (ft) = & - a8 = ft + | “s
{«8>“8}
for 1 < i < 8. Thus
.8 (/37 + /38) = |(-/31 -/32 -/33 -/34 -/35 -/36 + /37 + /38)еФ.
Since sj = 1 it follows that /37 + /38 e Ф. We then see that ±/3; ±/38 e Ф for
all i with 1 < i < 7. Thus the set of vectors
lies in Ф. We shall show this is the full root system Ф. In order to do so we
must verify that this set is invariant under s^,... , .ss. It is clearly invariant
under sx,... , s7 since these fix /38 and act by permutations together with an
even number of sign changes on /3X,... , /37. Thus it is sufficient to verify
that this set is invariant under .s8. Now we have
s8(ft-ft)=ft-ft for all/# j.
Thus the set of vectors of form /3; — f3j, i^j, is invariant under .ss. Also
*8 (ft + ft) = fii + fij + “s forallz#/
Thus .ss transforms vectors of form /3; + /3j, i^j, into vectors | (J2 £;/3;) with
two £; equal to 1 and six equal to —1. Moreover all vectors |(J2eift)
with this property arise in this way. Similarly such vectors with six £, equal
to 1 and two equal to —1 have the form .ss (—/3; — ft). Thus vectors of form
ft + /3j or —ft — /3j with i # j are transformed by \s into the given set, and
so are vectors | EiPi °f type (2, 6) or (6, 2). The vectors of this form of
type (0, 8) or (8, 0) are as and — as, which are transformed into one another
by s8. It remains to show that .ss transforms vectors |J2eift of type (4, 4)
into the given set. However, since such vectors have four positive signs and
four negative signs they are orthogonal to a8, hence .ss transforms each such
8.7 Lie algebras of types E6, E7, Es
143
vector into itself. Thus the given set of vectors is invariant under ,... , .s8
so is the full root system Ф.
There are 4- Г ) = 112 vectors of form ±/3;±/3; with i^j and 27 = 128
\2 /	J
vectors of form | with £; = ±1 and П£/ = 1- Thus the total number
of roots is
|Ф| = 112+ 128 = 240.
Finally we have dim L = 8 + | Ф| = 248. Thus we have proved
Theorem 8.	8 The simple Lie algebra of type Es has dimension 248.
We now turn to the simple Lie algebra of type E7. Its Dynkin diagram is
О------О----О------<j>----о-----о
Thus the vectors a2, a3, a4, a5, a6, a7, a8 considered above form a funda-
mental root system of type E7. In order to obtain the full root system we
must transform these vectors repeatedly by s2, • • • , + until no new vectors
are obtained. Now the vectors a2,... , a8 are all orthogonal to /3, —/38. Thus
all their transforms by s2,... , .ss will also be orthogonal to /3, —/38. These
transforms are contained in the set of roots of E8 obtained above.
Now the roots of E8 orthogonal to Д — /3S are:
±/3;±/3p	2<z, j<7, i^j
±(ft+/38)
Si = ±l, П£/ = 1’ £1=£8-
Thus the required root system of E- is contained in this set. We shall show it
is the whole of this set.
We first consider the action of the subgroup of the Weyl group of E-
generated by s2,s3,s^, s5, s6, s7. Elements of this subgroup fix Д and /38 and
act on /32>/З3,/34,/35,/36,/37 by permutations combined with sign changes
with an even number of negative signs. By applying elements of this subgroup
to a2,... , as we see that the vectors
±/3;±/3p	2<z,j<7, i^j
=	П£/=1’ Ei = £8
144
The simple Lie algebras
are all roots of E7. It remains to show that ± (/3X + /38) are also roots of E7.
However,
*8 (ft + ft) = ft + ft + a8 = | (ft - ft - ft - fa - ft - ft - ft + ft)
is a root of Е7, thus so is ft + ft and —ft — ft.
There are 4| | =60 roots of form ±ft±ft, 2<i, j<7, i^j and 26 =
у 2 у	J
64 roots of form | 22e;ft with £; = ±1, П Ei = 1 and = s8. Thus the number
of roots of Е-/ is given by
|Ф| = 60 + 2 + 64= 126.
Also we have
dimL = 7+ |Ф| = 133.
Thus we have shown:
Theorem 8.	9 The simple Lie algebra of type E7 has dimension 133.
Finally we consider the simple Lie algebra of type E6. Its Dynkin diagram is
Thus the vectors a3, a4, a5, a6, a7, as considered above form a fundamental
root system of type E6. In order to obtain the full root system of E6 we
must transform these vectors successively by the fundamental reflections
+ +4+5+б+7+8-
Now the vectors a3,... , a8 are all orthogonal to both ft — /38 and ft — ft.
Thus the full root system of E6 is orthogonal to ft — ft and /32 — /3%.
Now the roots of ft orthogonal to both ft — ft and ft — ft are:
±ft±ft 3<z,j<7, i^j
(8	\
=	П£/=1’ £1=£2 = £8-
i=l	/
Thus the required root system of E6 is contained in this set. We shall show it
is equal to this set of vectors.
Consider the action of the subgroup of the Weyl group of type E6 gen-
erated by s3, s4, s5, s6, s7. Elements of this subgroup fix /3j,ft,/38 and act
on ft, ft, ft, ft, ft by permutations combined with sign changes with an
even number of negative signs. By applying elements of this subgroup
to a3, a4, a5, a6, a7, a8 we can obtain all vectors of form ±/3i±/3j with
8.8 Properties of long and short roots
145
Table 8.11 The simple Lie algebras
L	dim//	|Ф|	dimL
Л />1	/	/(/ + 1)	1(1 + 2)
В/ />2	/	2/2	1(21+1)
q />з	/	2/2	1(21+1)
A />4	/	21(1 — 1)	1(21-1)
E6	6	72	78
E,	7	126	133
Eg	8	240	248
F.	4	48	52
g2	2	12	14
3 < i, j < 7, i^j, and (up to sign) all vectors of form | J)£;e; with £; = ±1,
П £/ = 1, £i = e2 = £8. Hence the vectors in the above set are all roots of E6.
There are К J-4 = 40 vectors of type ±/3, ±/T with 3 < i, j <7, i^ f and
у 2 j	J
25 = 32 vectors of type | J) £;e; with £; = ±1, П Ei = 1 and £x = £; = £8. Thus
the total number of roots is
and we have
Thus:
|Ф|=40 + 32 = 72
dimL = 6+ |Ф| =78.
Theorem 8.10 The simple Lie algebra of type E6 has dimension 78.
We have now determined the dimensions of all the simple Lie algebras.
We summarise the information we have obtained in Table 8.11. In this table
L is a simple Lie algebra, H is a Cartan subalgebra and Ф the system of roots
of L with respect to H.
8.8 Properties of long and short roots
Proposition 8.12 In the simple Lie algebras of types Al,Dl,E6,E1,Ei all
the roots have the same length. In the Lie algebras of types Bh Ch F4, G2
there are two possible lengths of roots. These are called the long roots and
short roots.
Proof. This is clear from the preceding results.
□
146
The simple Lie algebras
Proposition 8.13 (i) Let Ф be a root system of type If with fundamental
system
О
О
«1
«2
Then the long roots form a subsystem of type If with fundamental system
О----О
i
a2
and the short roots form a subsystem of type (Л, )' with fundamental system
О
"1 +---haz a2 + --'+al
О
О
al
(ii)	Let Ф be a root system of type Cl with fundamental system
о-----о
a2
<1 < <>
Then the long roots form a subsystem of type (Л, )' with fundamental
system
О о
О
О
2'i + •  • + 2<iz + az 2а2 + • • • + 2qz_^ + ctz
2azl + az
and the short roots form a subsystem of type If with fundamental system
О------O-
n	a2
al-l
(iii)	Let Ф be a root system of type F4 with fundamental system
o----о > о---------о
“1	а2 а3	а4
Then the long roots form a subsystem of type D4 with fundamental system
ce2 2ск3 И- 2q4
q2 + 2a3
O
q2
8.8 Properties of long and short roots
147
and the short roots form a subsystem of type l)4 with fundamental system
»2 + «3
a2 + a3
-O
«3
(iv)	Let Ф be a root system of type G2 with fundamental system
<) ) <>
Then the long roots form a subsystem of type A2 with fundamental system
О------О
and the short roots form a subsystem of type A2 with fundamental system
О------О
al + a2 a2
Proof, (i) We saw in Section 8.3 that the roots of type have form
±/3i±/3j, i^j, and ±/3;. The former are the long roots and the latter the
short roots. The long roots form a system of type /), with fundamental sys-
tem /3Х-/32,/32 -/33,... ,/3^-/3;,/3;_!±/3;. These are a3, a2, ... , at_3,
ai_r+2ai respectively. The short roots form a system of type (Ax); with
fundamental system /3X,... , /3;. These are a14-h ah a2 -I--h ah ... , al
respectively.
(ii)	We saw in Section 8.4 that the roots of type C; have form ±/3; ±/3j, i^f j,
and ±2/3;. The former are the short roots and the latter the long roots.
Thus the short roots form a subsystem of type /), and the long roots a
subsystem of type (Ax);.
(iii)	We saw in Section 8.6 that the roots of type F4 have form
±/3;
i(±/31±/32±/33±/34).
Roots of the first type are long and those of the second and third types are
short. The long roots form a subsystem of type Z>4 with fundamental sys-
tem /34 — /Зх, /Зх — /32, /32 — /З3, /32±/33. These are a2 + 2a3 + 2a4, ar, a2,
148
The simple Lie algebras
a3 + 2.a3 respectively. The short roots also form a subsystem of type O4,
with fundamental system /32, /?з, | (—/3j — /32 — /З3 + /3^). These are
а1 + а2 + аз> «г + ^з, a3, a4 respectively.
(iv)	The long and short roots of type G2 are evident from Section 8.5.	□
Let П = {aj be a fundamental system of roots in a simple Lie algebra whose
Dynkin diagram has a double or triple edge, and let Ф be the root system
with fundamental system П. Consider the simple Lie algebra whose Dynkin
diagram is obtained from that above by reversing the direction of the arrow.
Let I lv = {«'} be the corresponding fundamental system, labelled as before,
and Фу be the root system with fundamental system IP.
System Фу is called the dual root system of Ф. The possible types of
Ф, Фу are as shown.
ф	фу
Ct	Bt
F.	F4
g2	g2
We note that ai is a short root in П if and only if et[ is a long root in IP.
We suppose as usual that we have symmetric scalar products {,} on 1КП
and Iii lv such that
for all i, j, where A=(Atj), A'r = (A'!j) are the Cartan matrices of Ф, Фу
respectively.
We consider the free abelian groups ZII, ZIP generated by П, IP. We
define a homomorphism
в : ZIP ZD
by
0(0 =
/7 O';
ai
if a; is a short root
if a; is a long root
where p is the ratio of the squared lengths of the long and short roots. (Thus
/7 = 2 in types Bh Ch F4 and /7 = 3 in type G2.)
8.8 Properties of long and short roots
149
Lemma 8.14 2-r—-—  z ' = Ay; for all i, j.
{0(«a0(«D} i]J J
Proof. We write в (ay) = £;a; where =p if at is short and = 1 if at is
long. Then
{0 (ay), в (ау)!	. (ai, a;!
2 v 1 	j' >  г-i? 2 1 J	A
{0 (ay), 0 (ay)}	1	{a;, aj 1
However, ^r1^.Aij = Atij. This follows from the following observations.
If Atj 0 and a;, ctj have the same length then we have and A^ =
V '•
If 0, a; is long, aj is short then = 1,	= p, Atj = — 1 and A}. = — p.
If Atj t^O, a; is short, aj is long then = p, = 1, A^ = — p, A]j = — 1.
Thus in all cases we have АД.. = Av..	□
lj lj
Lemma 8.15 The diagram
commutes.
Proof. On the one hand 6s] (aj) = 0 (a] — A^a]) = 0 (aj) — A]j6 (ay). On the
other hand
sie (aj) = (£jaj) = £jsi (aj) = ^ (Uj-A^)
= ^i~ &ai) = # (“J) - W Ю •
Thus Os] = sf).	□
Let W, H'v be the Weyl groups of Ф, Фу. There is a natural isomorphism
W = IT' under which st corresponds to s], since the root lengths are irrelevant
as far as the structure of the Weyl group is concerned. We shall use this
isomorphism to identify IT" with W. Then Lemma 8.15 shows that 6w = w6
for all w e W.
Proposition 8.16 Given cd e Фу there is a unique a e Ф such that
0(«v) =
pa
a
if a is short
if a is long.
150
The simple Lie algebras
Proof. We have oT = w (ay) for some w e W, a] e IIV. Thus
0 (av) = Ow (aY) = wO (aY) =
pw (a;) if a; is short
w (a;) if a; is long.
Let a = w (a;) e Ф. Then a is uniquely determined since
w (aY) = w' (aj) => w' 1 w (aY) = aj => w' 1 w (a;) = a , => w (a;) = w' (a J .
Thus 0(av) =
pa	if a is short
a	if a is long.
□
This proposition determines a bijection Фу Ф under which av a. av is
called the dual root of a. av is long if and only if a is short.
Proposition 8.17 Let a e Ф satisfy a = 'ffniai. Then a is a long root if and
only if p divides nt for all i for which at is a short root.
Proof. Suppose a is long. Then в (av) = a and so
aV = 22 ига^+ У? niP laVi-
a.j long	a, short
Since av e JT Za’ we deduce that p divides n; whenever at is short.
Now suppose conversely that p divides n; for all i for which a; is short.
Let и; = pm for such i. Suppose if possible that a is short. Then в (av) = pa.
Thus
av = 22 Рп1а1 + У? п1а]
a.j long	a, short
= Р I 22 nia] + У? mia^ I •
\az- long	a, short /
This gives av e pTLa] which is impossible. Thus a is a long root. □
Proposition 8.18 (i) The abelian group generated by the short roots in Ф is
eLi
(ii) The abelian group generated by the long roots in Ф is longZa; +
Ea, short P^ai-
8.8 Properties of long and short roots
151
Proof. By considering the fundamental system of the subsystem of short
roots described in Proposition 8.13 it is clear that the abelian group generated
by the short roots contains ax,... , al so is £ ^ai-
The abelian group generated by the long roots lies in E„i„„gZa,) +
short pZa; by Proposition 8.17. However, this group contains a; for a;
long and pai for a; short, again by Proposition 8.17, so must be long Ztf +
E«, short pZa;.	□
9
Some universal constructions
9.1 The universal enveloping algebra
Let L be a Lie algebra over C. We shall show in this section how to construct
an associative algebra 1I(L), the universal enveloping algebra of L, such that
the representation theory of II (L) is the same as the representation theory of
the Lie algebra L. Even if L is finite dimensional its enveloping algebra II (L)
will be infinite dimensional.
We begin by forming the tensor powers of L. We define T° to be the
1-dimensional vector space Cl, Г1 =L, T1 = L®CL and, in general,
Tn = L®CL®C• • • ®CL (и factors)
Tn is a vector space over C of dimension (dimL)".
We next form the tensor algebra T = 7(7.) of L. We define T as the direct
sum of vector spaces
Г=Г°фГ1фГ2ф-- - .
Thus elements of T are finite sums of elements, each of which lies in some
Tn. We may define a bilinear map
Tm x Tn___> Tm+n
satisfying
(xl®---®xm)-(yl®---®yn) = xl®---®xm®yl®---®yn
for .v,, yj e L and then extend this map by linearity to give a multiplication
map
T x T T.
In this way T becomes an associative algebra called the tensor algebra of L.
The element 1 e T° is the identity element of T.
152
9.1 The universal enveloping algebra
153
Let J be the 2-sided ideal of T generated by all elements of the form
x®y — y® x — [xy] forx, yeL.
J is in particular a subspace of T. Let II (L) = Т/ J. Then II (L) is an associative
algebra over C called the universal enveloping algebra of L.
Example 9.1 Let L be an и-dimensional abelian Lie algebra over C. Then L
has basis xp... , xn and we have [х;х^] = 0 for all i, j. Thus J is the 2-sided
ideal of T generated by all elements of the form x®y—y®x for x, yeL.
Thus II (L) is a commutative algebra, and is generated as an algebra by the
identity 1 and the elements xp... ,x„. In fact II(L) is isomorphic to the
polynomial algebra C [xp ... , хл].	□
In general we have linear maps
L^T1 ^T^U(L)
and we denote by cr : L^ II (L) the composite linear map. We now show
that II (L) has a certain universal property which justifies its name.
Proposition 9.2 Let A be any associative algebra with 1 over C and [A]
the corresponding Lie algebra. Then given any Lie algebra homomor-
phism в : L^> [A] there exists a unique associative algebra homomorphism
ф : II (L) A such that фосг=0.
Note Associative algebra homomorphisms will be understood to be homo-
morphisms of associative algebras with identity in this chapter. Thus the
homomorphism will map identity to identity.
Proof. We first observe that the linear map в : L A can be extended to an
associative algebra homomorphism from T to A. If x;, i e I, are a basis for L
then the set of all monomials x; ... x; for ,... ,irEl form a basis for T.
The case r = 0 gives the identity element. The map
xil---xir^e(xii)...e(xi)
can then be extended by linearity to give an associative algebra homomor-
phism from T to A. Let this map be O' : T A. Let x, у e L. Then we have
O'(x®y — y®x — [xy])
= 0(x)0(y)-0(y)0(x)-0[xy]
= [0(x), 0(y)]-0[xy] = o
154
Some universal constructions
since в : L [A] is a Lie algebra homomorphism. Thus all the generators
of the 2-sided ideal J of T lie in the kernel of O'. Since the kernel is a 2-sided
ideal, J lies in the kernel of в1. This shows there is an induced homomorphism
ф : T/J^A
such that the diagram
т----► TH
A
commutes. When we restrict the domain to T1 we deduce that фосг = 0. This
proves the existence of a homomorphism ф : II (L) -a- A of the required type.
We now prove the uniqueness of ф. Let ф' : U(L)—> A be another such
homomorphism. Now T is generated by Г1 as an associative algebra with 1.
Thus its factor algebra II (L) is generated by cr(L), which is the image of Г1
in 1I(L). Let xeL. Then
^'(cr(x)) = 0(x) = ^(cr(x)).
Thus ф, ф' agree on cr(x) for all x e L. Since cr(L) generates II (L) it follows
that ф, ф' agree on 1I(L), so ф' = ф.	□
Using this universal property we can relate representations of the Lie
algebra L to representations of the associative algebra II (L). If V is a vector
space over C the set End V of all linear maps of V into itself forms an
associative algebra with 1, and the corresponding Lie algebra is [End V].
A representation of L is a Lie algebra homomorphism L -a- [End V] and a
representation of II (L) is an associative algebra homomorphism II (L) -a-
End V.
Proposition 9.3 There is a bijective correspondence between representations
0 : L [End V] and representations ф : II (L) -a- End V. Corresponding
representations are related by the condition
ф(сг(хУ) = 0(x) for allxeL.
Proof. Let 0 : L -a- [End V] be a representation of L. Then by Proposition 9.2
there exists a unique associative algebra homomorphism ф : II(Lj -a- End V
such that фосг = 0.
Conversely, given an associative algebra homomorphism ф : U(L)—>
End V we wish to define a corresponding Lie algebra homomorphism
9.2 The Poincare-Birkhoff-Witt basis theorem
155
в : L^> [End V]. Now we have a linear map cr : L^ 1I(L). Since 1I(L) =
Т/J and x0y — y0x — [xy] e J for all x, у e L we see that
ФИу) - сг(у)сг(л') - сг[л'у]=о
for x, у e L. This gives
[cr(x),cr(y)] = cr[xy]
andsocr : L^ [11(E)] is aLie algebra homomorphism. We now define 0 : L^-
[End V] by в = ф о cr. Then в is a Lie homomorphism of the required type.
It is clear from the definitions that the maps 0^ <!> and ф^ в are inverse
to one another.	□
We shall find this result very useful in the subsequent development, when
we shall obtain information about representations of finite dimensional Lie
algebras by considering the representation theory of the corresponding uni-
versal enveloping algebra.
9.2 The Poincare-Birkhoff-Witt basis theorem
We shall now describe how to obtain a basis for the universal enveloping
algebra II (L).
Theorem 9.4 (Poincare-Birkhoff-Witt). Let L be a Lie algebra with basis
{x; ; i el}. Let < be a total order on the index set I. Let cr : L—>U(L)
be the natural linear map from L into its enveloping algebra. Let cr (x;) = y;.
Then the elements
rl	rn
Уц yi.
for all и > 0, all r; > 0, and all f,... , inel with q < z2 < • • • < in form a
basis for II (L).
Proof, (a) We first show that the above elements y}}.. .yf span 1I(L). We
know that the elements of form x;| 0... 0 x^ for all k and all ,... , jke I
span T. By applying the natural homomorphism T II (L) it follows that the
elements of form • • • yjt span II (L). It is therefore sufficient to show that every
product y h ... yjt is a linear combination of the given elements of form y}}... yf.
We shall prove this by induction on k. It is obvious if k = 1. For arbitrary k it is
clear when jr < • • • < jk. If this is not so we may use relations of form
156
Some universal constructions
We note that [x;xj is a linear combination of elements xt for t e I and so
ст [х;х^] is a linear combination of yt for t el. Thus we may interchange the
order of two consecutive terms yh yj in a monomial of degree k provided we
introduce a certain linear combination of monomials of degree less than k.
By performing such interchanges a finite number of times we may express
the terms y; in the monomial with the i in the given order < on I. Thus
y^ ... yj = У;‘ • • • y[" + a linear combination of monomials of degree
less thar4
where 4-------h rn = k and ix< - • • <in. By induction we may assume that all
monomials of degree less than k are expressible as linear combinations of
monomials with terms in the given order c. The required result then follows.
(b) We now show that the given monomials of form y-‘ ... yf are linearly
independent. This is not so easy to see, and we shall prove it by an indirect
argument. We introduce the polynomial ring R = C [z; ; i G /1 and shall make
use of the following lemma.
Lemma 9.5 There exists a linear map в : T R satisfying the conditions
0 (xit ®   • ® xj = zh	if z'i < • • • < in
в ( X; ®	® X ®X:	®	® X; ~X: ®	® X; ® X; ®	® X; )
\ ll	lk 4+1	ln ll	4+1 lk	ln 1
= в (X; ®	® X; X; ®	® X; ) for all , . . . , L and all k
with 1 < k < n.
Proof. We define the index of the monomial x, • •®xi to be the number of
pairs (r, s) with \ <r < s <n satisfying ir > is. Thus the monomials of index 0
are those whose terms appear in their natural order. Let Г"’-' be the subspace of
Tn spanned by all monomials x; ® • • • ® x; of index at most j. Thus
yn,0 грп,1	yn
We define в : T° R by 0(1) = 1. Suppose inductively that в : T° ф • • • ф
T" ' R has already been defined satisfying the required conditions. We
shall show that в can be extended to в : T° ф- • -®Tn R. We define
0 : Tn’° -+ R by
0 (X;	®Xi)=Zi.  - Zj
if the monomial xt ® -  -®х^ has index 0. We suppose 0 : Tn,‘ R has
already been defined, thus giving a linear map from T° ф • • • ф 7'" 1 ф Гл,! to
9.2 The Poincare-Birkhoff-Witt basis theorem
157
R satisfying the required conditions. We wish to define 0 : Tn’l+1 R. Thus
suppose the monomial x; ® • • • ® xt has index i + 1. Then there exists к with
1 < к < n such that x; ® • • • ® xt ®xit	2®-  - ®xt has index i.
We then wish to define в (xt ® • • • ® xt ) by the formula
в ( X: ® • • • ® X: ®Xj, , ® • • • ® Xj )
\ ll	1к 4+1	ln /
= 6 ( Xj, ® • • • ® Xj ® Xj ® • • • ® Xj )
\	1	lk+l lk	ln 1
+ 0 ( Xj ® • • • ® X,- Xj ® • • • ® Xj )
\ ll	4 4+1	ln 1
noting that the terms on the right-hand side have already been defined. How-
ever, there may be more than one possible choice of к and we must check
that if we choose a different one the linear map 0 : Tn’l+1 R will still be
the same. So suppose к' also satisfies 1 <k' <n. We may without loss of
generality assume that к < к’.
We suppose first that k+l<k'. Let x; = a, xt = b, xt = c, xt =d.
Then the definition using the integer к gives
	- ®a®b®-  - ®c®d®- 
= 0 -®b®a®--®c®d®-
+в (• • • ® [ab] ®---®c®d®---)
= 0 -®b®a®--®d®c®-
+в (• • -® b® a® • • -® [cd]® • • •)
+в (• • -® [ab]® • • • ®d® c® • • • )
+в (• • • ® [ab] ® • • • ® [cd] ® • • •)
using the inductive assumptions.
The second definition using the integer k' gives
	- ®a®b®-  - ®c®d®- 
= 0(---®a®b®---®d®c®---)
+в (• • -® a® b® • • - ® [cd]® • • •)
= 0(---®b®a®---®d®c®---)
+в (• • - ® [ab]® • • • ®d® c® • • • )
+в (• • -® b® a® • • - ® [cd]® • • •)
+в (• • • ® [ab] ® • • • ® [cd] ® • • •)
158
Some universal constructions
using the inductive assumptions. These two expressions using integers k, k'
are the same.
Now suppose that k' = k+l. Let xit=a, xt =b, xt =c. We compare
the two ways of calculating 0(---®a®b®c®---}. The first method, using
the integer k, gives
0(- • • ®a® b® c® • • •)
= 0{-- ®b® a® c®--’) + (){--® [ab] ® c ® • • • )
= 0(- • -® b® c® a® • • -) + 0(- • -® b® [ac]® • • •)
+0(- • -® c® [ab] ® • • •) + $(• • -® [[ab]c] ® • • •)
= 0(- • -®c®b®a®- • •) + 0(- • -® [be] ® a ® • • •)
+0(- • -® b® [ac] ® • ••) + $(• • -® c® [ab]® • • •)
+0(-• • ® [[ab]c] ® • • •)
= 0(- • -®c®b®a®- • •) + 0(- • • ® a ® [be] ® • • •)
+0(- • -® b® [ac] ® • • •) + $(• • -® c® [ab]® • • •)
+0(- • • ® [[ab]c] ® • • •) + 0(- • • ® [[bc]a] ® • • • )
using the inductive assumptions. The second method, using the integer k' =
k+ 1, gives
0(- • -®a®b® c® • • •)
= 0(---®a®c®b®---) + 0(---®a® [be] ® • • • )
= 0(- • • ® c® a® b® • • - ) + 0(- • • ® [ac] ®b® • • • )
+0(- • -® a® [be] ® • • •)
= 0(- • -®c®b®a®- • •) + 0(- • • ® c ® [ab] ® • • • )
+0(- • - ® b® [ac]® • • •) + $(• • • ® [[ac]b]® • • •)
+0(- • - ® a® [be] ® • • •)
again using the inductive assumptions. Comparing the two expressions
obtained we see that they are equal since
[[ac]b] = [[ab]c] + [[bc]a].
Thus в : Tn’l+1 R is now defined and this gives
в : Г°ф---фГл-1фГл’;+1^А.
9.2 The Poincare-Birkhoff-Witt basis theorem
159
Since Tn = Tn,r for r sufficiently large we have
О : /'"©фГ^А1.
Since T = С°ф С1 ф С2ф- • • we have defined в : T^R satisfying the
required conditions.	□
We now return to part (b) of the proof of Theorem 9.4. We have II(C) = Т/ J
and the elements
X; ®	® X: ®Xj ® • • • ® Xj — Xj ® • • • ® Xj ®Xj ® • • • ® Xj
4	lk lk+l	ln 4	lk+l lk	ln
-V; ®---® X; Xj
ll	lk (k+1	ln
= Xj ... Xj I Xj ® Xj — Xj ® Xj — Xj Xj I Xj ... Xj
ll lk—l \lk lk+l lk+l lk L lk lk+l J J lk+2 ln
all lie in J. In fact the definition of J shows that each element of J is a linear
combination of such elements. Thus the linear map в : T R of Lemma 9.5
annihilates all elements of J, and so induces a linear map в : T/J^R,
that is в : 11(C) R. Now the monomial y]) ... yf e 11(C) for q < • • • < in
is mapped by в to г? ...zf eR. Since the elements г? ...z'f are linearly
independent in the polynomial ring R it follows that the elements y-‘ ... yf
given in the statement of Theorem 9.4 must be linearly independent in 11(C).
This completes the proof.	□
We now deduce some consequences of the Poincare-Birkhoff-Witt basis
theorem. (We shall subsequently call it the PBW basis theorem.)
Corollary 9.6 The map <j : L—>U(L) is injective.
Proof. The elements x;, i e I, form a basis for L and cr (x;) = y;. By the PBW
basis theorem the elements y;, i e I, are linearly independent. Thus the kernel
of cr is zero.	□
Corollary 9.7 The subspace cr{Lj is a Lie subalgebra of [11(C)] isomorphic
to L. Thus cr identifies L with a Lie subalgebra of [11(C)].
Proof. By Corollary 9.6 we know that cr : C^cr(C) is bijective. The
elements y;, i e I, form a basis of cr(C) and we have
vw wv ,r[xx\.
It follows that [y;yj 6<r(I) and so cr(C) is a Lie subalgebra of [11(C)].	□
160
Some universal constructions
It is often convenient to consider L as a subspace of II (L) without men-
tioning the map <j explicitly.
Corollary 9.8 II (L) has no zero-divisors.
Proof. Let a, b e 1I(L) have a 0, b 0. Then we have
a = У' A,.	; r r л/1 ... у"
; r r y? • • • y,r” •
We write
a = f (y;) + a sum of terms of smaller degree
where f (v,) is the sum of all terms A; r r y'1 ... y'" of maximal total
degree r = 4--------h rn. Similarly we have
b = g (y;) + a sum of terms of smaller degree.
Now we have
ytyj = y.yt -|- a sum of terms of degree 1
and so
f (У;) S (У/) = (/?) (У;) + a sum °f terms of smaller degree.
Hence
ab= (fg) (y;) + a sum of terms of smaller degree.
Now f is not the zero polynomial since a 0 and g is not the zero polynomial
since b 0. Thus fg is not the zero polynomial. The PBW basis theorem then
implies that ab ^0.	□
9.3 Free Lie algebras
It is well known how to define groups by generators and relations. One
first constructs the free group on the given set of generators and then forms
the factor group with respect to the smallest normal subgroup containing
the elements specified by the given relations. We shall show that something
similar can be done in the theory of Lie algebras. We first introduce the idea
of the free Lie algebra Г1ЛХ) on a set X.
9.3 Free Lie algebras
161
Let X = {x;, i e 1} be a set of elements parametrised by an index set I. We
first define the free associative algebra F(X) on the set X. F(X) is the set of
all finite sums of the form
* * * X.
k>0
with A; ,... ,j eC, summed over all non-negative integers k and all ordered
tuples q,... , ik from I (repetitions being allowed). When k = 0 the product
. xt is the empty product, and is written as 1. The operations of addition,
multiplication and scalar multiplication are defined in an obvious way and
make F(X) into an associative algebra over C with identity 1.
Let |/’(X) | be the Lie algebra obtained from the associative algebra F(X)
in the usual manner. X is a subset of |F(A')|. We define FL(X) to be the
intersection of all the Lie subalgebras of [ Z’(A') | containing X, i.e. the Lie
subalgebra of [ Z’(A') | generated by X. FL(X) is called the free Lie algebra
on the set X. It is clear that X is contained in FL(X) so we have an injective
map i : A'^FF(A').
In order to justify its name, we show that the free Lie algebra FL(X) has
the following universal property.
Proposition 9.9 Let О : X L be any map from the set X into a Lie
algebra L. Then there is a unique homomorphism ф : FLfX) L such that
фо1 = в.
x——^FL(X)
L
Proof. Consider the maps хДтД 11(F). Let O' : X 11(F) be given by
0' = cro0. The map O' from X into 11(F) can be extended uniquely (in an
obvious way) to an associative algebra homomorphism ф' : F(A')^ 11(F).
The same map gives a Lie algebra homomorphism ф' : [F(X)]	[11(F)].
Now we have ф'(Х) ccr(L) and we know from Corollary 9.7 that cr(F) is
a Lie subalgebra of [11(F)] isomorphic to F. The set of elements of [FfX)]
mapped by ф' into cr(F) is therefore a Lie subalgebra of [F(X)] containing
X, and this contains FI.(X). Hence we have ф' : FLfX) cr(L). We define
ф : FI.(X) F by ф = <r ! о f//'. We check that ф о i = 0. For if x e X we have
ф о /(л) = a' f'i(x) = сг^ф'^х) = cr^O'lx) = 0(x).
162
Some universal constructions
Thus we have a homomorphism ф of the required type. Finally we show that ф
is unique. Let ф : FL(X) L be another such homomorphism. Then we have
<7и(л) = 0(x) = <7> z Cv)	for all x e X.
Thus ф agrees with ф on X. Now the set of elements of FI AX) for which ф
agrees with ф is a Lie subalgebra of FI AX) containing X. Since X generates
FI AX) as a Lie algebra we deduce that ф agrees with ф on FL(X). □
We next identify the universal enveloping algebra of the free Lie algebra
FL(X). This turns out to be isomorphic to the free associative algebra F(X).
Proposition 9.10 The universal enveloping algebra l\(FI.(X)) is isomorphic
to Fix').
Proof. We have an inclusion map <j : FL(X) F(X). We shall show that the
universal property of enveloping algebras given in Proposition 9.2 is satisfied
by F(X). Thus we shall show that if A is any associative algebra with 1
over C and if 0 : FI AX) [A] is any Lie algebra homomorphism then there
exists a unique associative algebra homomorphism ф : F(X) A such that
фосг = 0.
Now the Lie homomorphism в : FL(X)—>[A] restricts to a map
в : X A. This map can be extended to a unique associative algebra homo-
morphism ф : IAX) A. This same map gives a Lie algebra homomorphism
ф : [F(X)] —> [А]. By restriction we obtain a Lie algebra homomorphism
ф : FL(X)^ [А]. However, ф agrees with в on X and X generates FL(X)
as a Lie algebra. Hence ф agrees with 0 on FL(X). It follows that фосг=0
as required. Thus there exists an algebra homomorphism ф of the required
kind. On the other hand ф is clearly unique since FL(X) contains X and
therefore generates the associative algebra F(X).
Thus F(X) satisfies the above universal property. Of course II (FL(Xf)
satisfies it also. This implies that 1I(FL(A')) is isomorphic to F(X). For
suppose we are given a Lie algebra L and two associative algebras II, II' with
maps <j : L—> II, <j' : L—> H' both satisfying the universal property. Then
we obtain unique algebra homomorphisms ф : II II' and ф' : II' II such
that <j' = фосг and сг=ф' о a'.
9.4 Lie algebras defined by generators and relations
163
It follows that
ф' ф cr(x) = a( x),	фф'сг'(х) = <j' (x)
for all .it/.. Now <t(L) generates II and cr'(L) generates II' as associative
algebras, by the uniqueness of ф and <!>'. It follows that
</>'</> = Idu,	<^' = Idu,
and so ф, ф' are inverse isomorphisms between II and II'.	□
9.4 Lie algebras defined by generators and relations
Let X = {x;, i e 1} be a given set. A Lie monomial in the elements of X is
a finite product of elements of X bracketed by Lie brackets in any manner.
For example
[[[x3 [x3x2]] x3] [х2[хЛ]]]
is a Lie monomial on the set X = {xp x2, x3}. A Lie word in the elements
on X is a finite linear combination of Lie monomials on X with coefficients
in C. For example
3 [[[x3 [xxx2]] x3] [x2 [x^]]] +2 [x2 [[xxx2] [x3x2]]]
is a Lie word on the set X = {xp x2, x3}.
Let R = {Wj, j e j) be a set of Lie words in the elements of X. We shall
define a Lie algebra I.(X; A) called the Lie algebra generated by X subject
to relations R.
Now the elements of X all lie in the free Lie algebra FI.(X) and all the
Lie words Wj also lie in FLfX). Of course different Lie words can give the
same element of FI AX) because of relations such as [x;x;] =0 and the Jacobi
identity. Let (R) be the ideal of 1'IAX) generated by R. Thus (R) is the
intersection of all ideals of 1'IAX) containing R. We define I AX; R) by
L(X; R) = FL(X)/(R).
Lemma 9.11 Let R, R' be sets of Lie words in X such that R'cR. Then
L(X; R) is isomorphic to a factor algebra of L(X , R'f
Proof Since R' cR we have
(7?')c(7?)cFL(X).
164
Some universal constructions
It follows that
T(Y. FL(X) ~ FL(X) , (R)	L(X;R')
(	)	(R)	(R')	(R') I
where I = (R) / (R').	□
Example 9.12 Let A be a Cartan matrix on the standard list 6.12. In Sec-
tion 7.4 we defined a Lie algebra L(A) associated with A, and ISA) was
subsequently shown in Proposition 7.35 to be a simple Lie algebra. In fact
all the finite dimensional non-trivial simple Lie algebras over C have form
L(A), as A runs over all Cartan matrices on the standard list. The definition
of L(A) given in Section 7.4 shows that /.(A) can conveniently be described
in terms of generators and relations. In fact we have
L(A) = L(X ; A)
where X = {ех,... , ehhx,... ,hl,fi,... , and R is the set of Lie words
in X given by
lhihi\
lAe?] ~Aijej
[eifi\ ~ hi
[ejJ for г#7
[e;[e;[...[e;e7]]]]	for i^j
[/;[/>[•• •[/>/;]]]]	for/#7
where the number of occurrences of e;, ft respectively in the last two words
is 1-Ay.
Example 9.13 Again let A be a Cartan matrix on the standard list 6.12. In
Section 7.4 we also defined a certain Lie algebra L(A) depending on A which
contains L(A) as a factor algebra. The algebra L(A) is infinite dimensional.
It can also conveniently be described by generators and relations. In fact we
have
L(A) = L(X;R')
where X = {er,... , ehhx,... ,hh	... , ft} and R' is the set of Lie words
on X given by
9.5 Graph automorphisms of simple Lie algebras
165
lhihj\
[hiej]-Aijej
[e;/J fori# j.
We observe that R' is a proper subset of the set R of relations in Example 9.12.
This explains why 1.(Л) is isomorphic to a factor algebra of L(A), as in
Lemma 9.11.
9.5 Graph automorphisms of simple Lie algebras
Let A be a Cartan matrix on the standard list 6.12 and <j be a per-
mutation of {1,...,/} such that A^j^j = Ay for all i,j. Let L(A) be
the simple Lie algebra associated with A. L(A) can be generated by
ex,... , e;, /ij,... ,hl,f1,... , f[. We define a permutation of this generating
set by
Under this permutation of the generators each of the defining relations of
L(A) in Example 9.12 is transformed into a defining relation. Let
L(A) = L(X; R)=F-^.
The given permutation of X extends to a Lie algebra homomorphism of
FL(X) into itself, and this homomorphism maps the ideal (R) into itself. It
therefore induces a Lie algebra homomorphism of L(X; R) into itself. Since
the permutation of X is invertible, so is this Lie algebra homomorphism. It
is thus an isomorphism of L(X; R) into itself, that is an automorphism of
L(A). This automorphism is called a graph automorphism of L(A) and will
also be denoted by a. The possible non-trivial graph automorphisms can be
described in terms of the action of <j on the Dynkin diagram of L(A). These
possibilities are listed below.
1	2	к
О-----О----О-----О-----О-----О
Туре A2jt
О-----О----о-----о-----о-----о
2к 2к-1	к+1
cr(i) = 2k+ 1 — i
166
Some universal constructions
Type Alk_r
cr(i) = 2k — i
Type Dk+1
O------O-
cr(z) = i for 1 < i < k — 1
<r(k) = k + 1
cr(k + 1) = k
Type /)4
cr(l) = l cr(2) = 3	cr(3)=4	cr(4) = 2
(The inverse of cr is also a graph automorphism, which can be obtained from
cr by renumbering the vertices.)
Type E6
cr(l) = 6	cr(2) = 5	cr(3) = 3
cr(4) = 4	cr(5) = 2	cr(6) = 1
Our main aim in the present section is to determine the fixed point subalgebra
L(A~)a = {.rt L(A~); cr(x) = x}.
We begin by considering the action of cr on V = given by cr (a;) = aa^
and extending by linearity. Let V1 = {и e V; cr(v) = u}. For each orbit J of
cr on {1,... , /} we define a7 = Ту J)^e7 aj- Then a7 e V1 and the a7 form
a basis of V1 as J runs over the сг-orbits on {1,... , /}. a7 is simply the
projection of ctj on to the subspace V1 of the Euclidean space V. We see
from the above diagrams that the orbits J have the following possible types.
9.5 Graph automorphisms of simple Lie algebras
167
(a)	|J| = 1 and J = {j} with a(j)=j.
(b)	|J|=2 and J = {jj} where a(f) = j,a(f) = j and и. + а.^Ф.
(c)	|J|=3 and	where a(j) = j, a(j) = j, a (j) = j and
Uj + ар Uj + a-., a-j + a-, do not lie in Ф.
J J J J J J
(d)	|J|=2 and J = {jj} where a(f) = j,a(j) = j and
These four will be called orbits of types A1^.A1, Ar xAjxAj and A2
respectively.
We next consider the possible pairs J, К of distinct orbits.
Lemma 9.14 The vectors a}, aK for distinct a-orbits J, К form a fundamen-
tal system of roots of rank 2. The type of this rank 2 system is as follows.
(vi)
If no node in J is joined to any node in К then
the type of{ctj, a^} is AjXAp
168
Some universal constructions
Proof. This is straightforward. Suppose for example we have case (v) with
roots numbered
1 О----0 2
4 0----0 3
Then
+	“г + ^з
tf7 =	2	' Uk =	2	'
We have
aK) = i (aj, aj)
(“л	=
Thus (a}, а7)=2(ак, aK} and 2 (a}, aK} / (a}, af) = -1.
Hence we have a fundamental system with diagram
«7 О >
Corollary 9.15 Let П1 be the set of vectors а} for all <j-orbits J on {1,... , /}.
Then П1 is a fundamental system of roots of the following type:
Туре П	Order of cr	Type П1
	2	Bk
A-lk-l	2	Ck
^к+1	2	Bk
d4	3	G2
Е6	2	F.
Proof. This follows immediately from Lemma 9.14.
□
The relationship between П and П1 may be illustrated in the following
diagrams.
1	2	к
п о----О--о—	• • •	---о---р
о-----о---о—	• • •	---о----б
2к 2к-1	к+1
12	к-1 к
п1 О------о---о—	• • •	---о > о
9.5 Graph automorphisms of simple Lie algebras
169
Now let Ф1 be the root system in V1 with fundamental system П1. Let W1
be the Weyl group of Ф1. Then Ф1 = W1 (П1). Let A1 be the Cartan matrix
of Ф1.
Proposition 9.16 Let I, J be distinct <j-orbits on {1,... , /}. Then
Eiei Aij for any j e J, if I has type Аг, Агх Аг or Ar x Ar x A
2Eie/Aij for any j e J, if I has type A2.
Proof. This follows from Lemma 9.14.
□
Proposition 9.17 Let W<7 = {weW ; wcr = crw onV}. Then there is an
isomorphism W1 -+ Wa under which the fundamental reflection s7 e W1
170
Some universal constructions
corresponding to a, maps to (w0)7 e И"7, the element of maximal length in
the Weyl subgroup Wj of W generated by the st for i e J.
Proof We first observe that Wa acts on V1. For let w e W'7, v e V1. Then
a(wv) = w(a v) = wv, thus wv e V1.
Secondly we note that (w0)7 e Wa for each сг-orbit J. For j eJ we have
asja-^s^
thus aW,a! = Wj. Since a preserves the sign of each root we have
o-Cwo^o-1 (ф+) = ф;
and hence
o- (w0)7 cr-1 = (w0)7
by Proposition 5.17. Thus (w0)7 e И"7.
Thirdly we note that the element (w0)7 e И"7, when restricted to V1, coinc-
ides with Sj. For
(wo)y (“7) = (»о)у (тут E “J =“T77 E aj = ~aJ
уИуе/ / Иуе/
since (w0)7 (ф+) = ф;.
Also if v e V1 satisfies (a7, v) = 0 then it satisfies (uj, v) = 0 for all j e J. It
follows that (w0)7 (u) = v. Thus (w0)7 coincides with s} on restriction to V1.
We next show that the elements (w0)7 for all сг-orbits J generate И"7.
Let weW<7 satisfy w^l. Then there exists a fundamental root a, with
w (ctj) e Ф . Let J be the сг-orbit containing j. Then
wa (ctj) = aw (ctj') e Ф-
since a preserves the sign of each root. Thus w (a;) e Ф- for all i e J. Now
(w0)7 changes the signs of all roots in Ф7 but of none in Ф — Ф7. Hence
/ (w (w0)7) = /(w) - / ((w0)7) < /(w).
We assume by induction on /(w) that w (w0)7 lies in the subgroup generated
by the (w0)7 for all сг-orbits I. It follows that w has the same property. Hence
the (w0)7 generate И"7.
We may now define a homomorphism Wa -+ W1, by restricting the action
of we Wa from V to V1. Since Wa is generated by the elements (w0)7 and
(w0)7 restricted to V1 is s7, the image of the homomorphism is generated by
9.5 Graph automorphisms of simple Lie algebras
171
the Sj and so is W1. Finally we show our map is injective. Suppose w e Wa
and ti'f 1. Then there exists a сг-orbit J such that w (a;) e Ф for all i e J.
Thus w (a7) aj and so w acts non-trivially on V1. Thus our map Wa —> W1
is an isomorphism under which (w0)7 e Wa corresponds to s} e W1.	□
We next consider the relation between the root systems Ф and Ф1. For
each a e Ф we denote by a1 its projection into V1.
Proposition 9.18 (a) For each a e Ф, a1 is a positive multiple of a root
in Ф1.
(b) Let ~ be the equivalence relation on Ф given by /3 if and only if a1
is a positive multiple of /31. Then the equivalence classes are the subsets
of Ф of form w (Ф7) where J is a <j-orbit in {1,... , 1} and w e H"r.
(c) There is a bijection between equivalence classes on Ф and roots in Ф1
given by w (Ф7) ** w1 (a7) where w1 is the restriction of w to V1.
Proof. We first show that each a e Ф lies in w (Ф7) for some сг-orbit J and
some w e Wa. Consider the element w0 e W of maximal length. w0 transforms
each positive root to a negative root. Since cr does not change the sign of any
root we have
а woa~r (Ф+) = ф •
Since <гпу<г! e W it follows that f/u1,/; = w0, that is w0 e W1. By Propo-
sition 9.17 the elements (w0)7 for all сг-orbits J generate Wa and so
wo = (»о)л • • • (»о)л
for some , Jr. Let a e Ф+. Then w0(a) e Ф . Thus there exists i such
that
(w0)7,+1 • • • (wo)yr («) e ф+
(w0)7_ (»о)л+1 • • • (wo)yr («) e ф •
Since the only positive roots made negative by (w0)7 are those in Ф7 we
have
(w0)7_+i... (w0)7r (a) e Ф+,
that is ae(w0)7...(w0)7.+i (Ф+) and -a e (w0)7r... (w0)7;+i (w0)7_ (ф+).
Hence each root in Ф lies in w (Ф7) for some сг-orbit J and some w e Wa.
172
Some universal constructions
Now consider the projection a1 for аеФ7. If J has type A1,A1 xAt or
AjxAjxAj then Фу = П7 and so a1 = aJ for аеФ7. If J has type A2,
however, then П7 = сгД and Ф7 = {ap ap ctj + сгД. We have
a
a, when a= a; or a-;
J	J J
2a, when a = o'; + a-;.
J	J	J
Thus a1 is a positive multiple of a7 when a e Ф7. Hence for a e w (Ф7) with
w e Wa we see that a1 is a positive multiple of w (a7) e Ф1.
Now consider the equivalence relation on Ф defined in (b). The elements
of each set ш(Ф7) for weW'7 lie in an equivalence class. Suppose
ir (Ф7 ), iC (ФА ) lie in the same equivalence class for сг-orbits J, К and
w, w e Wa. Then
w(a7) = w' (а^еФ1.
Hence w' 1 w (a7) = aK.
Consider the root w'^w (tt;) e Ф for j e J. This root has the property that
Since К is a сг-orbit this implies that iC 1 w (aj is a non-negative combination
of the ak for к e К. Hence
w'-1»|11;)СФ1,
and so w'^w (Ф7) С Ф^. By symmetry we also have
w’w-1 (ф+) с Ф+.
Hence we have equality, that is
ш(Ф7+) = ш' (Ф+).
Hence the equivalence classes are the subsets of Ф of form w (Ф7).
Now any root in Ф1 has form w (a7) for some w e W1 and some сг-orbit J.
The set of roots a e Ф such that a1 is a positive multiple of w (a7) is w (Ф7),
as shown above. Thus
ш(ф+)о»(а7)
is a bijective correspondence between equivalence classes on Ф and elements
of Ф1.	□
Theorem 9.19 Let a be a graph automorphism of the simple Lie algebra
L(A). Then the subalgebra 1ЛЛ)'' is isomorphic to the simple Lie algebra
L(A^.
9.5 Graph automorphisms of simple Lie algebras
173
Proof. For each сг-orbit J on {1,... , /} we define elements e7, h}, f, of
L(A)a by
e7 = Ee; // = ЕЛ hJ = I2hj
jeJ	jej	jej
if J has type Aj, Aj x Ax or AjxAjX Ax and
e7 = V2££p /7 = V2£/p ^ = 2£^
7e7	jej	jej
if J has type A2. Then we have
= hi
[e7/7] = 0 if 7# J.
[h^j] = 0
We consider [h/eJ]. If I, J have type Ab Ax x Ax or AjxAjX Ax we have
lhieA =
Y.hi^ej
iel jeJ
We also have [/i7e7] = A|7e7 if one or both of I, J has type A2. Similarly
[hIfJ] = -A1IJfJ for all I, J.
We also check the relation
[e7[e7[...[e7e7]]]]=0 for 1^ J
where there are 1 — A|7 factors e7. This follows from the following observa-
tions, which can be checked from Lemma 9.14.
If A)7 = 0 then [e;ej = 0 for all i el, j e J.
If A)7 = — 1 then [e; [e;,e7]] =0 for all i, i' e I, j e J.
If A)7 = —2 then [e; [e;, [е;„е7]]] =0 for all i, i', i" el, j e J.
If A)7 = —3 then [e; [e;, [e;„ [e;,„ eJ ]]] = 0 for all i, i', i", i'" el, j e J.
Similarly we obtain the relation
with 1 — A\j factors /7.
174
Some universal constructions
We now consider the generators and defining relations for the simple Lie
algebra L (A1) given in Example 9.12. All these relations are satisfied by the
elements e7, f}, h, of L(A)<7. Thus there is a homomorphism
L (A1) L(A)a
under which the generators of L (A1) map to the elements e7, fj, hj of L(A)<7.
Since L (A1) is simple this homomorphism is injective. We show it is also
surjective and that the map is therefore an isomorphism. It will be sufficient
to show that
dim L(Ay = dim L (A1).
We consider the decomposition of Ф into equivalence classes given in Propo-
sition 9.18. For each equivalence class 5 let
Ls =
aeS
Then <j (Ls~) = Ls and
l = h®^ls
s
L- = Hr	•
s
Now dim (l.g)r < 1 for each equivalence class S. This is clear if 5 has type
A^AjxAj or AjxAjxAp Suppose then that 5 has type A2. Then S =
{a, /3, a + /3}. We have
for some Л e C. Hence
67 [е«еуз] = [е/зеа] = - [еае!з] 
Thus (Ls)a consists of all multiples of ea + Xep and dim (l.g)r = 1. It follows
that
dim// < dimHa + no. of equivalence classes 5
= dim//1 + |ф'
= dimL(A1).
Hence dim/A <dimL (A1).
This shows that the homomorphism L (A1) La is suijective and hence is an
isomorphism. We note in particular that dim(Ls)<7= 1 for each equivalence
class S.	□
9.5 Graph automorphisms of simple Lie algebras	175
Thus we have shown that L(A)<7 is isomorphic to L^1). To be specific
we have:
L{Alky = L{Bk)
L{Alk_iy = L{Ck)
L(Dk+1)a = L(Bk)
L(Dff = L(Gf)
L(E6y = L(Ff)
where cr is a graph automorphism of order 2, 2, 2, 3, 2 respectively.
10
Irreducible modules for semisimple
Lie algebras
In the present chapter we shall determine the finite dimensional irreducible
modules for a semisimple Lie algebra over C. We begin by investigating
certain important modules for such algebras known as Verma modules.
10.1 Verma modules
We begin with a lemma on universal enveloping algebras. Let L be a finite
dimensional Lie algebra over C and К a subalgebra of L.
Lemma 10.1 There exists a unique algebra homomorphism в: П (W)	11(7.)
such that the diagram
K^UfKj
i Ф	4-0
L	1I(L)
commutes, where i is the embedding of К in L and <jk, <jl are the embeddings
of K, L in II(K), 11(7.) respectively.
Also в is injective.
Proof. Let x e К. Then we must have
Thus ()(<tk(x')) is uniquely determined. Since ll(A') is generated by crK(K)
as algebra with 1 we see that в is uniquely determined.
We now show that в exists. We recall that
n(L) = T(L)/J(L), U{Kj = T{Kj/J{Kj
176
10.1 Verma modules
177
where ./(/.) is the 2-sided ideal of '/(/.) generated by the elements
x®y — y® x — [xy] for x, y&L.
Now the map i : К I. induces an algebra homomorphism i : T(K) /(I.)
and we have
z(x® у — у ® x — [xy]) = z(x) ® z(y) - z(y) ® z(x) - [z(x)z(y)]
for all x, у e К. This shows that
z(J(/f))cJ(L).
Thus there is an algebra homomorphism О : II (К) II (L) such that the
required diagram commutes.
Finally we show that 0 is injective. This follows from the PBW basis
theorem 9.4. Let xp ... , xr be a basis of K. Suppose if possible there exists
и e 11(K) such that и 0 and в(и) = 0. Then и is a non-zero linear combination
of monomials x^1 ... xf. However, since xp ... , xr can be chosen as part of
a basis of L, the PBW basis theorem for L shows that such a combination of
monomials cannot be zero in L. Hence 0(и)^О, a contradiction. Thus в is
injective.	□
This lemma shows that II (Д') may be regarded in a natural way as a
subalgebra of II (L).
We now suppose that L is a finite dimensional semisimple Lie algebra over
C. Let H be a Cartan subalgebra of L and
L = H®^La
аеФ
be the Cartan decomposition of L with respect to H. Let Ф+ be the positive
system of roots in Ф. Then we have a triangular decomposition
I. = N ®II®N
where /V = ф,,фLa, N = ф,, ф Т,ь.. We recall that H, N, V are all sub-
algebras of L.
Let B = H®N.
Lemma 10.2 (i) В is a subalgebra of L.
(ii) N is an ideal of B.
(iii) B/N is isomorphic to H.
178
Irreducible modules for semisimple Lie algebras
Proof, (i) We have
[BB\ = [H + N,H + N\gH + N = B
since II. N are subalgebras and [HN] G. N.
(ii) [VB] = [V,V + #]cV.
(iii) B/N = (H + N)/N = H/H C\N = H
since //n,V = O, using Proposition 1.7.	□
Definition 10.3 Let hcJI', i.e. A be a linear map from H to C. We recall
that L has a basis
{ea, a e Ф ; ht, i = 1,...,/}.
We define
K,= E n(L)ea + EnW(VA(M).
аеФ+	i=l
Thus К f is the left ideal ofU(L) generated by the elements ea, a e Ф+, and
ht — X (hf) for i = 1,... ,
(We are as usual here embedding L in 1I(L).) We also define
M(A) = U(L)/7Ca.
Л7( A) is a left H(L)-module called the Verma module determined by A. It is
our aim in this section to describe some of the properties of M(A).
We note that the elements ea, a e Ф+, and ht — A (/i;) for i = 1,... ,/ all lie
in 11(B). We define
K[= E П(в)еа+Е11(в)(й;-А(о
аеФ+	i=l
to be the left ideal of 11(B) generated by these elements. Let
ф+^...../U-
Then the set
/ii,... , /i;,	,... , epN
is a basis of B. It follows from the PBW basis theorem that the elements
III... hf e‘i .. ,e‘o st > 0 t; > 0
form a basis of 11(B).
10.1 Verma modules
179
Proposition 10.4 (i) dim 11(B)/A") = 1.
(ii) The elements
(h1-X(h1^...(hl-X(hl)rel...e^N
with st > 0, t; > 0, excluding the element with si = ti = 0 for all i, form a
basis for Kf
Proof It is not difficult to see that the elements
(^-лсмг ...(^-лсмг
with st > 0, t; > 0 also form a basis for 11(B). This can be seen, for example,
by defining a partial ordering on the basis elements If ... If e!f ...	. We
say that	is lower than If ...If e‘f ... e‘f if s) < .jj,... ,
s[ < sh t\=f,... ,t'N = tN. Then there are only a finite number of basis
elements lower than a given one, and the element
is the sum of if ... if e‘f ... ef with a linear combination of strictly lower
basis elements in the partial order. An induction argument on the partial order
will then show that the elements
(/ll-A(/ll)r...(/l;-A(/l;))s,411---eX
st > 0, t; > 0 span 11(B) and are linearly independent. Now all these elements
clearly lie in Kf with the exception of the element with st = 0, t; = 0 for all i.
This is the unit element 1. However, 1 does not lie in Kf, as the following
argument shows.
Consider the representation A of H mapping /i; to A (/i;) for i = 1,... , /.
Since B/N is isomorphic to H there is a 1-dimensional representation A of
В with N in the kernel agreeing with the above representation on B/N = H.
This in turn gives a 1-dimensional representation p of 11(B) under which
ea 0 a e Ф+
/i;^A(/i;) z = l,...,/
1 -+1
Now kerp is a 2-sided ideal of 11(B) containing ea, a e Ф+ and ht — A (/i;) so
containing Kf Thus we have
A'/ckerp I 0kerp
hence 1 fKf.
180
Irreducible modules for semisimple Lie algebras
It follows that the elements
for Si > 0, t; > 0, excluding 1, form a basis of K'A and that
dim 11(B)/7^=1.
Proposition 10.5 II (N~) = O.
Proof. We are here regarding II (/V) as a subalgebra of 1I(L) as in
Lemma 10.1. Now we have
L = N~®B.
Regarding 11(B) as a subalgebra of II (L) also we assert that
1I(L) = II (ДТ) 11(B),
i.e. each element of 1I(L) is a finite sum ZTyy, with xt e II (N) , yt e 11(B).
This follows from the PBW basis theorem, choosing bases for N and for В
and combining them to give a basis of L. We then have
K,= E nw^+bwft-AW)
аеФ+	i=l
I
= e wwk+ewwxvao
аеФ+	i=l
= 11 (AT) ( E П(В)С„ + Еад(А;-А(М) =U(r)^.
\аеФ+	i=l	/
It follows that each element of К f is a linear combination of terms of form
...fi (hl - л (M)S1 ...(h-x (hl)y ... e‘i
where fa = e_a, a e Ф+, and r; > 0, st > 0, t; > 0 with (s-j, ... sh L, ... tN) Ф
(0,... 0, 0... 0). No non-zero element of II (/V ) can be a linear combination
of such terms by the PBW basis theorem. Hence
/ГАП11(ЛТ) = С>.	□
Let mA e M(A) be defined by mA = 1 + KA. Thus 1 maps to mA under the
natural homomorphism
1I(L)^ 1I(L)/ATa = M(A).
10.1 Verma modules
181
Theorem 10.6 (i) Each element of A/( A) is uniquely expressible in the form
um, for some и e II (N).
(ii) The elements f^ ... f^ m,. for all r; > 0 form a basis for A/( A).
Proof Each element of II (L) has form и -1 for some и e II (L). Thus each
element of A/(A) = 1I(L)///A has form итА for some и e 1I(L).
Now/ft,...	,hh	, epN are a basis of L so the ele-
ments
f'N ifl ifl Jl plN
Jh-'-Jh "i nl epl---ei3N
ri > 0, st > 0, t; > 0 form a basis of 1I(L) by the PBW basis theorem. Thus и
is a linear combination of such elements, and итл is a linear combination of
elements
/J!-” fZ
Now this element is 0 if any t; is positive. Suppose then that all t; = 0. Then
h-i ... hfm^ = ymf for some у e C
since /i;mA = A (/i;) mA. Thus итА is a linear combination of elements of form
Thus elements of this form for r;>0 span A/(A). They are also linearly
independent. For if we have
E
ri,...,rN
with Ёг r e C then it follows that
Vi:,. r E1 ...feN еТСАПИ(ЛТ)-
Hence this element is 0 by Proposition 10.5. Thus each £	=0 by the
PBW basis theorem for II (N ).
Thus the elements /J1 .. .f^m^ for >0,... , rN >0 form a basis for
A/(A). It follows that each element of A/(A) is uniquely expressible in the
form umK for и e II (N).	□
We now regard A/(A) as an //-module. For each 1-dimensional represen-
tation pc of H we define
M(A))1 = {meM(A) xm = p.(x)m for all x e H}.
A/(A)M is a subspace of A/(A).
182
Irreducible modules for semisimple Lie algebras
Theorem 10.7 (i) М(А) = ф/хеЯ.М(А)/х.
(ii) M(A)M 0 if and only if A — p. is a sum of positive roots.
(iii) dim M(A)M = ^J(A —/z), the number of ways of expressing X — p as a
sum of positive roots.
(Л — /л) is the number of vectors (rx,... , rN) with rteZ, r; > 0 such
that
A-ju = r1/31H----\-rN/3N.
Proof. We know from Theorem 10.6 that the elements fp ...f^m^ with
r; > 0 form a basis for M(A). We show that
Vft	=	------rNfIN) (x)f^ ...f^np for all xeII.
We prove this by induction on p 4---h rN, the result being clear if all r; = 0.
So suppose not all r; are 0 and let i be the least integer with r; > 0. Then we
have
xfp    ^т^=^.х^	   fpNm^
It follows that
= (A - oft-------Lftv) Wf,    .Грт>
as required.
This implies that frp\ ...f^rrpe M(A)g where p = A - p/31----rN/3N.
Since these elements form a basis of M(A) we see that
M(A) = £M(A)M.
We now show this sum is direct. To see this we must show that if a finite
sum 22 ’2 is 0 with e A/(A);7, then each is 0.
It is sufficient to show that
М(А)мп(М(А)М1+... + М(А)л) = О
where the elements p, pr,... ,	are all distinct.
Let v lie in this intersection. Then we have
v = v^+... + v^
where v e M(A)M, v^. e A/(A);7. Thus
(x — /x(x))v = 0
(x-pfx')') uM.=0
10.1 Verma modules
183
for all x e H. Hence
(x -	(x))... (x - pk(x)) (nMi 4-h	= 0,
that is (x — /Zj(x)) ... (x — pk(x)) v = 0. Since the vector space H over C
cannot be expressed as the union of finitely many proper subspaces we can
find x e H such that
Thus the polynomials
t - (x), (t -	(x)) (t - m2(x))... (t - ^(x))
in C[t] for this element x are coprime. Thus there exist polynomials
p(t), q(t) e C[t] such that
- ^(x)) + q(t) (t - рг (x)) ... (t- pk(x)) = 1.
Thus we have
p(x)(x - ^(x)) + q(x) (x -	(x)) ... (x- pk(x)) = 1.
It follows that
/?(х)(х —/х(х))и+ <?(x) (x — /x1(x)) ... (x — pk(x)) v=v.
The above conditions show that the left-hand side is zero, hence we have
v = 0. Thus
м(л)=ем(л)м.
Let Л = {p e H* ; A — p is a sum of positive roots}. For each A e Л let
be the subspace of M(A) spanned by the basis vectors
with A —----------rN/3N = p. Since these vectors for all such p form a
basis of M(A) we have
М(А) = ф^.
/леЛ
On the other hand we know that
^cM(A)g
and Л/(А) =	''W(A);7. It follows that M(A)M = for all p e Л, and that
MA)(J = О for all peH* with p g Л.
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Irreducible modules for semisimple Lie algebras
Thus we have dim /V/(A);7 = dimVlz = the number of vectors (rx,... , rN)
with r; e Z, r; > 0 such that
-------rN/3N = p.
This gives dim Л/(А)М = (A — /x) as required.	□
Definition 10.8 /z e H* is called a weight of M(X) if Л/f A)?J 7^ O, and Л/(А)М
is called the weight space of A/(A) with weight p.
We note that since Л/(А) = ФМЛ/(А)М an element m of A/(A) satisfying
the condition that, for all xeH, (x —/z(x))Z:m = 0 for some k>0 can have
no non-zero component in any M(Af. for r^/z, and must therefore lie in
A/(A);7. Thus we have
M(X.) = {me M(A) ; for each x e H there exists k such that
(x — pfxffm = 0}.
This shows that our definitions of weight and weight space here in the context
of //-modules are compatible with the definitions in Chapter 2 in the context
of representations of nilpotent Lie algebras.
Theorem 10.7 asserts that a Verma module is the direct sum of its weight
spaces. There are infinitely many weights, but each weight space is finite
dimensional.
We now proceed to another very important property of Verma modules.
Theorem 10.9 A/(A) has a unique maximal submodule.
Proof. Let V be a H(L)-submodule of A/(A) with V Let ve V. By
Theorem 10.7 we have
i
summed over a finite set of distinct weights /z;. We aim to show that each
v„ lies in V also. We have
P-i
xv„ =р.;(х)с„ xeH.
P-i • i \
Hence
П (x “ Pj (*)) v = П (x “ Pj (*)) vr.= П (pi (*) “ Pj (*)) 
j	j	j
j&	j&	j&
10.1 Verma modules
185
Since H is not the union of finitely many proper subspaces we can find x e H
with /z;(x) f p.fx) for all j i. For such an x we have
~-v
j
Hi
and
j
Hi
It follows that v e V.
We now define = УПМ(А)Д. We have shown that V = Since
we know that M(A) = ф^ША^ it follows that the sum in V must be direct,
that is
v=evM.
м
Thus every submodule V of M(A) is also the direct sum of its weight spaces.
Now VA = O. For if VA ф О then VA = М(А)Л since dimМ(А)Л = 1. This would
imply that mA e V. But then
,M(A) = lU7.)ffl,cV'
so V = M(X), a contradiction. Thus VA = O and we have
/z/Л
Thus every proper submodule V of M(A) lies in the subspace ф * M(XV
of codimension 1 in M(A). Let ./(A) be the sum of all the proper submodules
of M(A). ./(A) lies in the above subspace of codimension 1, so is a proper
submodule of M(A). Thus ./(A) is the unique maximal submodule of M(A),
since it contains all proper submodules of M(A).	□
Definition 10.10 Let A, fie H*. In view of Theorem 10.7 it is natural to make
the following definition.
We say that X > p. if X — p. i s a sum of positive roots. This is a partial order
on HL
Theorem 10.7 shows that the weights of M(X) are precisely the fieH* with
p. < X. Thus X is the highest weight of M(X~) with respect to this partial order.
VI (X) is called the Verma module with highest weight X.
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Irreducible modules for semisimple Lie algebras
We also define L(A) = A/(A)/J(A). Since J(A) is a maximal submodule
of A/(A), L(A) is an irreducible II (L)-module. In subsequent sections of this
chapter we shall determine under what circumstances the irreducible module
L(A) is finite dimensional. We note that A is a weight of L(A), since J(A)a =
O. Thus dimL(A)A = 1 and A is the highest weight of L(A).
10.2 Finite dimensional irreducible modules
Now let V be any finite dimensional irreducible L-module where, as usual in
this chapter, L is a finite dimensional semisimple Lie algebra over C. Let H
be a Cartan subalgebra of L and
[ea, a e Ф+; ht, i= 1,... , / ; fa, a e Ф+}
be a basis of L adapted to H. We may regard V as an //-module. Now H is
abelian, so in particular nilpotent, thus we may apply the representation theory
of nilpotent Lie algebras developed in Chapter 2. By Theorem 2.9 we have
v=evA
л
where VA = {ueV ; for each x eH there exists k such that (.v — A(r))/ i> = 0}.
We also know from Chapter 2 that each non-zero VA contains a non-zero
vector v such that
xu = A(x)u for all xe//.
We shall show that in our present situation the weight spaces VA can be
defined more simply.
Proposition 10.11 Let WA = { v e V; xv = A (x) v for all x e H}. Then WA = VA.
Proof. It is clear that WA c VA and that WA О whenever VA О. Let W =
J2aWa. Since V = ®AVA and WA G VA we see that W = ®AWA. We shall
show that W is a submodule of V. To see this it is sufficient to show that
h/W, eaw, faw lie in W for all w e WA, all i = 1,... , / and all a e Ф+. Now
we have
htw= A (/i;) w e W
x(eaw) = ea(xw) + a(x)eaw
= X(x)eaw + a(x)eaw
= (A + a)(x)effw.
10.2 Finite dimensional irreducible modules
187
Hence e„weWi+(IcW. Similarly we have /aweWA_acW. Thus W is a
II (L)-submodule of V. Since IT 7^ О and V is irreducible we have W = V. It
follows that WA = VA for each XeH*.	□
Thus the irreducible module V is the direct sum of its weight spaces VA
and VA is the set of all v e V such that xv = X(x)v for all xeH.
We now consider the set of all weights A for V, that is the set of all XeH*
for which VA ф О. This is a finite set, so will contain at least one weight
maximal in the partial order >- defined in Definition 10.10. Let A be such a
weight of V. If fi >- A and fifX then fi is not a weight of V.
We may choose uA e VA with uA 7^0.
Proposition 10.12 (i) xvx = A(x)ua for all xeH.
(ii)	e„vx =0 for all a e Ф+.
(iii)	V = U(AT)va
(iv)	A is the highest weight of V.
Proof. Condition (i) is clear. We have
X M = ea C™A) +
for all xeH. Now if eauA^0 this implies that A + a is a weight of V. But
A + a >- A so this cannot be the case. Hence eavx = 0 for a e Ф+.
Now V = 1I(L)va since	7^ 0 and V is an irreducible II (L)-module. Thus
each element of V is a linear combination of elements of the form
fgW-fZ hsf...hsf e^...elvx.
This element is 0 unless all t; are 0. In that case it is a scalar multiple of
Hence V = II (N) vx.
Finally we have
x (/Ji • • • /X = (A - ri/?i----rN/3N) (x)/J; ... ff vA
as in the proof of Theorem 10.7 ; thus all weights of V have form
ju = A-r1/31------rN/3N.
Thus A >- fi for all weights fi of V.	□
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Irreducible modules for semisimple Lie algebras
It follows from Proposition 10.12 that the set of weights of V has a unique
maximal element A with respect to the partial order >-.
We now compare the finite dimensional module V with the Verma module
M(A).
Proposition 10.13 There exists a surjective homomorphism в : Л/(Л) V
of U(L)-modules such that в (mf) = vA.
Proof We recall from Theorem 10.6 that each element of M(A) is uniquely
expressible in the form umA with и e II (/V). We define a linear map
в : М(Л) -+ V
by 0(umA) = uvA nellf/V). Then в is surjective by Proposi-
tion 10.12 (iii). We must check that в is a homomorphism of II (L)-modules.
Thus we must show
0{yumA) = yuvA for ally e II(L).
By the PBW basis theorem we know that the element у и of II (L) can be
written as a finite sum
VII = У^а :Ь:С:
where a; e II (V ), bt e II (//), c; e 1I(V). Thus
yumA = У) a;b;c;mA.
i
Now bicimA = tfmA for some eC. Hence
yumA =	тл-
Since £;a; e II (N ) we have
0 {yum Aj = Q	^at j mA j =	^at j vA.
On the other hand we have
vA=
since Ь;с;ил = Gyy. Hence 0{yumf) = yuvA for all yell(L). Thus в is a
homomorphism of II(L)-modules.	□
10.2 Finite dimensional irreducible modules
189
Corollary 10.14 V is isomorphic to L(X).
Proof. Since V is irreducible the kernel of 0 must be a maximal submodule
of M(A). But M(A) has a unique maximal submodule ./(A), by Theorem 10.9.
Thus kerd = ./(A). Hence V is isomorphic to M(A)/J(A) = L(A).	□
Thus we have seen that every finite dimensional irreducible L-module is
isomorphic to one of the irreducible modules L(A) obtained as irreducible
quotients of Verma modules. However, we shall see that by no means all the
L(A) are finite dimensional.
Proposition 10.15 Suppose L(Xi) is finite dimensional. Then A (/i;) is a non-
negative integer for each i=l,... , I.
Proof. Let uA be a highest weight vector of L(A), that is wAe£(A)A and
uA^0. As in Section 7.1 we shall choose elements eteL ,fEL such
that [e;/;] = /i;. We consider the sequence of elements
Lo //Lo	•••
of L(A). We have
x (fivx) = (A - kat) (*) (ftvx)
for all x e H. Thus we have
/;иле£(А)л_а_, ftvK eL(A)A_2a_
and so on. Now L(A), being finite dimensional, has only finitely many distinct
weights. Thus there exists p e Z, p > 0 such that
X*Li#0 forA</?
/Г^л = 0-
Let M = CuA + C/;uA -I--h C/f uA.This sum is direct since uA, /;uA,... , /fr’A
all lie in different weight spaces. We show that M is a submodule with
respect to the subalgebra (e;, hh f) of L. It is clear from the definitions that
htM с M and ftM с M. We shall show that etM с M also.
We verify that e M by induction on к. If к = 0 we have e;uA = 0. If
к > 0 we have
eJiV^feJ^v. + hJ^v,.
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Irreducible modules for semisimple Lie algebras
Now eJf^v^eM by induction, hence е;/*илеМ also. Thus M is an
{et, ht, -submodule. We consider the trace of ht on M. This can be
calculated in two ways. On the one hand we have
traced = trace,, [e;/;] = trace ,; (e;/; -/;e;) = 0.
On the other hand we have
traceM/i; = A (/i;) + (A — a;) (/i;) -I-h (A —pal) (hf)
= (/’+l)A(/i;) -p(p+1)
since a; (/i;) = 2. Hence
traceM/i; = (p + 1)A (hf)-p(p + 1).
It follows that
(/7+l)(A(/l;)-/7)=0,
that is A (/i;) = p. Thus A (/i;) e Z and A (/i;) >0.	□
The condition A (/i;) e Z, A (/i;) > 0 for all i = 1,... , / is therefore necessary
for L(A) to be finite dimensional. In the next section we shall show that this
condition is also sufficient.
10.3 The finite dimensionality criterion
We consider the set of A e H* such that A (/i;) eZ, A (/i;) > 0 for i = 1,... , /.
Definition 10.16 Let ы: e H* be the element satisfying ы: (ht) = l,a>i (hj) = 0
if j i. The elements w,,... , w, e // are called the fundamental weights.
We note that co,,... , a>l are linearly independent, since this is true of
hx,... , /i; e H. Thus co,,... , a>l form a basis of H*. Let X = {n1a>1 -I-
+ и;<У; ; и,,... , n;eZ}. X is a free abelian subgroup of H* with basis
the set of fundamental weights and is called the lattice of integral
weights or, briefly, the weight lattice. It is clear that an element A e H*
lies in X if and only if A(/i;)eZ for z=l,... ,/. Let А'+ = {и1<у1Ч--------
+ nl(i)l ; w;eZ, и;>0 for i = 1,...,/}. X+ is called the set of dominant
integral weights.
An element A e H* lies in X+ if and only if A (hf) e Z and A (hf) > 0 for
/=1,... ,/.
We have seen, therefore, that if L(A) is finite dimensional then A is a
dominant integral weight, and wish to prove the converse.
10.3 The finite dimensionality criterion
191
We shall first explore the connection between the fundamental weights
Wj,... , a>l and the fundamental roots ax,... , a;.
Proposition 10.17 a Y* . A. <•> .. Thus the matrix expressing the fundamen-
tal roots as linear combinations of the fundamental weights is the transpose
of the Cartan matrix.
Proof. Since Wj,... , <y; are a basis for H* there exist c^ e C such that
j
Then we have
ai (hj) = cij-
Hence
Thus we obtain a,- = V; A w;.	□
In particular we note that all the fundamental roots are integral combinations
of the fundamental weights, so lie in the weight lattice X. However, it is not
true that the fundamental weights are, in general, integral combinations of the
fundamental roots. We have
j
For example, when L has type Ax we have
a1=2w1,	a>1 = ^a1.
When L has type A2 we have
aj = 2(i)1 — (i)2
a2 = —a>,+2a>2
and so
W1 = +
«2= |“1 + |“2-
192
Irreducible modules for semisimple Lie algebras
We show that in general the coefficients expressing the w, in terms of the
aj are non-negative rational numbers.
Proposition 10.18 (i) (<y;, a>^ > 0 for all i, j.
(ii) w, is a non-negative rational combination of c^,... , a;.
(iii) The coefficients of the inverse A^1 of the Cartan matrix are non-negative
rational numbers.
Proof. We shall show that condition (i) implies the others, and prove (i) in
a subsequent lemma. Since the coefficients of A are integers the coefficients
of A1 are rational numbers. We show they are all non-negative.
We know that <y;	=8tj. This condition is equivalent to
where (,) is now the Killing form on II as defined in Section 5.1. Thus we
have
,	!	(“i,
а\ = 8и—-—— .
\ I’ JI	IJ	2
Now let <y; = V; c,;»;. Then we have
1	^-“'J lJ J
/	\	/	\	aj)
\шсшй = си\аршй = сц—^-
Thus
since (<y;,	> 0 and (ctj, > 0.
We must now show that (<y;,	> 0. This will follow from the fact that
and the fact that
by Proposition 5.4. The following lemma on Euclidean spaces will give us
what we need.
10.3 The finite dimensionality criterion
193
Lemma 10.19 Let V be an n-dimensional Euclidean space with a basis
, vn satisfying <0for all ifij. Let ... ,wnbe the dual basis
of V uniquely determined by the conditions (vt, w^ = 8tj. Then (w;, >0for
all i, j.
Proof. We use induction on n. If и=1 there is nothing to prove. So
assume n > 1 and let U be the (и — 1)-dimensional subspace of V spanned by
Uj,... , u„_1. Let wf ... , it/ -! be the dual basis of , vn_1 in U. Thus
we have
(vcw'j} = 8ij i, j=l,... ,n — l.
Let I! = {и e V ; (v, u) =0 for all и e U}. Then dim I! = 1 and I! is
the subspace of V spanned by w„. We see also that w; — wfiUL for
i = 1,... , n — 1. Thus we have
w; = w. + A;wn for some A; e IR,
for i = 1,... , n — 1. Taking the scalar product with vn we have
0= (vn, W;) +A;
hence A; = — (vn, uf). We wish to determine the sign of A;. By induction we
know (u/, uQ > 0 for i, j = 1,... , n — 1. This implies that wl is a non-negative
combination of ... , Since
{vn,vfi<0, ...(v„,
we see that (v„, w-) < 0 and so A; > 0. Hence for i, j=l,... , n — 1
(wi, Wj'j = (w. + A;wn, w'j + Xj
= (w., w'j'j + XiXj (wn, wfi >0
since (wl, w'J>0, A;>0, A;>0, (w„, w„)>0. It remains to show that
{wt, wfi > 0 for i = 1,... , n — 1. We have
(Щ, wfi = {w'i,wfi+Xi{wn, wfi > 0
since (w., wn} =0, A; >0, {wn, wn} > 0.	□
By applying this lemma in the case where vt = (^aia , wt = а>ь we deduce
that (<y;, and so Proposition 10.18 is proved.	□
We now turn to the main theorem of the present section.
194
Irreducible modules for semisimple Lie algebras
Theorem 10.20 Suppose XeH* is dominant and integral, that is A e X.
Then the irreducible L-module L(X) is finite dimensional.
Proof. We have L(A) = M(A)/./(A). We know from Theorem 10.7 that M(A)
is the direct sum of its weight spaces and from Theorem 10.9 that any
submodule of M(A) is also the direct sum of its weight spaces. This applies
in particular to ./(A). It follows that L(A) = M(A)/J(A) is also the direct sum
of its weight spaces. In fact the same proof as given in Theorem 10.9 shows
that any //-submodule of /.(A) is the direct sum of its weight spaces.
Let uA be a highest weight vector of L(A). Thus uA e L(A)a and wA^0. We
consider the sequence of elements
/X,	...
We wish to show that terms in this sequence eventually become zero. In fact
we show
ft‘ uA = 0 where kt = A (/i;) + 1.
Let mA be a highest weight vector of the Verma module M(A) such that
mA + J(A) = ua. We consider the submodule H(L)/*‘mA of M(A). As usual
we choose elements e; eL„ , f eL „ such that [e;= ht. We have
eifi=fiei + hi
= fiei + 2fi(hi-l)
and inductively we obtain
etf? =fi‘et + nf"~1 (ht —(ji-lf).
Thus we have
eifiims=fiieims + kifii^i (hi -fki-\f)mK = Q
since e;mA = 0 and hjm^ = A (/i;) mA = (A; — 1) mA. Also if ji then
Thus ejfi'm^ = 0 for all j = 1,... , It follows that eafj‘ т^ = 0 for all a e
Ф+, since ex,... , e; generate N, by Proposition 7.7. We also know that
hjfi’m^tX-kitti) (h^fi'm^
since f^m^ is a weight vector with weight A — A;a;.
10.3 The finite dimensionality criterion
195
We now consider an arbitrary basis vector of 1I(L) applied to f^‘mA:
fri fr': ir' i+fi ...fiN (fkim A
J Pi • • JpNni ni ePi epN \ Ji ms J
is zero unless all t; = 0, in which case it will be a scalar multiple of
This shows that
П(Т)/*тА = П(^)/*™А.
Now U(2V—)y*' is a proper subspace of II (/V ) since kt = A (/i;)+ 1 > 0.
It follows from Theorem 10.6 that П(Л^)/*‘тА is a proper subspace of
M(A). Hence H(L)/*‘mA is a proper submodule of M(A). It therefore lies in
the unique maximal submodule ./(A) of M(A). Hence fiimA e fifi) and this
implies J*‘TA = 0.
Now let К be the finite dimensional subspace of L(A) given by
K = CvA + CfivA + --- + Cf^1vA.
We clearly have UK с К since each /?vA is a weight vector. We have J, К с К
since J*‘ uA = 0. We also have etK cK since
ej?= /еЛ + п/Г1 -(n- 1)) vA
=/i (A (ip) — (/i — l )) fi." 1 t'A.
Thus К is a submodule of L(A) for the subalgebra (e;, II, fi) of L of dimension
1 + 2. We shall consider non-zero finite dimensional {et, H, j))-submodules
of L(A). К is such a submodule. If U is any finite dimensional {et,H, f))-
submodule of L(A) we claim that LU is also. For LU is finite dimensional
and we have, for и e U, z&L, ye (e;, H, j))
y(zu) = z(yu) + [yz]« e LU
since у и e U and [yz] e L.
Let V be the sum of all finite dimensional (e;, II, j))-submodules of L(A).
Then V fi О since V contains К. V is an L-submodule of L(A), since if U is
a finite dimensional (e;, H, j))-submodule of L(A) so is LU. Since L(A) is
an irreducible L-module we see that V = L(A). Thus L(A) is a sum of finite
dimensional (e;, H, j))-submodules.
Now each such finite dimensional {et, H, /;)-submodule of L(A) is the
direct sum of its weight spaces, as observed above. Thus we may choose a
196
Irreducible modules for semisimple Lie algebras
basis for it consisting of weight vectors, that is vectors spanning 1-dimensional
//-modules. Hence we can find a basis of L(A) consisting of weight vectors,
each of which lies in some finite dimensional (e;, H, /;)-submodule of L(A).
This fact will give useful information about the set of weights of L(A).
Let Л be the set of all weights of L(A). Thus /х e Л if and only if L(A)M O.
Of course all weights of L(A) are weights of A/(A) so have form
A — r1/31	 — rN/3N
by Theorem 10.7. In particular Л с X, since A e X and each /3; e X by Theo-
rem 10.7. Let p. be any element of Л. Then there is a weight vector e L(A)
for p. such that lies in a finite dimensional {et, H, f)-submodule U of
L(A).
We consider the vectors
... , fv v e2u„,...
These vectors all lie in U and have weights
... , /z — 2a;, /z — at, /л, /л + at, /х + 2а;,...
Since dim U is finite U has only finitely many weights so there exist p, q > 0
such that
forQ<n<p,	/f+1uM=0
e"uM^0 for0<w<<?,	е’+1гу = 0.
Let V = Cf!’vt,-\--hC/;u„+Cu„+Ce;u„-I-----------hCe’u„. Then V is a
(et, H, /;)-submodule of /.(A). This follows readily from the relations
eifi‘=	(ht —(n—1))
fte" = e’if - ne^ (ht + (и - 1))
and the fact that ff’+1v/J.=Q, e^+1v/J.=Q. We consider the trace of ht on V.
On the one hand we have
tracey/i; = (jx (hj-paf (/i;))d-l-ju. (/лг)Н-h(ju, (hi) + qai (ht))
( , , 14 n 4 , (?(?+l) P(j>+W\ n 4
= (p+q+i)g(M+ (—---------------------2— J ‘
= (p+q+i)p(hi)+(q-p)(p+q+i)
10.3 The finite dimensionality criterion
197
since a; (/i;) = 2. On the other hand we have
tracey/i; = tracey [e;/;] = tracey (e;/; — /;e;) = 0.
Hence p (/i;) =p — q.
We may make an important interpretation of this result in terms of the
Weyl group W. We recall from Section 5.2 that W is the group of linear
transformations of /С generated by the reflections sa with respect to the roots
a e Ф. We write st = sa. and recall from Theorem 5.13 that W is generated
by Sp ... , S[. We have
z x	o	Z; X
= fJL-2---------at = fjL — fjL (Д) at
since
/ 2h'a
p(ht) = p 777—
2а‘ \_2<'Q'i’^

Choosing p as above, where /z (/i;) =p — q, we have
M - (p - <?)“/ = M + (q - Р)а,-
Now p + (q — p)at is one of the weights in the list
p — pat,... , p — at, p, p + at,    , p + qat
of weights of V. Thus we have shown that if p is any weight of V then st (p)
is a weight of L(A) also. Since 57,... , sl generate W it follows that for any
p e Л and any we W we have w(/z) e Л also. Thus the set of weights Л of
L(p) is invariant under the Weyl group. We recall also from Proposition 5.8
that W is finite.
We now claim that for each p e Л there exists w e W such that w(p) e X.
To see this we consider the finite set of weights {w(/z) ; weW} and pick
one maximal in the partial order > on 77*. Let v be such a weight. Then
sfiv) = v — v (/i;) a;.
We know that veX since AcX, hence r(/i;)eZ. If p(/i;)<0 we would
have sfiy) > v, a contradiction to the choice of v. Hence v (/i;) > 0. This holds
for all i = 1,... , / and so v e X+. Thus each weight in Л has a W-transform
which lies in X.
We shall now concentrate on the set ЛПХ+. For any weight i'EATiX
we have i' < X We express A and v in terms of the fundamental roots at.
Since A, v e X+ these weights are non-negative integral combinations of the
fundamental weights Wj,... , <y;. By Proposition 10.18 they are therefore
198
Irreducible modules for semisimple Lie algebras
non-negative rational combinations of the fundamental roots ar,... , at. Thus
we have
i
X = '^2iqiai	Qt>0
1=1
I
v = '^Jl'iai <? E Q <? > 0.
i=l
The condition A > v means simply that — <? is a non-negative integer for
each i = 1,... , /. Now given q, there are only finitely many q. such that <? > 0
and q, — q. is a non-negative integer. Thus given A e X there are only finitely
many I’cX such that p < A. Thus ATY is finite. Since every element of
Л can be transformed by an element of W into one of Л A X and since W
is finite we see that Л is finite. Thus L(A) has only finitely many weights.
However, each weight space L(A)M of L(A) is finite dimensional, since
dim L(A)M < dim A7( A);7
and dim /W(A);7 is finite by Theorem 10.7. Thus we have
L(A) = eL(A)M
with finitely many summands, each finite dimensional. Hence L(A) is finite
dimensional.	□
We conclude by summarising the main ideas in this somewhat lengthy
proof. In order to show that L(A) is finite dimensional it is sufficient to
show that L(A) has only finitely many weights, since each weight space is
known to be finite dimensional. This can be proved if the set of weights
is known to be invariant under the Weyl group, since each weight will be
W-equivalent to one in X, and there are only finitely many elements of
X lower than A in the partial ordering. It is therefore necessary to show
that, for any weight pc of L(X),sfp.) is a weight also. This can be shown
provided we know that any weight pc comes from a weight vector lying in a
finite dimensional {et,H, f) -submodule of L(A). We therefore have to show
that L(A) is the sum of its finite dimensional {et, H, f)-submodules. This
comes from the irreducibility of L(A) provided L(A) has a non-zero finite
dimensional (e;, H, f)-submodule. The existence of such a submodule К is
proved above.
We have now completed the determination of the finite dimensional irre-
ducible L-modules where L is a finite dimensional semisimple Lie algebra
over C.
10.3 The finite dimensionality criterion
199
Theorem 10.21 Let L be a finite dimensional semisimple Lie algebra over
C. Then the finite dimensional irreducible L-modules are the modules L(X)
for A e X+. These modules are pairwise non-isomorphic.
Proof. The fact that any finite dimensional irreducible L-module is isomor-
phic to L(A) for some A is proved in Corollary 10.14. The fact that A must lie
in X+ is proved in Proposition 10.15. The fact that L(A) is finite dimensional
when A e X is proved in Theorem 10.20. The fact that the L(A) are pairwise
non-isomorphic follows from the fact that A is the highest weight of L(A).
Thus if A^/z, L(A) and L(/z) have different highest weights so cannot be
isomorphic.	□
A property of L(A) which will be very useful subsequently is given by the
following proposition.
Proposition 10.22 Let A e X and w e W. Then
dimL(A)/x = dimL(A)u,(/x).
Proof. Since W is generated by the fundamental reflections ,... , it is
sufficient to show that
dimL(A)M = dimL(A)s,w.
We recall from Section 7.5 that there is an automorphism d, of L such that
OfH) = H and 0;(/i) = sfli) for all h e H. We define an L-module L(A) which
is the same space L(A) as before but with a different L-action. For DeL(A)
we have
xv = Offilv
where v is the corresponding element of L(A). It is clear that this action
makes L(A) into an L-module.
Now let veL(A)M. For xeH we have
xv = 0fx)v = sfx)v = p (sfx)) v
= Mp)) СФ-
Thus veL(A) ( j. A similar argument shows that if veL(A) ( ) then ve
Т(Х)^. Hence
dimL(A)s,w=dimL(A)M.
200
Irreducible modules for semisimple Lie algebras
Now L(A) is an irreducible L-module, since L(A) is irreducible. For if
M were a submodule of L(A) the corresponding subspace M would be a
submodule of L(A). Let Л be the set of weights of L(A). Then we have
seen that ^(A) is the set of weights of L(A). But we showed in the proof
of Theorem 10.20 that w(A) = A for all w e W. Hence the set of weights of
L(A) is also A. In particular the highest weight of L(A) is A. Thus L(A) is a
finite dimensional irreducible L-module with highest weight A. Hence L(A)
is isomorphic to L(A) by Theorem 10.21. Thus we have
dim L(A)M = dim L(A)S.W = dim L(A)s.(/x).
Since each we W is a product of elements st we deduce that
diniL(A);7=diinL(A)„i;7l
as required.	□
11
Further properties of the universal
enveloping algebra
11.1 Relations between the enveloping algebra
and the symmetric algebra
Let L be any finite dimensional Lie algebra over C. Let T be the tensor
algebra of L. We recall that the enveloping algebra II (L) is defined by
U(L) = r/J
where J is the 2-sided ideal of T generated by all elements of the form
_>'0y - y 0r - [xy]
for x, у e L. The symmetric algebra S(L) is defined by
5(L) = T/I
where I is the 2-sided ideal of T generated by all elements of the form
x0y—y0x
for x, у e L.
S(L) is isomorphic, as C-algebra, to the polynomial ring C[zi,... , z„]
where n = dimL. We have
к
where Sk(L) = (Tk+l) /I.
Sk(L) is the set of homogeneous elements of S(L) of degree k. In particular
we have an isomorphism
L=T1^S\L)
thus L can be regarded as a subspace of S(L).
201
202
Further properties of the universal enveloping algebra
I f л ,,... , xn are a basis of L then the elements
form a basis of S(L).
We now explain how S(L) can be regarded as a left L-module. In the first
place L is an L-module under the adjoint action. Then T may be made into
an L-module by means of the action
у ®	® Xt) = [yx; ] ®	®	® xt 4---h xt ®	® Xt ® [yx; ].
The ideal I of T is then a submodule, and so S(L) = T/I can be given the
structure of a left L-module. We have
У (.К; . . . Л ( ) = [ VX( ] X, ... X; -I-h X ... X Г VX ]
J \ И lkJ [y ?! J l2 Ik '	?1 lk-l 1У tyj
where у eL and the xt are basis vectors of L. We note that each Sk(L) is an
L-submodule of S(L).
Similarly 1I(L) = T/J can be made into a left L-module. For the ideal J of
T is also a submodule since, for a,b eL, we have
y(a ®b — b®a — [ab]) = [ya] ® b + a ® [y b] — [yL>] ® a — b ® [ya] — [y [ab] ]
= [ya] ® Ь — b ® [ya] — [[ya]b] + a® [yb]
-[yb]®a-a[yb]
since
[y[ab]] = [[ya]b] + [a[yb]].
We shall find it useful to compare the enveloping algebra II (L) with the
symmetric algebra 5(L). We first compare their C-algebra structures. Of
course they need not be isomorphic as C-algebras since S(L) is commutative
whereas II (L) is in general non-commutative. However, there is a relation
between these two algebras: it is the relation between a filtered algebra and
the corresponding graded algebra.
A filtered algebra is an associative algebra A with a chain of subspaces
A0CЛ с A2 c •• •
such that U;A; = A and AtAj c Ai+j.
A graded algebra is an associative algebra A with a decomposition
A = Ао ф A2 ф A2 Ф * * *
into a direct sum of subspaces such that A,A; C Ai+j for all i, j.
11.1 Relations between enveloping and symmetric algebra
203
Given any filtered algebra we may obtain a corresponding graded algebra
as follows. Let А = ЦА; be a filtered algebra. We define vector spaces Bo,
В^Вг,... by
=	B1=A1/Ao, B2 = A2/A1,
and define the vector space В by
В = В0фВ1фВ2ф... .
We define a multiplication on В to make it into a graded algebra. It is
sufficient to define xy when xeBt, у eBj and to extend this multiplication by
linearity. Thus let xeAi/Ai_1,yeAj/Aj_1. Let x = At_x + at, y = Aj_x+ aj.
Then, for any pair of elements и e At_r, v e Aj_x we have
(u + at) (u + a) = uv + uat + atv + atat e A;,+a,a;.
Thus the coset in Ai+j/Ai+j_x containing the product of any element in x
with any element in у is the same. Thus we may without ambiguity define
xy^Bi+j by
ТУ = Л+у-1 + а>ау-
It is readily checked that this multiplication when extended by linearity makes
В into a graded algebra. В is called the associated graded algebra of the
filtered algebra A.
We may regard 1I(L) as a filtered algebra as follows. Let 11,(7.) be the
subspace of 11(7.) generated by all products axa2... aj for j < i, where ak e L.
We also define 1TO(22) = Cl. Then we have
Un;(L) = lI(L)
i
and
Moreover 1I;(L)1Ij(L) c 1I;+2(L). Thus 1I(L) is a filtered algebra. We con-
sider its associated graded algebra.
Proposition 11.1 The associated graded algebra of the filtered algebra II (L)
is isomorphic to
204
Further properties of the universal enveloping algebra
Proof. Let В = Bo ф Bx ф B2 ф • • • be the associated graded algebra of П(В).
We first observe that В is a commutative algebra. В is generated as an algebra
by 1 and Bb and Bx = П1(В)/П0(В). The natural map
В^П1(В)/П0(В)
is an isomorphism of vector spaces. For elements x, у e L we have
xy-yx= |.vv| in П(В).
Thus
(Uo(L) + x) (П0(В) + y> (По(Й + у) (По(b) + x)	mod П, (B).
Hence any two elements of Bx = П1(В)/П0(В) commute in B, where their
product lies in П2(В)/П1(В). It follows that В is a commutative algebra.
We now compare В with the symmetric algebra S(L). Let xr,... , xn be a
basis of L. Then it follows from the PBW basis theorem that the elements
ri + • • • + r„ < i
form a basis of П;(В). Moreover the elements II,, (B)+ .V|'... л)" with rx +
---\-rn = i form a basis for П;(В)/П;_1(В) = B;. Now we have
OWBR*!1 • ••<") =n;+>_1(L) + x(1 ...x’fxl1 ...xs~.
This is equal to
u,+H(L)+^'.---es"
since multiplication in В is commutative. This shows that the linear map
S(L) В defined by
extends to an isomorphism of algebras. Thus the associated graded algebra
of П(В) is isomorphic to S(L~).	□
We now wish to compare the enveloping algebra П(В) and the symmetric
algebra S(L) as left В-modules. We shall show that they are isomorphic as
В-modules. In order to do so we shall first find a complement to n;_j(B) in
П;(В).
We have Tl = L®- • -®L (z factors). The symmetric group S, operates
on Tl by
ст (yx ® • • • ® y;) = ya-i (1) ® • • • ® ya-! (;)
11.1 Relations between enveloping and symmetric algebra
205
and extending by linearity. A tensor in T' is called symmetric if it is fixed by
all a e S;. The natural map T II (L) induces a map Tl 1I;(L). Let 1T(L)
be the image under this map of the space of symmetric tensors in T'.
Proposition 11.2 (i) 1I;(L) = 1I;_1(L)®1I!(L).
(ii) These spaces are all L-submodules ofVt(L~).
Proof. We first show that
n;(L) = n;_j(L) + n;(L).
Let ...xf be a basis element of 1I;(L) with rx4-----------\-rn = i. For each
a e Sj we define а (x)1 ... x(”) to be the element obtained from x? ... xr% by
permuting the factors by the permutation <j. Since multiplication in the graded
algebra of II (L) is commutative we have
where и e llj-^L). Since the sum lies in 1T(L) we have
1I;(L) = 1I;_1(L)-|-1I;(L).
We next show that II,, (L) nlT(L) = O. Any element of 1T(L) has the
form
We express this element as a linear combination of basis elements of II (L).
We obtain
ri+-+r„=i	ri+-+r„=i
where	since multiplication in the graded algebra of 1I(L) is
commutative. This element can only lie in II,|(L) if each Л ,... ,r is 0.
Thus Uj_ j (L) П IT (L) = O. Hence we have
1I;(L) = 1I;_1(L)®1I;(L).
Finally these subspaces are all L-submodules. The subspaces 1I;(L) and
!!;_! (L) are evidently submodules by the definition of the L-action. 1T(L) is
an L-submodule since the L-action commutes with the .S',-action on T'.	□
Let T' be the subspace of symmetric tensors in T'.
206
Further properties of the universal enveloping algebra
Proposition 11.3 There is a commutative diagram of vector space isomor-
phisms
a	у
/ W) \
> Sl(L)
where a is induced by the map 7 (1.) 11(7.), /3 is induced by / (I.) S(I.),
у is induced by 1I;(L) 1I; (£)/!!;_! (L) and 8 is the map of Proposition 11.1.
Example
\	„	7^+lx^
Proof. It is sufficient to show that ya(t) = 8/3(t) where t = Уо-1 (i) ®
• • • ® ya-i ft and yk e L. We have
О'ЕЕмч-'-М.)
a
ya(t) = n;_j (L) +	ya-r (1)... ya-r (;)
a
a
sp(f) =
a
since the difference between • • -Уо-Щ) and the corresponding element
in canonical form lies in II,, (7.). Hence ya(t) = 8/3(t).	□
We now define в : Sl(L) 1I!(L) by в = у13, and extend this map by lin-
earity to give в : S(L) II (L). в is called the operation of symmetrisation.
We have
в (У1У2 • • • Уд = E У<7-1 (!)••• У<7-1 (о •
l' creSj
Proposition 11.4 в : S(L)^ 11(7.) is an isomorphism of L-modules.
Proof. We know that в is an isomorphism of vector spaces and so must show
that
x- 0(P) = 0(x-P) for all x e L, P e S(L).
11.2 Invariant polynomial functions
207
A derivation of an associative algebra A is a linear map D : A A such
that
D(ab) = D(a)b + aD(b)
for all a, b eA. It follows from the definition of the L-action that the maps
5(L)^5(L)	H(L) —> H(L)
P X-P	u^-x-u
for xeL are derivations. Now L may be identified with a subspace of S(L)
and the map P^> x-P when restricted to L is adx. Similarly L may be
identified with a subspace of II(L) and the map u^-x-u when restricted
to L is again adv. Now .S'(L) is generated as an algebra by L and 1. We
have D(l) = 0 for any derivation of S(L). Thus there is a unique derivation
of S(L) extending adv on L. Similarly u^ x-u is the unique derivation of
II (L) extending adv on L.
Let D : 1I(L)^1I(L) be this derivation. D transforms 1T(L) into 1T(L)
for each i. Using the isomorphism у of Proposition 11.3, D determines a map
T1WL) 7ПН(£)
which is still a derivation. Using the isomorphism 3 of Proposition 11.3 we
obtain a map S(L) S(L) that is still a derivation and which acts as ad x on L.
Thus it is the map P^> x-P. Hence for P e S(L) we have
0 (x- 0(F)) = x-P.
Thus x  0(F) = 0(x• F) as required.	□
Note The L-action on II (L) considered here may be described simply by
x-u = xu — их хеЬ,иеУ1(Е)
For this is a derivation of 1I(L) which extends adv : L^- L.	□
11.2 Invariant polynomial functions
Let G = Inn L be the group of inner automorphisms of the Lie algebra L. We
recall from Section 3.2 that G is generated by automorphisms of the form
exp adv for elements xeL such that adv is nilpotent. We define an action
of G on L* by
= M #eG’ feL*, xeL-
208
Further properties of the universal enveloping algebra
The tensor algebra
T(L*) = (J)(L*® •••&£*)
£>0 к factors
may then be made into a G-module satisfying
g (Л ® ® fk) = gfr ®	® gfk
for g eG, fel.. Let I be the 2-sided ideal of T (7.) generated by all elements
of form
f®g-g®f
for f,geL*. Then I is a G-submodule of T (L*). Let
5 (L*) = T (L*') /I.
Then 5 (L*) may also be made into a G-module. 5 (L*) is the symmet-
ric algebra on L*. The algebra S(I.) may be identified with the algebra
of polynomial functions on L. The element I + fx ® • • • ®fk of 5 (L*) gives
rise to the polynomial function	on L. define P(L) = S(L*)
and Pm(L) = Sm (L*). This is the image of Tm (L*) under the natural homo-
morphism T (7/)^ 5 (L*). Pk(Lf is the space of homogeneous polynomial
functions of degree к on L. In particular /Jl(7j may be identified with L*.
Each subspace Pk(L) is clearly a G-submodule of P(L).
We now prove some lemmas which will help in understanding the action
of G on P(I.).
Lemma 11.5 The linear map a : 7™ (//) ^ (7 '"(7.)) uniquely determined
by
(« (Л ® • • • ® fmf) (xj ® • • ®xm)=f1 (xj f2 (x2) ... fm (xm)
is an isomorphism of G-modules. Here xx,... , xm lie in L and fi,... , fm
in L*.
Proof The linear map a is clearly injective. Since Tf" (I. ) and (rm(L))*
have the same dimension, a must also be surjective. Thus a is an isomorphism
of vector spaces. We must also show that
а (y -/i ® • • • ® fm) = у (a (/i ® • • • ® /J)
for all у eG. Now we have
у (Л ® • • • ® fm) = y/i ® • • • ® yfm.
11.2 Invariant polynomial functions
209
Thus
= (y/i) (л-i) • • • (y/m) (xm)
= /i (? 4) •••./„,(? 4,).
On the other hand
У («(fi ® ® fm)) (x1®---®xm) = a(f1®---®fm) (y-1x1 ® • • • ® y-1xm)
= fl(y^Xl)---fm(y^Xm)-
This gives the required equality.	□
Lemma 11.6 Consider the maps
Tm{L)*—>Tm (L*) ~^Sm (L*)
a-1	ft
and let в : (Tm(L))* —> Sm (L*) be given by 0 = [3a!. Thus в is a homo-
morphism of G-modules. Then we have (0f)x=f(x®-  - ®x) with m factors,
for xeL.
Proof It is sufficient to prove this when f has the form
f (*1 ® • • • ®Xm)=fl (*1) '''fm (Xm)
that is when or1f = fl ® • • • ®fm. In this case we have
(V> = /i(v)/2(x).../m(x)=/(x®---®x).	□
An element f e (rm(L))* is called symmetric if
f (*1 ® • • • ® Xm) =f (-M) ®   • ® Xa(m)}
for all , xm e L and all creSm. The set of symmetric elements of
(rm(L))* will be denoted by (Tm(L))*ym.
An element of Tm (L*) is called symmetric if it is invariant under the linear
maps which transform ® • • • ® fm to fa^ ® • • • ®fa(m) f°r all a e Sm.
The set of symmetric elements of Tm (L*) will be denoted by Tm (L*)sym.
Lemma 11.7 The subspaces Tm(L~)*ym and Tm (L*)sym are G-submodules.
Moreover the maps a^1, /3 give isomorphisms
Tm(LfsyB1^T”'{L*')syB1—,Sm(L*').
J a-1	P
210
Further properties of the universal enveloping algebra
Proof. The subspaces are G-submodules since the G-action commutes with
the 5m-action on Tm(L)* and Tm (L*). The map a transforms Tm (P*)sym into
rm(L)*ym and, since a is an isomorphism and these two spaces have the same
dimension, we have
«(rm(L*)sym) = r'"(L):ym.
Again the spaces Tm (P*)sym and Sm (L*) have the same dimension and
the map
is surjective, since it transforms
1 v-
j" / у Уо-(1) ®	® fa(m)
<reSm
into fxf2.. ,fm. Thus this map is also an isomorphism.	□
The G-module isomorphism
P'"(L)^r'"(L)s*ym
is useful in determining the G-action on Pm(L), since it is often easier to
calculate the action on the linear functions in Tm (Р)*уш than on the polynomial
functions in Pm(L).
We shall now assume that the Lie algebra L is semisimple. The group G of
inner automorphisms is called the adjoint group of L. A polynomial function
P e P(I.) is called invariant if y(P) = P for all ye G. The set of invariant
polynomial functions on L is denoted by P(I.f '. This is clearly a subalgebra
of P(L). We shall investigate the algebra of invariant polynomial functions on
L by relating it to the algebra of polynomial functions on a Cartan subalgebra
of L invariant under the Weyl group.
Let H be a Cartan subalgebra of L and P(II) = 5 (//) be the algebra of
polynomial functions on H. Let W be the Weyl group of L. Then we know
that both H and II are W-modules. (We recall from Section 5.2 that an
action of W was defined on the real subspace /Р of IP. and this gives
rise to a H'-action on H* by linearity.) The H'-actions on H and H* are
related by
w £ W, f E H*, h £ H.
There is then a H'-action on T (//) satisfying
w(Jl®---®fm) = wfl®---® wfm.
11.2 Invariant polynomial functions
211
This in turn induces a И'-action on 5 (H*) = T (H*) /1 since I is a
W-submodule of T (H*). Thus P(H) = S(H*) may be regarded as a
W-module.
A polynomial function P e Iff!) is called И-invariant if w(P) = P for all
w e W. The set of И'-in variant polynomial functions on H will be denoted by
PlJPjw. This is a subalgebra of P(Hj.
Now we have an algebra homomorphism
ф : P(L)^P(tf)
given by restriction from L to H. We consider the image of P(LjG under this
restriction map. We show first that this image lies in the subalgebra P(Hjw.
Proposition 11.8 (Д (P(P)G) cP(//)w.
Proof. We use the element d, e G given by
d, = exp ad e; • exp ad (—/;) • exp ad e;.
We recall from Proposition 7.18 that d,(7/) = H and that 0;(/i) = sfh) for all
h e H, where steW is a fundamental reflection. Thus d, acts on H in the
same way as st. It follows that d, and st also act in the same way on ll', and
on S(H*) = P(H).
Let PeP(L)G. Then 0;(P) = P. We have i/t(P) eP(H) and so s;(t/t(P)) =
|Д(Р). However, the Weyl group W is generated by its fundamental reflections
Sp ... , S[, by Theorem 5.13. Thus we have
w(t/t(P)) = <Д(Р) for all w e W
and so |Д(Р) eP(H)w.	□
Proposition 11.9 The map if : P(LjG P(Hfw is injective.
Proof Let R be the set of regular elements of L. We recall from the proof of
Proposition 3.12 that there is a polynomial function F e P(L) such that x e R
if and only if F(x)^0. We also recall from Theorem 3.2 that every regular
element lies in some Cartan subalgebra and from Theorem 3.13 that any two
Cartan subalgebras are conjugate. Thus given any regular element x e R there
exists у eG such that yCv) e H.
Now suppose P e P(IfG satisfies i//(P) = O. Let x be a regular element and
let у e G be such that y(x) e H. Then
|Д(Р)(у(х)) = 0,
212
Further properties of the universal enveloping algebra
that is P(y(x)) = 0. Hence (y XP) (x) = 0. Since P e P(If)G we have у VP = P
and we may deduce that
P(x) = 0.
Thus P annihilates all regular elements of L. Hence P(x) = 0 whenever
F(x) 0. By the principle of irrelevance of algebraic inequalities we have
P(x) = 0 for all x e L,
that is P = O.	□
Finally we show that the map j/ is also surjective.
Theorem 11.10 Ф : P(L~)G Pill)" is surjective, and is therefore an iso-
morphism of algebras.
Proof We make use of ideas from the representation theory of L. Let Л eH*
be a dominant integral weight and L(A) be the finite dimensional irreducible
L-module with highest weight A. We can choose a basis of L(A) with respect
to which L(A) decomposes into a direct sum of 1-dimensional //-modules.
Let p be the representation of L afforded by this basis. Consider the function
P : L C given by
P(x) = tr((p(x))m)	xeL.
We claim that P e Pm(L). For let bx,... , bn be a basis of L and let
x = £iM------^eC.
Then we have
P(*) = E&P (bi)
(p(p-jj"= E ^...^тр(Ь^...р(Ь1т)
h,--,im
trace (p(x))m = £ tr(p(b;i)...p(b;J)e;i...^.
ii,... ,im
This is evidently a polynomial function on L which is homogeneous of
degree m. Thus PeP”(L).
We wish to show that P is an invariant polynomial function, that is
P e (Pm(Lf)G. We shall make use of the isomorphism
Pm(L)^T'"(L):ym
11.2 Invariant polynomial functions
213
obtained in Lemma 11.7. The element /eTm(L)*ym corresponding to
P e Pm(L) is given by
f (%! ® • • • ® xm) =	tr (p (x^d) ... p (xa(m))).
m' <^sm
For f certainly lies in Lm(L)*ym and
/(x® • • - ®x) = tr(p(x)m) .
Lemma 11.6 now shows that f corresponds to P.
We recall that Tm(L) may be regarded as an L-module under the action
X • (Xj ® • • • ® xm) = 52 X1 ®	® [xX;] ®	® xm-
i
Its dual space Tm(I.y then becomes an L-module under the action
(X/) (xx ® • • • ® xm) = -f (x (Xj ® • • • ®xm))
for xeL, /eTm(L)*.
We now consider xf where f e Lm(L)*ym is the function defined above.
We have
(x/) (xx ® • • • ® xm) = - 52 f Cl ® • • • ® [xx;] ® • ••® xm)
i
= - Д E E tr (p kb)) • • • p ([^w]) • • • p
aeSm i
= E E tr (p kb)) • • -pWp k(o) • • • p kw))
aeSm i
+ A E E (P (*<,(!)) • • • P (x^)) P(X) . . . p (x^))) •
m' aeSm i
All the terms in these expressions cancel except those for which p(x) occurs
at the beginning or the end of the product. Thus we have
(x/)(x1®’--®xm)	=	—J E2 (tr(p(x[7(i))...p(x[7(m))p(x))
m' <y<=~sm
-tr(p(x)p (x[7(1))...p(x[7(m))))
=	0 since tr(AB) = tr(BA).
Thus xf = 0 for all xeL.
214
Further properties of the universal enveloping algebra
We now compare the L-action on Tm(L)*ym with the G-action.
Let x be an element of L such that adx is nilpotent. Then exp ad x e G and
G is generated by all such elements. Let
r(x) : T'"(L):ym^T'"(L):ym
be the linear map given by
r(x)/' = x/'.
Then we have
r(x)/ (xx 0 • • • 0 xm) =	(xx 0 • • • 0 ad (-x) -x; 0 • • • 0xm) .
Thus
(~(xy \ ,	„	„	((ad —xY1	(ad —xY™ \
-77-/) (x10---0xm) =	f\---П---Xj0---0-i——------xm .
—к
Since adx is nilpotent the right-hand side is 0 for k sufficiently large.
Hence
(expr(x)-/) (XJ0---0XJ
(ad —x)!1
(ad — x)!™
= f (exp ad — x • Xj 0 • • • 0 exp ad — x • xm)
= (exp ad - x • /) (xj 0 • • • 0 xm) .
Thus we see that
exp ad — x • f = exp r(x) • f.
Now we have shown that x/ = 0, hence r(x)/ = 0. Thus ехрт(х)-/ = /. It
follows that
exp ad —x-f=f.
Since this holds for all x e L with ad x nilpotent we deduce that
/e(rm(L):ym)G.
By Lemma 11.7 it follows that P e Pm(L)G.
11.2 Invariant polynomial functions
215
The restriction <Д(Р) therefore lies in Pm(H~)w. Let Ab A2,... , Лк be the
weights of L(A) with A, = A. Then we have
1гр« = АГ(х) + --- + АГ(х).
Hence ^(P) = A'"b-----hA”.
We shall show that polynomial functions of this kind span Pm(H)w. In the
first place we know that H* is spanned by the lattice X of integral weights.
It follows that Pm (77) is spanned by the set of monomials of degree m in the
integral weights. However, it is well known that the process of polarisation
can be used to express such a monomial as a linear combination of mth
powers. (For example the formula
AxA2 = j (Ax + A2)2 — |Aj — |A|
expresses the monomial A, A2 as a linear combination of squares.) Thus the
elements Am for A e X span Pm(PP). It follows that every W-invariant element
of Pm (77) is a linear combination of elements of form
52 w(A)m AeX.
weW
Since each H'-orbit of integral weights contains a dominant integral weight
we see that elements of form
52 w(A)m AeX+
weW
span Pm(Hf
Now we have <Д(Р) = A'" b---b A"' where Ax = A. A appears with multiplic-
ity 1 in the set {Ab ... , A^} and each w(A) also appears in this set. Moreover
this set is W-invariant, so is a union of W-orbits.
It follows from these facts that <Д(Р) = w(A)m + a linear combination
of terms	for p. e X with p. < A. There are only finitely many
216
Further properties of the universal enveloping algebra
weights p. e X with < Л. Therefore we may invert these equations and
express w(A)m as a linear combination of functions of the form <Д(Р)
coming from representations with highest weight /x<A. Thus Рт(НУ^ js
spanned by functions of the form <Д(Р). Hence Pm(I[y' lies in the image
of if. Since this is true for all m the image of if must be the whole of P(FL)W.
Thus if is surjective.
We therefore have an isomorphism of algebras
if : P(L)G -+ P(H')W.	□
11.3 The structure of the ring of polynomial invariants
In this section we shall prove a theorem of Chevalley which shows that the
ring P(FL)W of W-invariant polynomials on H is isomorphic to a polynomial
ring in / variables over C.
We write I = P(H)W and define в : P(FT) P(H) to be the operation of
averaging over W. Thus
I I weW
It is clear that 0(Р(Я)) = I, that в acts as the identity on I, and that в2 = в,
i.e. в is idempotent.
Let P(II) be the set of polynomial functions with constant term 0, and
let I+ = 1Г\Р(Н)+. Let P(H)I+ be the ideal of P(FT) generated by I+. The
elements of P(H)I+ have form
ЛЛН-------\-P)A
with P,e P(7/), ./,e/.
Lemma 11.11 Suppose Jx, ... ,Jk are elements of I such that f does not
lie in the ideal of I generated by J2,... , Jk. Let Pk,P2,    ,Pk£P(H) be
homogeneous polynomials such that
p1j1+p2j2 +    + pkjk = o.
Then Pr e P(H)I+.
Proof. We shall show that does not lie in the ideal of P(II) generated by
J2,... , Jk. Suppose this were false. Then we have
Л = Q2J2 + • • • + QA with e Р(Я).
11.3 The structure of the ring of polynomial invariants
217
Applying w e W we obtain
Л = w (2г) Л H Ь w (Qk) к
and therefore
However, в (<2;) e I and so lies in the ideal of I generated by J2,... , Jk.
This gives the required contradiction.
We now show that Px e P(H)I+ by induction on the degree of the homo-
geneous polynomial Px.
If degPj =0 then Px is constant. Since P1J14-= О and is not
in the ideal of P(PT) generated by J2,... , Jk this implies that Px = 0. Thus
Pj e P(H)I+ in this case.
Now suppose deg Px > 0. We recall that W is generated by its fundamental
reflections Sp... , s^ In the H'-action on H each Sj has a fixed point set
which is a hyperplane in H given by an equation H- = O where Hj e P(II) is
a homogeneous polynomial of degree 1. We have
(fj{Plf')x=Pi(sj{x')) = Pi{x')
where /7;(л) = 0. Thus the polynomial Sj (P^-Pf vanishes at all xeH for
which Hj vanishes. It follows that
Sj^-P^HjP,
for some Pi eP(H).
Since P; is homogeneous, Sj (P;) is also homogeneous of the same degree,
hence Sj (P;) — P; is homogeneous. Thus P; is also homogeneous with degP; <
degP;.
Now the relation
Р1Л + --- + Р^=О
implies
and so
Я/Р171 + --- + Р^) = О.
Since Hj is not the zero polynomial this implies that
Р1Л + --- + Р^ = О.
218
Further properties of the universal enveloping algebra
Since degPj < deg Px we may deduce by induction that Px G P(H)I+. Hence
^(Pj)-Pj еР(Я)/+ also.
Now P(7/)/ is a W-submodule of P(H), thus P(H)/P(H)I+ is also a
W-module. We have
sj(P1) = P1 mod P(7/)/
and since W is generated by ,... , it follows that
w(P1) = P1 mod P(7/)/
for all w G W. Hence
0(P1) = P1 modP(7P)/+.
Now Pj is a homogeneous polynomial of positive degree, therefore 0(PX) G
I+. In particular в (PJ G /’(//)/ and so Px G /’(//)/ as required.	□
Now the ideal P(t/)l of P(7/) is generated by the homogeneous elements
of I of positive degree. By Hilbert’s basis theorem there is a finite subset
of this generating set which generates P(H)I+. Let f,... ,In be a set of
homogeneous polynomials in I such that f,... , In generates P(t/)l but no
proper subset generates P(t/)l .
Proposition 11.12 The polynomials Ir,... ,In are algebraically independent.
Proof. Suppose the result is false. Then there is a non-zero polynomial P in
n variables such that
P (Л,..., If)=o.
We may assume, by comparing terms of a given degree, that all monomials
in f,... , In which occur in P have the same degree d in ... , x;. Let
P; = dP/df. Then
Р;(Л,...,/Л)	i=\,...,n
are elements of I and not all the P; are zero.
Let J be the ideal of I generated by Px, P2,... , P„. We may choose the
notation so that Px,... , Pm but no proper subset generate J as an ideal in I.
Thus there exist polynomials Qtj GI such that
Pi = '^2,Qi,jPj i = m+l,... ,n.
7=1
11.3 The structure of the ring of polynomial invariants
219
Now each P; is homogeneous in . , xt of degree d — degf. Thus, by
comparing terms of the same degree in ... , on both sides, we may
assume that each Qtj is homogeneous of degree degP; — deg Pj.
Now P (7j,... ,If) = O thus дР/дхк = 0 for к = 1,... , /. Hence
Д dP dL
У------- = 0,
У dxk
that is
n
УР,дЦ/дхк = Ъ.
i=l
It follows that
m	n m
уР,дЦ/дхк + у yQijPj9Ii/dxk = 0
i=l	i=m+l j=l
that is
m	/	n	\
УРДдЦ/дхк+ У Q^df/дхД^.
i=l	\	j=m+l	/
We now apply Lemma 11.11. Px,... , Pm are in I and Px is not in the ideal
of I generated by P2,... , Pm. Each of the polynomials
n
dljdxk+ У Qjidlj/dxk i=l,...,m
j=m+l
is homogeneous in xr, ... , of degree deg /, — 1. For
deg Qj.i = deg Pj - deg pi = deg A - deg •
It follows from Lemma 11.11 that
n
dljdxk+ У <2^/дхкеР(Н)1+.
j=m+l
We now multiply this polynomial by xk and sum over k= 1,... , /. For a
homogeneous polynomial f in ... , we have, by Euler’s formula,
' df
Y.Xkfy=^IfIj-
k=l 0Xk
Thus we have
n	n
deg/1-/1+ £ degf-Q^yiff
j=m+l	i=l
220
Further properties of the universal enveloping algebra
where each R, eP(H)+. We note that all the terms on the left-hand side are
homogeneous polynomials of degree deg/,. Comparing terms of this degree
on the two sides we obtain
n
degA-A+ E dQSIj'QjAIj = ^IiRi
j=m+l	i
where the sum on the right extends over a subset of 1,... , n not including
i = 1, since I1R1 has degree greater than deg f. It follows that f is in the ideal
of P(77) generated by I2,... ,In. However, this contradicts the definition of
f,... , In. Thus the proposition is proved.	□
Proposition 11.13 Every element of I is a polynomial in f,... , In.
Proof. It is sufficient to prove this for homogeneous polynomials in I. Let
J el be homogeneous. We use induction on deg J, the result being clear if
deg./ = 0. Suppose deg J> 0. Then J el+ and in particular J e P(H)I+. Thus
we have
/ = Р1ДЧ-----\-PnIn
for certain polynomials Pr,... , Pne P(H). Since J, f,... , In are all homo-
geneous we may clearly assume that each P; is homogeneous also, with
deg P; = deg J — deg I.
Then we have
j = 0(p1)/1 + --- + 0(p„)4.
в (Pj),... ,0 (Pn) are homogeneous polynomials in I of degree less than
deg J. Thus they are polynomials in . , In by induction, and so J is also.
□
Corollary 11.14 The algebra P(H)W = C[Ir,... , /„| is isomorphic to the
polynomial ring in n generators over C.
Proof. This follows from Propositions 11.12 and 11.13.	□
The set f,... , In is called a set of basic polynomial invariants of W. We
now determine the number of invariants in a basic set.
Proposition 11.15 The number n of invariants in a basic set is equal to the
dimension I of H.
11.3 The structure of the ring of polynomial invariants
221
Proof Let K = C (xb ... , x;) be the field of rational functions in xb... ,xt
over C. Also let k = C (Ir,... , If) be the field of rational functions in
f,... , I„ over C. Then we have inclusions
С С к с К.
Since xx,... , X[ are algebraically independent over C the transcendence
degree of К over C is given by
trdeg7C/C = /.
Since f,... , In are algebraically independent over C, by Proposition 11.12,
the transcendence degree of к over C is given by
trdeg£/C = w.
Since we have
tr deg A"/C = tr deg Л/С + tr deg К/к
we shall consider trdeg/f/A. Now К is generated over к by xx,... ,xt.
However, each x; is an algebraic element over к. For the polynomial
П (?-wCl))
weW
has x; as a root, and its coefficients are the elementary symmetric functions in
the w (x;) as w runs over W. These coefficients are W-invariants and therefore
lie in I. In particular this polynomial lies in к[t] and so x; is algebraic over к.
Thus К is generated by a finite number of algebraic elements over к and so
trdeg K/k = Q.
It follows that
tr deg K/C = tr deg k/C,
that is и = /.	□
Now the set f,... , f of basic polynomial invariants of W is not uniquely
determined. We show, however, that the degrees of these polynomials are
uniquely determined.
Proposition 11.16 Let f,... , f and If ... , f be two sets of basic poly-
nomial invariants of W in P(II). Then we may arrange the numbering so
that
deg /, = deg I-	for z = 1,... , I.
222
Further properties of the universal enveloping algebra
Proof Each of If...,I'l is expressible as a polynomial in f,... , It and
conversely. Consider the matrices
(df/df).
These are inverse matrices, thus the determinant
det (dljdl'^
is non-zero. It follows that for some permutation cr of 1,... , /
dlt
By renumbering If ... , if necessary we may assume a is the identity. Thus
df
~dTi
and so dlfdl'i ф 0 for each i. This means that It, as a polynomial in If ... , If
involves f and so
deg> deg/..
This implies that
i	i
EdesA>E degA'-
i=l	i=l
By symmetry we must have equality. This implies
deg/(=deg/' for each i.	□
We summarise the results of this section in the following theorem, due to C.
Chevalley.
Theorem 11.17 (a) The algebra Iff If' of W-invariant polynomials on H is
isomorphic to a polynomial ring in I variables over C.
(b) Iff If' may be generated as a polynomial ring by I homogeneous invari-
ant polynomials f,... , If
(c) The degrees dx,... ,dl of f,... , f are independent of the system of
generators chosen.
11.4 The Killing isomorphisms
In the preceding sections we have investigated the algebras P(I.)a and P(IT)W
of invariant polynomial functions on L and H respectively. Assuming again
that the Lie algebra L is semisimple we show now how to relate these algebras
11.4 The Killing isomorphisms
223
to algebras S(L)G and S(H)W of invariants on the symmetric algebras of L
and H.
The action of G on the Lie algebra L may be extended to a G-action on
'/'(/.) satisfying
y(x1®---®xmj = yx1®---®yxm y&G.
We then obtain an induced action on S(L) = T(I.)j I since I is a G-submodule.
S(L)G is the subalgebra of all G-invariant elements of S(L). We shall relate
this to P(LfG by means of the Killing form.
We recall from Theorem 4.10 that the Killing form on the semisimple Lie
algebra L is non-degenerate. This implies that the linear map L —> L* given
by x^ л' where x*(y) = (x, y) is bijective. We wish to show that this is an
isomorphism of G-modules.
Proposition 11.18 Let у eG and x,y eL. Then (yx, yy) = (x, y). Thus the
adjoint group preserves the Killing form.
Proof. Since G is generated by elements exp ad z where z G L is such that
ad z is nilpotent, it is sufficient to show that
(exp ad z • x, exp ad z • у) = (x, у).
We recall from Proposition 4.5 that
([xz],y) = (x, [zy]).
Thus (ad z • x, y) = (x, ad — z • y) Iterating we obtain
((adz)'.v, y) = {x, (ad-z)!y).
Now we have
(adz)2	(adz)*1
exp ad z = 1 + ad z + — ----!-•••+	-—
2!	k\
for some k, since ad z is nilpotent. Hence
(exp ad z • x, y) = (x, exp ad — z • y)
and so
(exp adz-.v, exp ad z-y) = (x, y).	□
Corollary 11.19 The Killing map L* is an isomorphism of G-modules.
224
Further properties of the universal enveloping algebra
Proof. We must show that (yx)* = yx* for all у e G, x e L. We have
(7^)*(у) = <Ж У) = (х, y-1y) = x* (у 'у) = (ух )(у).
Thus (yx)* = yx* as required.	□
The Killing map L* induces an isomorphism 7(7) T (/.) and then
an isomorphism S(L) S (L*) in an obvious way. This is again an isomor-
phism of G-modules. There is therefore an isomorphism between S(L)G and
5 (P*)G. We recall that 5 (L*) = P(L) and so obtain a Killing isomorphism of
algebras S(L) P(I.) which induces a Killing isomorphism S(I.)G P(I.)G
between the subalgebras of invariants.
We now consider the action of the Weyl group W on the Cartan subalgebra
77 of L. We recall from Proposition 4.14 that the Killing form of L remains
non-degenerate on restriction to 77. Thus the map II II given by x^ x*
where x* (y) = (x, y) for all у e 77 is bijective.
Proposition 11.20 The Killing map II II is an isomorphism of
W-modules.
Proof. We have
(w/i)*x = (wh, x) = (h, w-1x) = h* (»’ 1 x) = (w/i*) x for all x e 77.
Hence (wh)* = wh* as required.	□
The Killing isomorphism II II induces an isomorphism P(77) T (77*)
and then an isomorphism S(H) S(II ). This is again an isomorphism of W-
modules. Since 5 (77*) = P(77) we obtain a Killing isomorphism of algebras
S(H) P(II) which induces an isomorphism S(H)W P(H)W between the
subalgebras of invariants.
We now consider the relation between S(L) and S(II). We recall that
L may be identified with a subspace of S(L) and that L has a triangular
decomposition
L = N~®H®N.
Let К be the ideal of S(L) generated by N and К . Then we have S(L)/K
isomorphic to S(H). Let 17 : S(L) S(H) be the natural homomorphism given
in this way.
11.4 The Killing isomorphisms
225
Proposition 11.21 We have a commutative diagram of algebra homomor-
phisms
S(L) A P(L)
v 'l' Ф
S(H) -+ P(H)
where a, /3 are the Killing isomorphisms, j/ is restriction from P(Lj to P(H),
and t] is projection from S(I.) to S( t/).
Proof We must show jtafQj = fhjtff) for all QeS(L). It is sufficient to
prove this when
n—fri fNhSl hs,e‘l e‘N
where Ф+ = {(31,... , j3N}.
If r; = 0 and t; = 0 for each i then t](Q) = Q. Moreover /3(Q) = ifa(Q).
Thus the diagram commutes.
If not all the r; and t; are 0 then t](Q) = O. Thus /3t](Q) = O. We have
“(2) = a (/д )ri... a (f^ a (hff1 ...a (htf1 a (ед )'*... a .
Therefore, for x e H we have
(a2)x =	, xf1 ...	, xf (h^xf1 ... (hh, x}s‘ (eft, x}‘l ... (ерк, xf .
This is 0 since (N~, H) = 0 and (N,H) = 0, and some r; or t; is non-zero.
Thus the diagram commutes in this case also.	□
Corollary 11.22 We have a commutative diagram of algebra isomorphisms
S(LjG	p^g
VS(H)W	Р(П)" ''
Proof. We have seen that the Killing isomorphisms a, /3 map S(I. )G to P(L)G
and S(Hjw to P(Hf, respectively. We also know from Theorem 11.10 that
j/ : P(LjG P(H)W is an isomorphism of algebras. Thus 17 acts on S(L)G in
the same way as [3 !ilra. Hence 17 : S(L)G S(H)W is an algebra isomor-
phism.
We note by Theorem 11.17 that the four algebras S(L)G, P(LjG, S(Hjw,
P(H)W are all isomorphic to the polynomial algebra C [zx,... , z;].
226
Further properties of the universal enveloping algebra
11.5 The centre of the enveloping algebra
The centre Z(L) of II (L) is defined by
Z(L) = {z G 1I(L) ; zu = uz for all и e II(L)}.
Proposition 11.23 The centre Z(L) acts on each Verma module M(A) by
scalar multiplications.
Proof. Let mA be the highest weight vector of M(A). Let z G Z(L) and h e H.
Then
h (zmf) = z (hmf) = X(K)zmx.
Thus z.m,. еЛ/(А)л. Now the А-weight space of M(A) is 1-dimensional - in
fact Л/(А)л = CmA. Hence
zmA = £mA for some ^eC.
Now let и e II (L). Then we have
Z (umf) = и (zmf) = cumf.
Since M(A) = lI(L)mA we see that z acts on M(A) as scalar multiplication
by	□
We write yA(z) = Thus x,, : Z(L) C is a 1-dimensional representation
of Z(L). x,, is called the central character of M(A). We shall show how to
determine this central character.
We consider II (L) as an L-module, as described in Section 11.1. The
L-action on II (L) is given by
x-u = xu — их	x e L, и e II(L).
II (L) has basis
L'l frN l,sl l,sl <1 <N
JPi-'-JPN	epi-"ePN
where Ф+ = {/У, ... , jiN}. Ifxdl we have
h^-.-hf e,^...e,^ = (-r1l31----------rN/3N
+t1/31 + --- + tA,/3A,)(x)/;i.../^ hs'...hs‘ е^...^.
Thus ... /J" /ij1 ...hf e^ ...e^ is a weight vector with weight
(У — ri) /У H Ь — Tv) Pn-
11.5 The centre of the enveloping algebra
227
We consider the zero weight space П (/.),,. This has basis
...hs‘ e‘^...e‘^N where (tx - rj /Зх + • • • + (tN - Гд,)^ = 0. We have
1I(L)O = [u e II (22) ; хи — их = 0 for all x e H}
thus 1I(L)O is a subalgebra of 1I(L). It is clear that Z(L) c 1I(L)O.
Proposition 11.24 (i) II(L)NПII(L)o = /VII(L) ПII(L)o = K.
(ii) The subspace К of (i) is a 2-sided ideal ofU(L)0.
(iii) П(Л)о=^®П(Я).
Proof (i) II (Л)/V is spanned by the basis vectors of II(L) with some
t;>0. /V 11(72) is spanned by the basis vectors with some r;>0.
U(L)7VO1I(L)O is spanned by the basis vectors of 11(22) with =
and some t;>0. /V 11(72) П 1I(22)O is spanned by the basis vec-
tors of 1I(L) with J) t;/3; = J) r;/3; and some r;>0. These are clearly
equal.
(ii) U(L)7VO1I(L)O is clearly a left ideal of 1I(22)O and /V II (72) П II (72),, is
a right ideal of 1I(L)O. Thus К is a 2-sided ideal of 1I(L)O.
(iii) 11(77) is spanned by the basis vectors with all r; = 0 and all t; = 0. This
shows that II (72),, is the direct sum of its subspaces К and 11(77).	□
Let ф : U(L)0—> 11(77) be the projection map obtained from the decom-
position
U(L)0 = 27 ф 11(77).
Since К is a 2-sided ideal of II(L)o, ф is a homomorphism of algebras, ф is
called the Harish-Chandra homomorphism.
We can now determine the central character Xs- The weight XeH* deter-
mines a 1-dimensional representation of 11(77), also denoted by A.
Theorem 11.25 The central character x,, : Z(L)^C is given by yA(z) =
A(<A(z)J where ф is the Harish-Chandra homomorphism.
Proof. We have
1I(L)O = (U(L)7V П 1I(L)O) Ф 11(77)
and Z(L) с II (/.),,. Let z G Z(L). Then we can write
z=w—\-икпк+ф(е>
228
Further properties of the universal enveloping algebra
where и; e 1I(L) and n; eN. Thus
= (u1n1-{----h uknk + ф (z)) mA
= A(<)>(z))mA
since NmK = О and <)>(z)mA = A(</>(z))wa. Thus yA(z) = A(</>(z)).	□
We have seen that the Harish-Chandra homomorphism maps Z(L) into
11(77). Since the Lie algebra 77 is abelian we have 11(77) = 5(77). We shall
show that by combining the Harish-Chandra homomorphism with a ‘twist-
ing homomorphism’ we get a homomorphism from Z(L) into S(H) with
very favourable properties. The twisting homomorphism r : S(H) S(H) is
defined as follows. We recall that S(H) is a polynomial algebra over C with
generators h.^... , hj. Thus there is a unique algebra homomorphism
г : 5(77)^5(77)
such that г (hj) = ht — 1. 7 is in fact an automorphism of algebras. Its inverse
is given by 7 1 (/i;) = ht + 1.
Let p e X be the element of the weight lattice given by
P = co1+--- + a)l.
Thus p is the sum of the fundamental weights. We recall from Section 10.3
that
<y;(/i;) = l w;(/iJ=0 if j^i.
Thus p (hf) = 1 for each i = 1,... ,
Now any element A e 77* extends to a 1-dimensional representation of S(H).
A — p is also a 1-dimensional representation of S(H). We have
A7 (hf) = A (hj — 1) = (X — p~)hj.
Since A7 and A —p are 1-dimensional representations of S(H) and the ht
generate S(H) we have
A 7(2) = (A -p)(2) for all Q e 5(77).
The homomorphism
тф : Z(L)^ 5(77)
is called the twisted Harish-Chandra homomorphism. We wish to show
that the image of Z(L) under the twisted Harish-Chandra homomorphism lies
in S(H)W. To do so we first need a result on Verma modules.
11.5 The centre of the enveloping algebra
229
Proposition 11.26 Let XeH* and M(A) be the corresponding Verma module
with highest weight vector mK. Suppose (A + p) (/i;) G Z and (A + p) (/i;) > 0
for some i. Let
/•(A+p) (hj)
v = f>	m^.
Then the submodule of M(A) generated by v is isomorphic to M(p) where
p+p = Sj(X+p).
Proof We recall from Theorem 10.6 that there is an isomorphism of
II G'V )-modules between II (/V) and M(A) given by	Since
у.(А+р)(л,) q -n	we see t|lat in M(A). Since тлеМ(А)л we
have i’GW(+l(, where
p = A — (A + p) (/i;) at.
Thus we have
p + p = (A + p) - (A + p) (/i;) at = st(A + p).
We shall show that Nv = O. It is sufficient to show that ел = 0 for i = 1,... ,
If j ф i we have
/•(A+p) (hj)	/.(A+p) (hj)	r\
e-v = ejj> r'Vl'mx=j>	=
If j = i we have
etv =
= /;(Л+р)(А-)е; + (А + р)(Ш(Л+рЖМ№ - (Л + Р)№) - 1))тл
= (А + р) (й;)	(X (h) - А (й;) - 1 + 1) тл = 0.
Thus Nv=O.
Let V be the submodule of M(A) generated by v. Since Nv=O and htv =
p (hj) v for i = 1,... , there is a surjective homomorphism of II(L)- modules
from M(p) into V given by
um.^ur mgII(IW).
(See Proposition 10.13.) We consider the kernel of this homomorphism. Let
и GII (/V) be such that uv = 0. Then
и/;(л+р)(/!,)тл = 0.
Since и/.(л+р^-’ g II (/V ) this implies that uf^+p^=0. Since у.(л+р)+) _^q
and 1I(L) has no zero-divisors we have и = 0. Thus our homomorphism is an
isomorphism and so V is isomorphic to M(p)	□
230
Further properties of the universal enveloping algebra
Proposition 11.27 The twisted Harish-Chandra homomorphism тф maps
Z(L) into S(H)W.
Proof. We must show that тф(ф) eS(H)W for all zeZ(L). Since W is
generated by ,... , it will be sufficient to show that
ФФ(г)) = тф(г)-
Since S(H) = P (77*) it will be sufficient to show these elements take the
same value for all A G 77*, i.e. that
А(у;(т</>(г))) = А(т</>(г)) for all A g77*.
In fact it will be sufficient to prove this for elements of 77* of the form A + p
where A G X+ is dominant and integral. For such weights form a dense subset
of 77* in the Zariski topology, for which the closed sets are the algebraic sets.
Thus suppose A G X+. Then we have
(A + p)(r(<^(z))) = A(0(z)) = XaU)
using Theorem 11.25 and the definition of r. Similarly we have
(A + p) (л (т(Ф (z)))) = (p + p) (т(Ф (z))) = p (Ф (z)) =	(z)
where st(A + p) = p + p.
We now apply Proposition 11.26. Since AeX+ we have A(/i;)>0, so
(A + p) (hi) > 0. Thus the Verma module A7(A) contains a submodule isomor-
phic to A7(p). Now z eZ(7) acts on A7(A) as scalar multiplication by yA(z)
and on M(p.) as scalar multiplication by X^z). Since M(p.) is isomorphic to
a submodule of A7(A) we must have
Xa(z) = Xm(z).
Thus
(A + p)(r(<£(z))) = (A + p) (s;(r(<)>(z))))
and hence
T(^(z)) = ^W(z))).
Thus тф(ф) G S(H)W as required.	□
In fact we shall show that the twisted Harish-Chandra map
тф : Z(L)^S(H)W
is an isomorphism of algebras.
11.5 The centre of the enveloping algebra
231
To see this we first recall the operation в : S(L) II (L) of symmetrisation
which was shown in Proposition 11.4 to be an isomorphism of L-modules.
Now the adjoint group G acts on both S(L) and 1I(L). For the G-action on
L can be extended to a G-action on 7(1.) as described in Section 11.4 and
these induce G-actions on the quotients S(L) and 1I(L). Suppose xeL is
such that adx is nilpotent. Then exp adx e G. Let x induce the linear maps
a(x) on S(L) and /3(x) on II (L). The definition of the G-actions then shows
that exp adx acts as exp a(x) on S(L) and as exp/3(x) on II (L). Since в is
an isomorphism of L-modules we have
0a(x) = /3(x)0.
It follows that
a(x)1 B(xY
0——L = l——L0 for all z,
z! z!
and therefore that
в exp a (x) = exp /3 (x) 0.
(Note that both a(x) and /3(x) are nilpotent.) Since G is generated by such
elements exp adx it follows that в is an isomorphism of G-modules. We
deduce that в restricts to an isomorphism between S(I.)G and II(L)G.
Proposition 11.28 1I(L)G = Z(L).
Proof. We first note that Z(L) с II (L)G. Let z G Z(L). Let x e L be such that
adx is nilpotent. Thus exp adx e G. Since z G Z(L) we have
x-z = xz — zx = 0.
Hence /3(x)z = 0. Thus
exp ad x • z = exp /3(x) • z = I 1 + /3(x) +	-1---j z = z.
Thus z is invariant under exp ad x. Since such elements generate G we have
zgU(L)g.
Conversely we show that II(L)G c Z(L). Let и e 1I(L)G. Then exp ad x • и =
и for all xeL with ad x nilpotent. Suppose (adx)' ^0 but (ad x)(+1 =0. We
choose elements ... , £t+1 G C which are all distinct. Then ad ((jx) is also
nilpotent and
exp ad (£;x) = 1 + ad (£;x) 4----h (ad (£;x))(
£‘
= 14-£;(adx)4-------h-jy(adx)'.
232
Further properties of the universal enveloping algebra
ft
2!
2!
Now the determinant
i ei
i
F-
1 £	f+1
e(+i —
fl
t!
fl
t!
2!3!... t! П
is non-zero. Thus the vector (0, 1, 0,... , 0) is a linear combination of the
rows of the determinant. Thus there exist т/p ... , t]t+1 G C such that
adv =t]1 exp ad(^1x)-|--------|-Т7г+1ехр ad(£r+1x).
So ad x • и = (t/j Ч----\-t]t+1)u. Since ad x acts nilpotently on и it follows
that 4--------h t]t+1 = 0 and that ad x • и = 0. This means that хи — их = 0. This
holds for all xeL with ad x nilpotent, in particular for x=e; and x=ft.
However, ex,... , e;, fr,... , fi generate 1I(L), together with 1. It follows that
xu — ux = 0 for all x e 1I(L), that is и e Z(L).	□
Thus the operation в of symmetrisation gives an isomorphism of vector
spaces
6 : 5(L)G^Z(L).
Now we also have an isomorphism of algebras
T] : S(L~)g -+ S(H~)W
given in Corollary 11.22. Combining these maps we obtain an isomorphism
of vector spaces
гф"' :
Thus we have two maps p0! and тф from Z(L) into S(H)W. The first is an
isomorphism of vector spaces and the second a homomorphism of algebras.
We shall compare these maps, using the structure of Z(L) and S(H) as filtered
algebras.
We recall from S ection 11.1 that II (L) may be regarded as a filtered algebra
with filtration

11.5 The centre of the enveloping algebra
233
We define Z;(L) = Z(L)nlI;(L). This makes Z(L) into a filtered algebra.
S(H) also has a natural structure as a filtered algebra, where SfH) is the
subspace of S(H) generated by all products axa2 Oj, j < i, where ak eH.
We also define
(.ST//)l'); = .ST//)"n
This makes S(H)W into a filtered algebra.
We shall make use of the following lemma on filtered and graded algebras.
Lemma 11.29 Let A = ljj>o arld B = 'gji>0Bi be filtered algebras with
AoGAjCAjC---
and
в0 с вг с в2 c • • •.
Let
gr A = Ао ф A-JAo ©Aj/Aj Ф • • •
and
gr В = B0 Ф В, /В,, Ф B2/B, Ф • • •
be the corresponding graded algebras. Let a : A -+ В be a linear map such
that a (A;) c B; for each i. Then:
(a)	There is a linear map gr a: gr A gr В satisfying gr а (А;_х + a;) = B;-1 +
a (a;) for at e A;.
(b)	If a (A;) = B; for each i and a is bijective then gr a is bijective.
(c)	If gr a is bijective then a is bijective.
Proof, (a) We must show that gr a: Ai/Ai_x Bi/Bi_x is well defined.
Suppose Ai_1 + ai = Ai_1+a'i where ai,a'i&Ai. Then ai — a'i&Ai_x, so
a (a; — a.) eB;_P Thus В;_х + a (a;) = B;_1 + a (a.) and so gra is well
defined.
(b) Suppose now that a(A;) = B; for each i and that a is bijective. Then
the induced map gr a : Ai/Ai_x Bi/Bi_x is bijective. It follows that
gra : gr A gr В is bijective.
(c) Suppose conversely that gr a : gr A gr В is bijective. This implies that
gr a : A,/A, ,	B,/B, ,
is bijective for each i. We show first that a is surjective. Bo lies in the
image of a since a : A0^B0 agrees with gra : A0^B0. Assume by
234
Further properties of the universal enveloping algebra
induction that В;_х lies in the image of a. Let bt -Jf Then there exists
a; e At such that
Д-1 + a (ai) = -Sj-1 +
Thus bt — a (a;) e B;_P Hence bt — a (a;) lies in the image of a, thus bt
does also. Thus a is suijective.
Now let ae ker a. If aeA0 then a = 0 since a agrees with gra on
Ao. Otherwise there exists i>0 such that ae/l; but a^A^. But then
A;-1 + a 0 whereas gr a (A;-1 + a) = 0, a contradiction. Hence ker a = О
and so a is bijective.	□
Theorem 11.	30 The twisted Harish-Chandra map тф gives an isomorphism
of algebras Z(L) S(H)W.
Proof. We have maps тф : Z(L) S(H)W and p0! : Z(L) S(H)W.
Those induce maps
gr(r^) : grZ(/.)^gr.S'(//)"'
gr^fl-1) : grZ(/.)^gr.S'(//)11'.
We shall show that gr(r<^>) = gr (т/бН1). Let zeZ(L). Then there exists d
such that zeZd(L) but z^'Zd_1(L). Then z has the form
Z = E e(L s, t)/j; ... /X hf ...hf ... e'l
E ^+E^+E h—d
where Ф+ = {/3P ... , /3N} and £(r, s, t) e C. Then
<№) = E ^(o.s.oj/ij1 ...hf
^Si<d
тф(г) = E ^(o.s.oXTij-iy1 ,..(/i;-l)s'
Es,<d
o! (z)=Ee(Ls,t)/j;hf...hfe‘f...e‘fN modsd_,(l)
T]e~\z) = E^fes, o)/ij‘ ...hf mod Sd_fH).
Now it is apparent that
= mod ^(Я)
hence
тф^ = ув x(z) mod^jT/).
11.5 The centre of the enveloping algebra
235
Since t<A(z) and pd '(z) both lie in S(H)W they satisfy тф(г) = riO f r)
mod (S(H)w)dl. Thus gr(r^) = gr (т/б-1).
Now the maps d 1 : Z(L)^ S(L)G and 17 : S(L)G S(H)W satisfy
в-1 (Zd(L~f) = (S(L)G)d
V(S(L-)G)d=(S(Hr)d.
Thus we have
' (Zd(Lf) = (S(H)w)d.
We may now apply Lemma 11.29. The map
rf)" : Z (7.)^.S'(7/)11
is bijective and satisfies
7?0-1(Zd(L)) = №)w)d
for each d. Hence
grM ') : grZ(/.)^gr.S'(//)11'
is bijective. This is turn implies that
тф : Z(7.)^.S'(7/)11
is bijective. Since тф is known to be a homomorphism of algebras, it must
therefore be an algebra isomorphism.	□
We can deduce from this theorem a necessary and sufficient condition for
two central characters xK, x,j to be equal.
Theorem 11.	31 Let A, p.e H*. Then x,, = Xu. if and only if p. + p = w(A + p)
for some w e W.
Proof Suppose first that p. + p = w(A + p). Then, for z G Z(L), we have
XM(z) = m(<№)) = (w(A + p) -p)(0(z))
= w(A + p)(r(^(z))) = (A + p) (w^T^zf)).
Now t<A(+) G S(H)W and so is fixed by w!. Hence
AjJz) = (а + р)(т(<№))) = A(<£(z)) = TA(z),
by Theorem 11.25. Hence X/i = Xs-
236
Further properties of the universal enveloping algebra
Suppose conversely that p. + p w(A + p) for all w e W. Then the finite sets
W(A + p) and W(p. + p) do not intersect. Therefore there exists a polynomial
function QeP(H*) such that Q takes values 1 on W(A + p) and values 0 on
W(p.-vp). We have
Oe.S(//) = P(//).
By replacing Q by	w(2) we таУ assume Q lies in S(H~)W.
We now make use of the isomorphism тф : Z(L) S(H)W. There exists
z G Z(L) such that r<^(z) = Q. Thus we have
Xs (z) = Л (ф (z)) = (A + p) (тф (z)) = (A + p) Q = 1
Xy (z) = M (Ф (z)) = (м + P) (тф (z)) = (m + p) Q = 0.
Hence Xs^Xy	□
A second deduction from Theorem 11.30 is the following important
result.
Theorem 11.	32 The centre Z(L) of 1I(L) is isomorphic to the polynomial
ring over C in I variables, where L is semisimple and I = rank L.
Proof This follows from Theorem 11.30, Corollary 11.22 and Theorem 11.17.
□
As an example we consider the Lie algebra L of type AP The algebra L
has a basis f, h, e with
[M=2e, [hf] = —2f, [ef] = h.
The algebras
S(H)W, P(H)W, S(L)g, P(L)g, Z(L)
are all isomorphic to the polynomial ring over C in one variable. We find a
generator of each of these algebras.
We have W = (s) where s(h) = —h. Thus S(H)W is the polynomial algebra
generated by h2.
We now consider the isomorphism S(L)G S(TI)W given by projection.
The element of S(L)G mapping to h2 is homogeneous of degree 2 in e, h,f
11.5 The centre of the enveloping algebra
237
and has weight 0. It must therefore have form If + f/e for some feC. We
determine the constant (.We have
ade-/i=—2e,	ade-f = h, ade-e = 0.
Thus
(exp ad e)h = h — 2e
(exp ad e)f = f + h — e
(expade)e = e
(exp ad e) (h2 + gfe) = (h — 2e)2 + £;(f + h — e)e
= /i2 + e/s+(e-4)/ie+(4-e)e2.
Thus exp ad e fixes If + £fe if and only if £ = 4. Hence S(I.)G is the polyno-
mial ring generated by h2 + 4fe.
Next we consider the Killing isomorphism L*. L* has basis f* ,h*, e*
dual to f, h, e, that is y* (x) = 1 if у = x and y* (x) = 0 if у x. Now the Killing
form satisfies
{h,h) = 8,	(f,e)=4,	(h, f) =0, (h, e) =0,	(e,e)=0,	=
Thus under the Killing isomorphism L^-L* we have e Af*, h^ 8h*,
f -^Ae'. This induces a map S(L)^P(L) under which h2 + Afe maps to
64 (h*2 + f*e*y Thus P(L)G is the polynomial ring generated by h*2 + f*e*.
We also have a map S(I. )G	Z(G) given by symmetrisation. Under this
map h2 + 4/e is transformed into
h2 + 2/e + 2ef = h2 + 2h + 4/e.
Thus Z(L) is the polynomial ring generated by h2 + 2h+Afe. We also note
that the element of Z(L) mapping to h2 e S(H)W under the twisted Harish-
Chandra homomorphism is h2 + 2h + 1 4-4/e.
Thus we have:
S(H)W = C [/i2]
P(H)W = C[h*2]
S(L)G = C[h2 + 4fe]
P(L)g = C[/i*2+/V]
Z(L') = C[h2 + 2h + 4fe]
238
Further properties of the universal enveloping algebra
11.6 The Casimir element
We now introduce an element of the centre Z(L) of 1I(L) which has useful
properties. Let xx,... ,xn be a basis of L. Since the Killing form of L is
non-degenerate by Theorem 4.10 there is a unique dual basis yx,... , yn of L
satisfying
Let c e 11(7.) be defined by
n
c = ^xiyi.
i=l
Proposition 11.33 The element c is independent of the choice of basis
хъ... ,xn of L.
Proof. Suppose xf ... ,x'n are a second basis of L and у'х,... , y'n are the
dual basis. Let
x'i=Taijxi	у'!=ТтцУр
j	j
Then we have
(х'о y'j}=(E E )=E aikTji (xk, yt> = E
\ k	I I к,l	к
Hence if a = (of) , т = (ry) we have от1 = I. We then have
EE = EIEvJ (ЕТ/л) =E I EcrijT») х}Ук-
Now 0*7 = 1 so 'Sli0ij7ik = 8jk. Hence JTx.y. = JTx;y;.	□
Definition c is called the Casimir element of II (L).
Proposition 11.34 c lies in the centre Z(L) of 1I(L).
Proof. It is sufficient to show that ex = xc for all .i t A. We have
ex = E Х[У[X = E X (xyi + [y;x])
i	i
= E (.(.xxi + к;*]) Л + xt [y;x])
i
= xc + E ([*;*] УI + Xi [y;x]) .
11.6 The Casimir element
239
Let =	and [yix] = XjSince <[x;x], y7> = <x;, [xyj> we
have a;i = — It follows that
lJ 'J1
E (Ж ++xi 1+4) = E E aijxjyi+E E
i	i j	i j
= Е(а++/4)-ъ+=°-
Thus cx = xc and so ceZ(L).	□
We now recall from Proposition 4.18 that for each CX-L, we can find
f„e.L „ such that \e„f„} = h' and that we then have (e„, f„ ) = 1. Since
the Killing form of L remains non-degenerate on H we may choose a basis
hf ... , h'f of H and there will be a dual basis h'[,... , h" satisfying
Then ... , ea (a g Ф+) , fa (a e Ф+) are a basis of L and its dual basis
is
/а(аеФ+), еа(аеФ+).
Using this pair of dual bases we have
c = h'1h"-{--\-h'th"+ E eafa+ E fVL-
аеФ+	аеФ+
Thus we obtain:
Proposition 11.35 The Casimir element of Z(L) is given by
E 4+2 E /X
z=l	аеФ+	аеФ+
where hf ... , hf, h'[,... , h" are any pair of dual bases of H.	□
The properties of the Casimir element will be useful as we explore further
the representation theory of L.
Proposition 11.36 Let ceZ(L) be the Casimir element. Then
Xx(c) = (X + p,X + p')-(p,p')-
Thus c acts on the Verma module M(A) as scalar multiplication by
(A + p, A + p)-(p,p).
240
Further properties of the universal enveloping algebra
Proof. We consider the action of c on the highest weight vector mA of A7(A).
By Proposition 11.35 we have
cms = \ Y,h'ih"+ E h'a + 2 E faeajms
\i=l	аеФ+	аеФ+ /
= (£A(/l()A(/l") + £ A(/i'ff)jmA
\i=l	аеФ+ /
Now Е,, ф A (hf) = Е„Ф (A, a) = (A, E +,ф) = 2(A, pf
Let h'K e 77 be the element corresponding to A e 77* under the isomorphism
defined by the Killing form. Thus
K(hf=(hfhf
K(h'!)=(hfhf.
We express /i'A in terms of the dual bases hf ... , /i'; and hf ... , /i" of 77. Let
Ip = a^h^~\~ • • • T alhl
h'K = blh'[ + --- + blhf
Since (hf h"f = 8i; we have
fhf h'f) = ajfojd--Valbl
(hfhf = bi (hfhf) = ai.
It follows that
E A (hf A (hf) = jf(hf hf (hf h'f = (hf hf = (A, A).
i=l	i=l
Hence
cmA = «А, A)+2(A, p))mA
= ((A + p, A + p) — (p, p))mA.
Thus the value of the central character yA at c is given by
La(c) = M + p, A + p)-(p,p).	□
12
Character and dimension formulae
12.1 Characters of L-modules
Let V be an L-module where L is semisimple. We say that V admits a
character if V is the direct sum of its weight spaces and each weight space
of V is finite dimensional. Thus we have
V’= ф V'; dim VA finite
Лей*
where VA = {ue V ; hv=X(h)v for all heH}. The character of V is then
the function ch V:H*	Z given by
(ch V)(A) = dim VA.
We see that if V admits a character then the structure of V as an //-module
is determined by ch V.
In this chapter we shall obtain formulae for the characters of the Verma
modules A/(A) for Л e H* and for the finite dimensional irreducible modules
L(A) for AeX+.
We first identify a certain ring of functions H* Z in which it will be
convenient to work. Given a function f : H* Z we define Supp f, the
support of f, to be the set of A eH* for which /(A)^0. For example the
support of the function ch A/(A) is the set of all p. e H* which have form
/z = A — n1a1--- — nlal nie'^, nt>0.
This follows from Theorem 10.7. We define
5 (A) = Supp (ch A/(A)).
241
242
Character and dimension formulae
Definition St denotes the set of all functions f : H* Z such that there
exists a finite set Xr,... , XkeH* with
Supp/cS/AJU-’-US/Aj).	□
It is clear that chM(A) for XeH* and chL(A) for AeX lie in St. It is
also clear that if f, g e St then f + g e St, since
Supp (/ + g) c Supp f U Supp g.
Thus St is an additive group. We can also define a product on St which makes
it into a ring. Given f, g e St we define fg : H* Z by
Ш)(А)= E ЛмШО-
We note that the sum is finite, so that fg is well defined. For we may assume
p e Supp/ and v e Supp g. Suppose
Supp/c5(^1)U---U5(juJ
Suppgc5(r1)U---U5(pJ.
If p e S (pt) and v e S (rj we have
М = Мг —miai----'~miai mke^, mk>0
v=Vj-n1a1 — -—ntat nteZ, nk>0.
Since p + v = A we have
А = (мг + ^)-^1«1-----riai rkeZ,rk>0
where rk = mk + nk. However, given i,j and A the non-negative integers
mk, nk with mk + nk = rk can be chosen in only finitely many ways, thus our
sum is finite. Also we see that Supp(/g) c lj; jS (p-i + Vj), hence fg e St. It is
also readily checked that (Jg)h = f(gh), thus St becomes a ring.
For each XeH* we define ел : H* Z by ел(А) = 1, eA(/z) = 0 if p ф A.
Thus eA is the characteristic function of A. All such characteristic functions
lie in St. In fact if / is any function in St it is convenient to write
f= E /ОМ
ЛеЯ*
even though the sum may be infinite.
We note that eAeM = eA+/x.
12.1 Characters of L-modules
243
Lemma 12.1 Suppose that the L-module V admits a character and let U be
a submodule of V. Then both U and V/U admit a character, and
V
ch U + ch — = ch V.
U
Proof We have V = ®AV\- Also t/A = UCi VA. Thus the sum J2A t/A is direct.
Moreover U = J2A t/A since if и e U and и = £ us with иА e VA then иА e U,
as in the proof of Theorem 10.9. Hence we have
u=®ux
A
with c VA, so U admits a character. We also have
W=®(vA/t/A)
л
and VA/t/A can be identified with the Л-weight space (V/[/)A. Thus V/U
admits a character. Finally we have
(ch t/)(A) + (ch (V/t/))(A) = dim t/A + dim (VA/t/A) = dim VA.
Thus ch[/4-ch(V/[/) = ch V.	□
Lemma 12.2 Suppose Vx, V2 are L-modules which both admit characters
such that ch and ch V2 lie in St. Then 0 V2 admits a character and
ch (Vj 0 Vf) = ch V, ch V.
Proof Since Vi, V2 admit characters we have V1 = ®A(V1)A and
H2 = ®M(V2)A[. Hence
V1®V2 = ©((V1)A®(V2)J.
A, jj.
Vj 0 V2 may be made into an L-module by means of the action
X (Uj 0 V2~) = XUj 0 V2 + Uj 0 xv2
extended by linearity. In particular, if x e H, vr e (V1)A and v2 e (V2)M we have
x (i?j 0 v2) = (A(x) + pt(x))v10 v2.
Thus (Ti)A0 (V2)M C (Vj 0 V2)A+M. It follows that
Vr 0 V2 = ф (Vj 0 V2)
244
Character and dimension formulae
where (Vx ® V2) = Ц л,/х ((Юл ® (Юи)- Thus V1 ® V2 admits a character.
A+/z=r
Moreover we have
(ch(V1®V2))(p) = dim(V1®V2)r= E dim(V1)Adim(V2)/x
A,/z
A+/z=v
= E (chyi)(A)(chV2)(M) = (chV1ch V2)(p).
A,/z
A+/z=v
Thus ch (Vx ® V2) = ch Vr ch V2 as required.
□
12.2 Characters of Verma modules
We now consider the character of the Verma module M(A) where XeH*.
We recall from Theorem 10.7 that
(chM(A)) (/x) = ^J(A —/л)
where ^J(A — /л) is the number of ways of expressing А —/л as a sum of
positive roots. Thus we have
chM(A)= £ ^(A-M)eM= £ We2_v
veH*
= E	E W)e-r-
гей*	гей*
We write Г = Юей* ^(r)e-r- have Г e St since SuppT c 5(0). Then we
have
ch Л/(А) = елГ.
Lemma 12.3 Г has an inverse in the ring St given by
Г != n (l-e_J-
аеФ+
Proof. Let Ф+ = {/3j,... , f}N}. Then ЧЮ7) 0 if and only if there exist non-
negative integers rx,... , rN such that v = r1/31-\-\-rN/3N. In fact TO7) is
the number of such sets (rx, ... , rN). Thus we have
r = EWKr= E 4a--^
N /	\
= E ^...e-N = n E^, •
ri,...,rw>o	i=l \r,->0	/
12.2 Characters of Verma modules
245
This factorisation of Г in St gives us the required result. For the element
1 + S-уЗ; +	+ ’ ’ ’ Of Si
has an inverse 1 — e_(3. e St. Thus Г has an inverse
N
^ = 0(1-^.)= П	□
г=1	аеФ+
This gives us a useful formula for the character of the Verma module M(A).
Proposition 12.4 ch M(A) = e л+р where X = ep П„ ф (1 - e_a).
д
Proof. We have
chM(A) = eAr=^ = ^
by Lemma 12.3.	□
The denominator Д is an element of St which can be expressed in a number
of alternative ways.
We recall that p e X was defined by
р = Ш1-\-----h«;
i.e. p is the sum of the fundamental weights. This element can also be
expressed simply in terms of the roots.
Proposition 12.5 p = | ZL, <i. a- Thus p is one half the sum of the positive
roots.
Proof. Let p' = | ZL, <i. a- We can express p' as a linear combination of the
fundamental weights. Let
i
p' = ci0)i with c; e Q.
i=l
Now the fundamental reflection st e W transforms at to — at and transforms
every other positive root to a positive root, by Lemma 5.9. Thus we have
(p') = p' -“/•
On the other hand we have
246
Character and dimension formulae
by Proposition 10.18. This shows that Sj (w;) = w, if i Ф j and gj (wj = a>j — ctj.
Thus we have
si (p') = P' ~ciai-
Comparing this with the above formula for st (p') we deduce that c; = 1.
Hence p' = p as required.	□
Corollary 12.6 Д = e_p П„ ф (ea - 1)
Proof. We have
A = eP П (1-e-«) = ep П e-a(.ea~1) = ep[ П e-J П K-')
аеФ+	аеФ+	\аеФ+ / аеФ+
= еРе-2р П (еа-!) = е-р П (е»-1)-
аеФ+	аеФ+
There is a further useful expression for the denominator Д. Before proving
it we shall need some information about the geometry of the action of the
Weyl group W on the Euclidean space V = H^.
12.3 Chambers and roots
We recall that the Weyl group is a finite group of isometries of the Euclidean
space V generated by the reflections sa for a e Ф. We have
(a, v)
sa(v) = v — 2----a veV.
{a, a)
Let
La = {veV; sa(v) = v}
= {weV; (a, v) =0}.
La is the reflecting hyperplane orthogonal to the root a. We consider the
complement
of the set of reflecting hyperplanes. This is an open subset of V. The con-
nected components of this set are called the chambers of V. Two points of
12.3 Chambers and roots
247
V — lie in the same chamber if and only if they lie on the same side
of each reflecting hyperplane.
Let C be a chamber in V and 3(C) be the boundary of C. Then the
hyperplanes La such that Л„ПЗ(С) is not contained in any proper subspace
of La are called the bounding hyperplanes, or walls, of C.
Now let П = {dip ... , ce;} be a fundamental system of roots. Then the set
C = {и e V; (ce;, v) > 0 for i = 1,... , /}
is a chamber of V. For if a is any positive root we have (a, v) > 0 for all v e C.
Thus all elements of C lie on the same side of each reflecting hyperplane
La. Thus C lies in V — [_}ae®La and C is connected. Moreover any subset of
V —	larger than C would contain an element v with (ai, v) <0 for
some i, and so would be disconnected. C is called the fundamental chamber
corresponding to the fundamental system П. The bounding hyperplanes of C
are L,... , L . For La. ПЗ(С) consists of all v e V such that (ai,v)=0 but
(a,, v)>0 for ji. Since a1,... , ce; are linearly independent La. П8(С) is
not contained in any proper subspace of La,. On the other hand let a be a
positive root which is not fundamental. Then a = J2;=1 niai with each nt >0
and at least two n; > 0. If v e La П 3(C) then
22 ni (ai, v)=о
and so (ce;, v) = 0 whenever nt > 0. Thus La П8(С) lies in a proper subspace
of La. Hence the bounding hyperplanes of C are L,... , L . In fact the set
П = {dip... , U[} of fundamental roots may be characterised as the roots
orthogonal to the bounding hyperplanes of C which point into C, that is such
that а,, lies on the same side of L„ as C.
Now the Weyl group acts on V in a way which permutes the roots. It
therefore permutes the reflecting hyperplanes La, and so acts on V —
Since W is a group of isometries of V, W permutes the connected components
of V —	Thus the Weyl group W acts on the set of chambers of V.
Proposition 12.7 (i) Given any two chambers С, C of V there is a unique
element w e W such that w(C) = C.
(ii) The number of chambers of V is equal to the order of the Weyl group.
(iii) If C is a chamber in V its closure C contains just one element from each
W-orbit on V.
Proof. Let П be a fundamental system of roots and C be the chamber defined
by v e C if and only if (ce;, v) > 0 for i = 1,... , /. Let C be any chamber
and let veC. We recall from Section 5.1 that П is associated with a total
248
Character and dimension formulae
ordering > on V. We consider the set of transforms w(v) for w e W and let
v' be the one which is greatest in the above total ordering. Then we have
v (v) = v -2-----------a,. a, ell
ai)
and since sa. (if) < v' we must have (ai, v') > 0. This holds for all i= 1,... , /,
thus v' e C. Now let v' = w(v). Since v e C we have v' e w (C')- Thus w (C') is
a chamber which intersects C. However, the only chamber intersecting C is C.
Thus w (С') = C. Hence any chamber C' is in the same W-orbit as C. Thus W
acts transitively on the set of chambers. It follows that any chamber is associated
to some fundamental system of roots in the manner described above.
Now suppose w(C) = C. Then we have w(II) = П where П is the funda-
mental system determined by C, i.e. the set of roots orthogonal to the walls
of C and pointing into C. It follows that w (Ф+) = Ф+, so w makes every
positive root positive. Hence w(w) = 0. It follows from Corollary 5.16 that
/(w) = 0, i.e. w= 1. Thus W acts simply transitively on the set of chambers.
It is a consequence of this that the number of chambers of V is equal
to |W|.
We now consider the closure C of a chamber C. Since each vector lies
in the closure of some chamber and W acts transitively on the chambers
each orbit of W on V intersects C. We must also show that if v1,v2eC and
w (iij) = i>2 then iij = i>2. We prove this by induction on /(w). It is clear when
/(w) = 0, i.e. w=l. Thus we assume /(w)>0. Then w(w)>0 so there exists
a; e П with w (a;) < 0. Thus
0 < (iij, at) = (v2, w (a;)) <0.
Hence ai) =0 and sa. (vj = But now wsa. (vj = i>2. The only positive
root made negative by sa. is at. Thus the positive roots made negative by
w and wsa. are the same, apart from a;, which is made negative by w and
positive by wsa,. Thus
n(w) = n(wsa.) + l
and so
/ (ws ) = l(w) — 1
by Corollary 5.16. We can then deduce that vr = v2 by induction, as required.
□
We shall now suppose that П is a fixed fundamental system of roots and
C is the corresponding fundamental chamber.
12.3 Chambers and roots
249
Proposition 12.8 (i) v e C if and only ifv = J2;=1 «,w, with nt > 0 for all i.
(ii) I’eC if and only if v = J2;=1 и;<у; with nt>Q for all i.
Proof Since Wj,... , a>l are a basis of V we can write v = ^Jnia>i for each
v e V. Now v e C if and only if (a;, v) > 0 for i = 1,... , We recall from
the definition of the fundamental weights a>,,... , a>l that
= 0 ifz#J
= 2	.
Thus we have (a;, v) =2nt {at, at). In particular (a;, v) >0 if and only if
и; > 0. Similarly (a;, v) > 0 if and only if n; > 0. The required result follows.
We show in Figures 12.1, 12.2 and 12.3 the chambers for the 2-dimensional
root systems A2, B2 and G2.
Figure 12.1 Two-dimensional root system type A2
250
Character and dimension formulae
Figure 12.2 Two-dimensional root system type B2
Figure 12.3 Two-dimensional root system type G2
12.3 Chambers and roots
251
Proposition 12.9 Suppose Ф is the root system of a simple Lie algebra and
let C be the fundamental chamber.
(i) Suppose all roots in Ф have the same length. Then there exists a unique
root Oi = Ylli=iaiai in C. This root satisfies the condition that for any
root a = J2;=1 kiai we have к < at.
(ii) Now suppose there are two root lengths. Then there are just two roots
Oi = Y,aiai’ =
i=l	i=l
in C. is a long root and 0, is a short root. d. satisfies the condition
that for any root a = J2;=1 we have kt < at. (In particular ct < at.)
0/ is called the highest root and 0. the highest short root.
Proof. By Proposition 12.7 C contains just one root in each H'-orbit on Ф.
Now two roots lie in the same И'-orbit if and only if they have the same
length. For roots in the same orbit obviously have the same length; but any
root is in the same orbit as a fundamental root, and any two fundamental
roots of the same length can be joined in the Dynkin diagram by a sequence
of fundamental roots all of this length. Two fundamental roots of the same
length joined in the Dynkin diagram obviously lie in the same H'-orbit.
Thus in case (i) C contains a unique root d, and in case (ii) C contains one
long root and one short root ()..
We now introduce a partial order > on the set Ф+ of positive roots. Given
i	i
a = '<^jmiai, /3 = У^ n,u,
1=1	1=1
in Ф+ we write at>/3 if mi>ni for each i. We consider maximal elements
of Ф+ with respect to this partial order. Let a be maximal. Then (a, a;) >0
for each i, as otherwise a + at would be a root higher than a. We also have
(a, at} > 0 for some i. Let a = J).=1 m;a;. show that each mt > 0. Suppose
this is not so. Then there exist i, i' with mi^0,mi, = 0 and (ai,ai,)<0.
But then (a, af) = mj (ap at'} < 0, a contradiction. Hence each mt > 0.
We now show that a is the unique maximal element of Ф+ with respect
to >. Suppose if possible that /3 is also maximal and /3=fa. Then а + /3$Ф.
Also а — /З^Ф, as а — /З^Ф would imply at>/3 or /3>a. Hence (a, /3) =0
by Proposition 4.22. But
i=l
252
Character and dimension formulae
since each > 0, each (ah /3) > 0, and some (a;, /3) > 0. Thus we have a
contradiction. Hence a is the unique maximal element of Ф+ with respect to >.
Now aeC since (a, a;)>0 for each i. Thus а = в1 or 0S. We wish to
show a= вр To do so we show that if a' e ФП C then (a', a') < (a, a). By
the maximality of a, a — a' is a non-negative combination oi a1,... , al and
so (a — a', x) > 0 for all xeC. In particular we have (a — a', a) > 0 and
(a — a', a') > 0. Hence
(a, a) > (a, a') > (a', a').
It follows that a = 0;. Thus is the unique maximal element of Ф+ with
respect to > and the result is proved.	□
Definition The number /?= l+htd. is called the Coxeter number of L. It
is known to be equal to the order of the element sxs2  • .s;eW, and also
to |Ф|/|П|. (See, for example, Bourbaki, Groupes et algebres de Lie,
Chapters 4, 5, 6.)	□
In order to prove Weyl’s denominator formula we shall need some prop-
erties of the transforms w(p).
Proposition 12.10 (i) w(p) = p — a for some subset Cl of Ф+.
(ii) Given any subset Cl of Ф+ the vector p — a either lies in one of
the reflecting hyperplanes La or has the form w(p) for some w e W.
(iii) If p — a lies in the fundamental chamber then Cl is empty.
Proof We know from Proposition 12.5 that p = | ZL, <i. a- Let w e W. Then
w permutes the roots and so
W(P) = 5 E (±“) = P- E “
аеФ+	aefl
where fl is the set of positive roots made negative by w!.
Now suppose fl is any subset of Ф+. Suppose p — o lies in the
fundamental chamber. We write v = ^JaEna. Then (p— ;’)(/?,)> 0 for each
i=l,... ,1. Moreover (p — v)(ht)eZ and p(ht) = l, since <y;(/i;) = l and
a>.(hj) = Q if j^i. It follows that u(/i;)<0, that is that (v, a;)<0 for i =
1,... , However, v is a sum of positive roots so has form u=
where nt > 0 for each i. Hence
i
(v, v) = '^ni (v, Ctj) < 0.
i=l
It follows that v = 0 and so fl is empty.
12.3 Chambers and roots
253
Finally we must show that p — E„o a either lies in a reflecting hyperplane
or is a W-transform of p. Suppose it does not lie in any reflecting hyperplane.
Then it lies in a chamber. Thus there exists w e W such that
w I p — a I
\ aeO /
lies in the fundamental chamber. However, p — E„o a has the form
|Eae®+(±“), so w(p-Eaen“) also has form |E„ <i, ('±«) since w
permutes the roots. Hence
w [p-E “I =P~ E “
\ aeO /	aeO'
for some subset fl' of Ф+. Since this vector lies in the fundamental chamber,
fl' must be empty. Hence
w [p-E“I=p
\ aeO /
and so p — Eaeo a is a W-transform of p	□
We can now prove Weyl’s denominator formula.
Theorem 12.11 (Weyl’s denominator formula).
eP П (1-£-J= E £(WK(P)
аеФ+	weW
where e(w) = (—1)1<w\
Proof. Let Z [TT*] be the set of functions f : Tf*^Z of finite support.
This is the set of finite Z-combinations of the characteristic functions ел.
Weyl’s denominator formula is an identity in Z [TT*]. There is a natural action
of W on Z [TT*] given by
(w/)A = /(w^1A).
We define a map 0 : Z [TT*] Z [TT*] by
0(7) = E e(w)wf.
weW
It is clear that, for w' e W, Ow' = s (w') 0, hence
в (e (w') w') = в
254
Character and dimension formulae
and
Q2= |W|0.
We now consider the effect of a fundamental reflection si on
We have
A = eP П C1 — e-«) 
аеФ+
Si ( ep П (1 — e-a) j £s,(p) П О - e-s,(«)) •
\ аеФ+	/	аеФ+
Now sfp) = p — a; by the proof of Proposition 12.5. Also st transforms every
positive root to a positive root, except for at. Hence we have
= ~eP П С1-6-»)-
аеФ+
Thus s;(A) = — Д. It follows that w(A) = s(w)A for all w e W. Hence 0(A) =
|W|A.
We also have
A = eP П C1 —e «)
аеФ+
= eP E
ПсФ+
= £ (-i)lnlvs„„-
о- Ф
Now p — a either is of form w(p) for some w e W or lies in some
reflecting hyperplane, by Proposition 12.10. If v lies in a reflecting hyperplane
then 0(u) = 0 since the terms in 0(u) cancel out in pairs. For if v e La then
E (wWa) WWaV= —s(w)wV.
Thus we have
0(Д) = 0 E £(w)ewP
weW
12.4 Composition factors of Verma modules
255
since if p — o a = w(p) then |fl| = /(w) by the proof of Proposition 12.10.
Thus
0(Д) = 0(0(£р)) = 02 (£p) = |W(ep).
But 0(Д) = | W| Д as shown above. It follows that
A = 0W= E £(WKP-	□
weW
Corollary 12.12 chM(A) =-------------.
ZLir £(w)eiup
Proof. This follows from Proposition 12.4 and Theorem 12.11.	□
12.4 Composition factors of Verma modules
We shall show in this section that each Verma module M(A) has a composition
series of finite length and that all its composition factors are irreducible
modules of the form L(p2) where p. = w(X + p)—p for some weW. It will
be convenient to define
w- A = w(A + p) — p.
We shall use these results in the following section to prove Weyl’s character
formula for the finite dimensional irreducible modules L(A).
We begin with a lemma on filtered algebras and their corresponding graded
algebras. We recall the definitions as given in Section 11.1.
Lemma 12.13 Let A = be a filtered algebra with
AoGAjCAjC---
and let В = Bo® Bx® B2®   be the corresponding graded algebra.
/ч	Д-1 + (ЛП/)
(i)	if I is a left ideal of A then gr I =	------- is a left ideal of B.
Л-i
(ii)	If f C 4 then gr Д c gr I2.
(iii)	If 4 C 4 and gr Ii = gr 4 then 4=4-
(iv)	If В satisfies the maximal condition on left ideals so does A.
Proof. We recall that В = ф;В; where Bt = At/At_i. If x e At ПI, у e Aj then
we have
(A;-i+y) (Л-i+*)=Ai+j-i+ух
256
Character and dimension formulae
where yxGAi+jQI. Thus А;+^_х+yx e gr/. It follows that gr I is a left
ideal of B.
It is clear from the definition that if f GI2 then gr /, c gr Л.
We now suppose that f gI2 and gr /, =gr I2. Then
rlj-l + (А; A if) = A;_j + (А; A if)
for each i. Thus we have
A; A/j = (A; A/j) + (Aj-j A/j) .
We shall show that A; А = A; П f2 by induction on i. We know А0А/! =
AoA/2 since gr Zj = gr/2.
Assume inductively that Л, । А/, = Л, , А Л. Then we have
A;A/2 = (A;A/1) + (A;_1 A/1) = A;A/1.
Thus A; AI2 = A; A f for all i. Since A = U;A; it follows that f = I2.
Now suppose that
Лс/2с/3с---
is a chain of left ideals of A. Then
gr/iCgr/2cgr/3c---
is a chain of left ideals of B. Assume that В satisfies the maximal condition
on left ideals. Then we have gr/,=gr/; for all i,j sufficiently large. It
follows that /, = I] for all i, j sufficiently large. Hence A satisfies the maximal
condition on left ideals.	□
Proposition 12.14 II (L) satisfies the maximal condition on left ideals.
Proof II (L) is a filtered algebra whose graded algebra is the symmetric alge-
bra S(L). However, S(L) is isomorphic to the polynomial ring C[z3,... , z„]
where w = dimL, so satisfies the maximal condition on (left) ideals, by
Hilbert’s basis theorem. Thus 1I(L) satisfies the maximal condition on left
ideals, by Lemma 12.13.	□
Corollary 12.15 The Verma module Л/(Л) satisfies the maximal condition
on submodules.
12.4 Composition factors of Verma modules
257
Proof The left ideals of 1I(L) are the same as the lI(L)-submodules. Thus
1I(L) satisfies the maximal condition on submodules. We recall that
M(A) = H(L)/AfA
where K, is a submodule of II (L). It follows that M(A) satisfies the maximal
condition on submodules.	□
Theorem 12.16 The Verma module M(A) has a finite composition series
M(X~) =N03N12>N23"'lNr = O
where each Nt is a submodule of M(A) and Ni+1 is a maximal submodule of
Nt. Moreover Nt/Ni+1 is isomorphic to L(w-A) for some w e W.
Proof Since M(A) satisfies the maximal condition on submodules, every sub-
module of M(A) has a maximal submodule. Thus we have a descending
series
M(A) = N0^N^N2^---
of submodules, in which Ni+1 is a maximal submodule of Nt. We wish to
show that this series reaches О after finitely many steps.
Now M(A) is the direct sum of its weight spaces by Theorem 10.7. Thus
every submodule of M(A) is also the direct sum of its weight spaces, by the
proof of Theorem 10.9. It follows that each quotient NJNi+1 is the direct sum
of its weight spaces. Moreover each weight p. of NJNi+1 is a weight of M(A)
so satisfies pc -< A with respect to the natural partial order on weights. Thus
we can choose a weight p. of Nt/Ni+1 which is maximal in this partial order
among the set of possible weights. Let и be a non-zero vector in Nt/Ni+1 of
weight p.. Then we have etv = 0 and hv = pb(h)v for all h e H. Thus we have
U(L)v=U(AT)v.
However, Nt/Ni+1 is an irreducible lI(L)-module, thus U(L)v = Ni/Ni+1.
Thus we have a homomorphism M(jx) Nt/Ni+1 given by um..^uv for
all и e II (N) as in Proposition 10.13. This homomorphism is surjective and
its kernel is the unique maximal submodule of M(/z), since Nt/Ni+1 is irre-
ducible. It follows that Nt/Ni+1 is isomorphic to L(pd), the unique irreducible
quotient of Mlp.).
We now consider the action of the centre Z(L) of II (L). Z(L) acts on
M(A) by scalar multiplications. The element z G Z(L) acts on M(A) by scalar
multiplication by X^fz), as in Section 11.5. Hence z acts on each submodule
Nt and each quotient Nt/Ni+1 as scalar multiplication by yA(z). However,
258
Character and dimension formulae
z acts on Л7(р.) as scalar multiplication by xM(z), and so also on its quotient
Since Nt/Ni+1 is isomorphic to L(pd) we deduce that XA(z) = ^(z)
for all zeZ(L). Hence xA = xg. It follows from Theorem 11.31 that pc + p =
w(X + p) for some w e W. This is equivalent to p. = w • A for some w e W.
Now W is finite and so there are only finitely many possible composition
factors of M(A), up to isomorphism. Also each weight space of M(A) is
finite dimensional. Thus /.(p.), which contains p. as a weight, can appear as
a composition factor with multiplicity at most the dimension of the p. weight
space M(A)M. It follows that the series
M(A) = No D /V| D N2 D • • •
must reach О after at most J2weWdimM(A)w.a steps. Thus M(A) has a finite
composition series and each composition factor has form L(w-A) for some
we W.	□
12.5 Weyl’s character formula
We now find a formula for the characters of the finite dimensional irreducible
modules L(A) where A e X+.
Theorem 12.17 (Weyl’s character formula). Let XeX+. Then
k	£(w)eiu(A+p)
СП А/( Л ) — ~ ~	~ ~	.
(This is an equality in the ring 9? of Section 12.1 since the denominator
д = E £(w)ew(P)
weW
is an invertible element of St.)
Proof. Since A is a dominant integral weight we have A (/i;) > 0 for
i = 1,... , Hence (A + p) (/i;) = A (/i;) + 1 > 0 for i = 1,... , I. Thus A + p
lies in the fundamental chamber C. Hence w(A + p) lies in the chamber
w(C). It follows from Proposition 12.7 that the weights w(A + p) for we W
are all distinct.
Now the highest weight of the Verma module M(w • A) is w • A = w(A + p) — p.
Thus the characters ch M(w  X) e St are linearly independent as w runs over W.
Similarly the characters ch L(w • A) are linearly independent for w e W.
12.5 Weyl’s character formula
259
Now M(w-A) has a finite composition series with composition factors of
form L(y  A) for у e W, by Theorem 12.16. Moreover, since у • A is a weight
of L(y-A) and w-A is the highest weight of M(w-A) we have y-A <w’-A
whenever L(y  A) occurs as a composition factor of M(w  A). Moreover w  A
occurs as a weight of M(w- A) with multiplicity 1, thus L(w- A) appears as a
composition factor of M(w- A) with multiplicity 1. We therefore have
chM(w-A)= 22%, chL(y-A)
yeW
where awy e Z, awy > 0, a.... = 1, and awy ф 0 only if у • A -< w • A. If we write
the elements of W in an order compatible with the partial order у • A -< w • A
we see that the integers awy form a triangular |W| x |W| matrix with entries
1 on the diagonal. The determinant of this matrix is 1. Thus we may invert
the above equations to obtain
ch L(w-A) = bwy chM(y-A)
ye IV
where bwy e Z and bww = 1. (The bwy will no longer be non-negative.) In
particular we have
ch L (A) = 22 Sch М^У ’A)
yeW
where cy = bly. By Proposition 12.4 this gives
chL(A)=EyeW^£XA+p)
where cx = 1. We wish to determine the remaining coefficients c .
We recall from Proposition 10.22 that
diin/.(A);7=dim/.(A)„i;7l
for all w e W. Thus we have
w(chL(A)) = chL(A) for all we W.
On the other hand we have
%Д) = -Д
by Theorem 12.11 and thus
1/’(Д) = £(»)Д.
260
Character and dimension formulae
It follows that
W I 52 cyey(S+p) I =s(w) I 22 cyey(S+p)
Thus
22 суе»>у(\+р) = 22 £(w)cyey(A+P)
yeW	yeW
since weA = ewA. This is equivalent to
22 Cw-‘yey(A+p) = 22 £(w)cAey(A+p)-
yeW	yeW
Since the functions evlApj for у e W are linearly independent we deduce that
cw-iy = £(w)cr
In particular we have cw-i = s(w), thus cw = s (ir 1) = e(w). It follows that
E(w)ew(S+p)
ch L(A) =------------------
as required.	□
We note that in the special case A = 0 we have ch L(0) = e0 as L(0) is the
trivial 1-dimensional representation of L. Thus we have
22 £(wK(p) = Aeo = A-
weW
This gives an alternative proof of Weyl’s denominator formula, Theo-
rem 12.11.
We also note that while the character chL(A) is invariant under the Weyl
group both the numerator and the denominator in Weyl’s character formula
are alternating functions under the Weyl group, i.e. satisfy w(a) = s(w)a.
We may deduce from Weyl’s character formula a formula due to Kostant
for the dimension of the weight space L(A)p of L(A).
Theorem 12.18 (Kostant’s multiplicity formula). Let A e X and p.eX. Then
dimL(A)p= 22 s(w)^(w(A + p) - (p. + p))
weW
where T is the partition function defined in Theorem 10.7.
Proof. We have chL(A) = dimL(A)pep. Moreover we know that
A ' = е_рГ = e_p E ^(v)e_v
12.5 Weyl’s character formula
261
by Lemma 12.3. Thus Weyl’s character formula gives the identity
J2dimL(A)MeM = I 52 £(w) ^w(A+p)
/z	\weW	f v
= E
weW v
We compare the coefficients of e^ on both sides. This gives
dimL(A)M= £(w)^(w(A + p)-(^ + p)).	□
weW
We can also derive from Weyl’s character formula a formula for the
dimension of L(A).
Theorem 12.19 (Weyl’s dimension formula). Let A eX+. Then
dimL(A) = П^фИА + р, а) _
ПаеФ+ \P’
Proof. Let 9t0 be the subring of St consisting of all finite sums
with n„ e Z. Then the character formula
Д chL(A)= e(w)
ew(X+p)
weW
may be regarded as an identity in 9t0. Let A = IR[[t]] be the ring of formal
power series in the variable t with real coefficients. Then for each weight
f, e X we have a ring homomorphism
given by
= exP«£’ Ж = 1 + <£, PE+ 7^ (£ E'2 + • • • •
Consider 6^ (£,, „ s(w)ew^). We have
E I E £(WKJ = E £(w)exP(<E wfL)t)= 52 £(w)exp((p., w-^)t)
\weW	J weW	weW
= E £(w)exp«p,. w^}t) = Oll I 52 s(w)ew( j .
weW	\weW	/
262
Character and dimension formulae
In particular we have
др I E £(WK(A+p) j = 6x+p I E £(WKp j
yweW	/	\u+W'	/
= dAp (e-p П (е«-!))
\ аеФ+	/
= exp((A + p, -p)t) ]”[ (exp(A + p, a)t- 1)
аеФ+
= exp((A + p, -p)t) П ((A + p, a)t + -• •)
аеФ+
= tN I ]""[ (A + p, a)-I-j where /V = | Ф|.
\аеФ+	/
By putting A = 0 we obtain
0p (E E(w)euV) =tN( П (p> “>+•••) •
\weW	J \аеФ+	/
Thus by applying вр to Weyl’s character formula we obtain
tN ( П (p> «) + ••• )EdimL(A)pexP((P’^)0 = fA' ( П (A + p,a) + ---j
\аеФ+	/ M	\аеФ+	/
By cancelling tN and then taking the constant term we obtain
I ]“[ (p, a) j dimL(A) = ]“[ (A + p, a).	□
\аеФ+ /	аеФ+
12.6 Complete reducibility
We have now attained a good understanding of the finite dimensional irre-
ducible modules for a semisimple Lie algebra L. We now consider arbitrary
finite dimensional L-modules. Each of these turns out to be a direct sum of
irreducible L-modules.
Theorem 12.20 Let Lbe a semisimple Lie algebra and V a finite dimensional
L-module. Then V is completely reducible.
Proof. We shall prove this result in a number of steps. If V is itself irreducible
there is nothing to prove. Thus we suppose U is a proper submodule of V. It
12.6 Complete reducibility
263
will be sufficient to show that U has a complementary submodule U' in V,
that is a submodule such that V = U® U'.
(a)	Suppose dim V = 2, dim U= 1. Then U and V/U are 1-dimensional
L-modules. Since for x, у e L, и e U we have
\xy\u = x(yu)-y(xu)
and since the actions of x and у on the 1-dimensional module commute
we have
[ху]и = 0.
Thus [LL] acts as 0 on U. Since L is semisimple we have [LL] = L. Thus
U gives the trivial 1-dimensional representation L(0). Similarly V/U is
isomorphic to L(0).
Now let v e V. Then
[xy]u = x(yu)-y(xu).
Since L annihilates V/U we have xv e U and yv e U. Since L annihilates
U we have x(yu) = 0 and y(xv) = 0. Hence [xy]v = 0. This shows that
[LL] annihilates V, i.e. L annihilates V. But then any complementary
subspace U' of U is a submodule of V.
(b)	Suppose U is irreducible, dim U> 1, and dim(V/L) = 1.
Then U is isomorphic to L(A) for some A e X+ with A 0. We consider
the action of the Casimir element c on V. We recall from Proposition 11.36
that c acts on the irreducible module L(A) as scalar multiplication by
(A + p, A + p) — (p, p). In particular c acts on L(0) as zero, and c acts on
L(A) for AeX+, A^O, as multiplication by a positive scalar. For then
(A, A) > 0 and (A, p) > 0 since A e X+. Thus c has one eigenvalue 0 on
V and dim V — 1 eigenvalues yA(c) = (A + p, A + p) — (p, p) > 0. Let U'
be the eigenspace of c on V with eigenvalue 0. Then we have
v=u®u'.
Moreover U' is a submodule of V. For let x e L, u' e U'. Then
c (xu1) = x (cu'/ = 0
since c lies in the centre Z(L) of II (L). Thus xu' e U' and U' is the
required submodule of V.
264
Character and dimension formulae
(c)	Suppose dim(V/I7) = 1 but U is not irreducible. We prove the existence
of the required complementary submodule U' by induction on dim U. Let
U be a proper submodule of U. Then by induction we have
V/U=U/U®V/U
for some submodule V of V containing U. We have dim(V/t7)= 1 and
dim U < dim U. Thus we may apply induction again and conclude that
there exists a submodule U' such that
v=u®u'.
But then we have V = U ® U' as required	□.
(d)	We now consider the general case when U is any proper submodule of V.
We consider the set Hom(V, U) of all linear maps from V to U. We can
make this set into an L-module as follows. If xeL and 0eHom(V, 17)
we define хв eHom(V, 17) by
(x0)v = x(0(v)) — ff(xv) e U.
Then we have
(y(x0))u = y((x0)u) - (x0)(yv)
= y(x(0v)) ~ У(0(-™)) - x(0(yv)) + 0(x(yv)).
Similarly
(x(y0))u = x(y(0u)) - x(0(yu)) - y(0(xv)) + 0(y(xu)).
Thus
My0) - У(хбУ)» = x(y(0v)) - y(x(0u)) + 0(y(xv)) - 0(x(yu))
= [ТУ](^)-0([ху]и)
=
Thus Hom(V, 17) is an L-module. Let 5 be the subspace of Hom(V, 17)
of maps в such that 0\v is a scalar multiplication. Then .S' is a submodule
of Hom(V, 17). For suppose в e S, x e L. Then for и e U we have
(xff)u = x(6u) — 0(хи) = cxu — £xu = 0
where в acts on 17 as multiplication by 7. Thus .S' is a submodule of
Hom(V, 17). Moreover let T be the subspace of 5 of maps в such that
в\и is zero. Then T is a submodule of 5 and dim (.S'/7 ) = 1.
We know then from the earlier parts of the proof that there is a sub-
module T' of 5 such that S = T®T. We have dim I" = 1. Suppose T' is
12.6 Complete reducibility
265
spanned by the non-zero element f : V U. We may choose f so that
f(u) = u for all ueU. We have xf = 0 for all xeL since dimL'=l.
Thus
(x/)v = x(/v)-/(xv) = 0
for all v e V, that is
x(fv) = f(xv) for all x e L, v e V.
This shows that f : V U is a homomorphism of L-modules. Let U' be
the kernel of f. Then U' is a submodule of V. We have UC\U' =0 since
f acts as the identity on U, and
dim V = dim U + dim U'
since f is surjective. Hence we have
v=u®u'
and U' is the required complementary submodule.	□
Note The crucial step in the above proof of complete reducibility is the
fact that the Casimir element c acts on the irreducible module L(A) for
A e X+, A 0, as multiplication by a positive scalar.
The theorem of complete reducibility shows that every finite dimensional
L-module is a direct sum of irreducible L-modules each isomorphic to L(A)
for some A e X .
In particular the tensor product L(A)®L(/z) is a direct sum of irreducible
modules L(r) where A,/z, vfX . It is natural to try to determine the mul-
tiplicity with which L(r) occurs as a direct summand of L(A)®L(/z). This
multiplicity is given in a formula of Steinberg.
Theorem 12.21 (Steinberg’s multiplicity formula). Let A, p. eX+ and
L(X)®L(p)= £ c^Lfv).
PEX+
Then
c^v= E s(w)s(w')^w(X + p) + w'(p + p)-(v + 2p)).
w, w'eW
Proof. We have
chL(A)chL(^)= c^vchL(v)
veX+
266
Character and dimension formulae
by Lemma 12.2. We multiply both sides of this equation by the Weyl denom-
inator Д. By Weyl’s character formula, Theorem 12.17, we have
I E £(wK(A+p) I I Edim/dMhL j = E сл^ I E E(w~)ew(v+p) I •
Thus	x dim/.(^t)Ae„lA pi A = A, cXpvs(w)ew(v+p).
Now veX+, thus v + p lies in the fundamental chamber C. Thus w(y + p)
lies in the chamber w(C). Thus the elements w(y + p) are all distinct as w, v
vary, and so the elements ew(l,+p) are linearly independent. We may therefore
compare the coefficients of ew(l,+p) on both sides of the above equation. In
fact we compare the coefficients of ev+p on both sides. This gives
c^v = E E £(w)dim l(pE
weWf-_X
w(A+p)+£=v+p
= E £(w)dimL(M)r+p -w(A+p) •
weW
We now use Kostant’s multiplicity formula, Theorem 12.18. This gives
dim L(ji)v+p -w(A+p)	E £(w')^(w'(m + p) + w(a + p)-(z? + 2P))-
w'eW
Thus we obtain
сАрг= E £(w)£(w')^(w(A + P) + w'(M + p)-(^ + 2p)).
w,w'eW
13
Fundamental modules for simple
Lie algebras
13.1 An alternative form of Weyl’s dimension formula
Let L be a finite dimensional simple Lie algebra. The irreducible L-modules
L (<y;) whose highest weights are the fundamental weights Wj,... , a>l are
called the fundamental modules. In this chapter we shall determine the dimen-
sions of the fundamental modules for the various simple Lie algebras. We
shall first derive an alternative form of the Weyl dimension formula which
will be useful in this respect.
Theorem 13.1 Let A = J2;=1 т,ы, be a dominant integral weight. Then
dimL(A) = ]~[ d,,
аеФ+
where a = ^fli=ikiai and
< =-----^-7-------•
ЕЛ1";
Here the integer wt, called the weight of at, is defined by
where aio is a short fundamental root. Thus w; e{l, 2, 3} for each i.
Proof. We know from Theorem 12.19 that
dimL(A)= ]~[ d,,
аеФ+
(A + p, a)
where d„ = — ---;—.
(P,
267
268
Fundamental modules for simple Lie algebras
Since A = Emi<uo p = Ewo oi = ^fkiai we have
a~ <Х^кЛ) ’
Now we know from the proof of Proposition 10.18 that (<y;, cf} = 0 if
and (<y;, at) = | {at, ai). Thus
E; (m; + l)A;-|
ЕЛ;-1
_ E; (mi + tykjWj
E;A;W;
13.2 Fundamental modules for At
The fundamental weights a>,,... , a>l for a simple Lie algebra of type A; will
be numbered according to the vertices of the Dynkin diagram as shown.
О-----О— • • • —О------о
12	/ 1 Z
We shall use Theorem 13.1 to calculate dim Л (wj for je{l,... ,/}. We
have dimL (wj = ГТ, ф <4, where
, E; (^; + 1)Аг
a~ Kkt ’
(All weights w; are equal to 1.) Now mj= 1 and m; = 0 if i^j. Thus if a
does not involve the fundamental root a. we have da = 1.
So suppose a does involve c^. Then a = ai-\-l-a^H-bar for some
i with 1 < i < j and some к with j <k<l. For such a root a we have
к — i: + 2
<к = -,—“fm
к — i T 1
Thus
,	\	.-г	. к — i + 2
dimL(^) = П П p--7
i к K l~rL
j<k<l
(A+1)A... (A —j + 2) _	k+1
k k(k-l)...(k-j+l) k k-j+1
j<k<l	j<k<l
= (j +l)(j + 2)...(/+1) =	G+l)!	= fl +1\
1.2...(/+1-J)	j!(Z+1-7)!	V	j	Л
13.2 Fundamental modules for At
269
Thus we have shown
Proposition 13.2 The dimensions of the fundamental modules for the simple
Lie algebra of type are
□
These fundamental modules may be described in terms of exterior powers
of the natural A;-module of dimension 1+ 1. We recall from Theorem 8.1 that
A[ is isomorphic to the Lie algebra §[;+1(C) of all (/+1) x (/ + 1) matrices
of trace 0. The identity map gives an (/+ 1)-dimensional representation of A;
called the natural representation. Its weights are the maps fi1, p.2, • • • , Mz+i
given by
Ai
^/+1/
Then we have

Ml “I----h^;+1 =0.
On the other hand we have at = JT Ajfitj by Proposition 10.17. Hence
=2.6»! — w2
a2 = — и, + 2(i)2 — (i)3
“/-i = -w/-2 + 2w/-i-w/
“/=	-w;_i+2w;.
270
Fundamental modules for simple Lie algebras
Eliminating Op ... , we obtain
^i = «!
М2 — — Wl +W2
№•1—	W|-l+W!
Mz+i = ~Mi-
Now the weights of the natural module satisfy /z1>/z2>---> Mz+i since
/X; — Mi+i = ai- Thus the highest weight of the natural module is = <>j . It
follows that L (и, ) is one of the irreducible direct summands of the natural
module V. Since
dim V = dimL (wj = 1+ 1
it follows that V = L (coj. Thus we have shown
Proposition 13.3 The natural Armodule is an irreducible module with high-
est weight Wp
To obtain the remaining fundamental A;-modules we introduce exterior
powers of modules.
13.3 Exterior powers of modules
Let V be a finite dimensional module for a Lie algebra L. Let
L(V) = L°(V) ф T\V) ф T2(V) ф • • •
be the tensor algebra of V, where
L"(V) = V®---® V (и factors).
L(V) may be made into an associative algebra in which
(x1®---®xm) (y1®---®y„) = x1®---®xm®y1®---®y„
for хр... ,хт,У1,... ,y„eV.
T(V) may also be given the structure of an L-module satisfying
m
X (Xj ® • • • ® Xm) = Xj ® • • • ® X;_J ® XX; ® X; + j ® ’ ’ ’ ® Хщ
for all xeL.
13.3 Exterior powers of modules
271
Let J be the 2-sided ideal of Г( V) generated by the elements v 0 v for all v e V.
Definition 13.4 A(V) = l'(V)/J is called the exterior algebra of V.
Let v, v' e V. Then
(n + l/) ® (n + l/) = V ® V + v' ® v' + V ® v' + v' ® V.
Hence
v ® v' + v' ® v e J.
Now let ... , be a basis of V. Then J is the 2-sided ideal of T(V)
generated by all elements of form
i = 1, ... , n
vt 0 Vj + Vj ® vt i < j.
It follows from this that
7 = ф(^(У)П7)
k>Q
and that Г°(У) П J = О, Г1 (V) П J = O. Hence
Л(У) = Л°УфЛ1УфЛ2Уф---
where A.k V = Tk( V)/Tk( V) П J. In particular we have
A°V=r°(V) = Cl
A1V=T1(V) = V.
Thus we may identify the subspace Л°УфЛ1 V of A(V) with Cl ф V.
Let <j : T(V) -a A(V) = T(V)/J be the natural homomorphism. We define
<j (v®i/) = vAv'	for v, v' e V.
Then every element of A(V) is a linear combination of elements
f,... ,ike{l,... ,n}.
The relations defining J may be written
Vt A Vj = 0 i E {1,... , n}
Vj AVj = — (VjAVj) i<j-
тп
Fundamental modules for simple Lie algebras
By applying these relations we see that each element of Л(У) is a linear
combination of elements
IL A • • • Л IL for Z, < • • • < L
l[	Ik	1	K,
and that the relations cannot be used further. Thus we have shown:
Proposition 13.5 (i) Л(У) = Л°УфЛ1 Уф• • • фЛ"V
(ii)	dimA*V=Q)
(iii)	dimA(y) = 2"
(iv)	The elements i^A-Ati^ for subsets {z\,... , ik} c {1,... , n} with
z\ < • • • < ik form a basis of Л(У).	□
We now show that Л(У) has the structure of an L-module. We recall that
T(y) is an L-module and that its ideal J is generated by v®v for all veV.
For r eL we have
x(u ® и) = хи ® и + и ® xu.
Since the right-hand side lies in J we see that J is a submodule of Т(У).
Thus Л(У) = T(V)/J can be made into an L-module in the natural way. Each
exterior power Л* У is a submodule.
Proposition 13.6 Let V be a finite dimensional module for the simple Lie
algebra L. Then the weights of A.kV are all sums of k distinct weights of V.
Proof Let H be a Cartan subalgebra of L. We consider У as an //-module.
У is a direct sum of 1-dimensional //-submodules. Let	, vn be a basis
of У adapted to this decomposition. Let Ab ... , A„ e H* be the corresponding
weights. Then
xv = A;(x)v; for xeH.
Now А*У has basis v< A• • • Av< for all z, < • • • < i,. We have
Zj	Ifc	1	к
к
X (V;	Vi A • • • Л XV; A • • • A V;
\ И	1к/	И	lr	lk
r=l
= (\1(х)+---+Ч(Аг’.1л-”Лг’.)[ xeH-
Thus vt A- • • Avt is a weight vector with weight A; -I--Thus the
weights of A.kV are sums of к distinct weights of У.
13.3 Exterior powers of modules
273
Theorem 13.7 Let V be the natural module for the simple Lie algebra At.
Then the fundamental modules for are
Л1 V, Л2V,... , AlV.
Proof We have seen in Proposition 13.3 that Л1У= V is the fundamental
module with highest weight <>j . The weights of V are the maps fi1,... , /z;+1
given by
A
Pi-
\
Since /X; — /z;+1 = a; we have /z; > ju,;+1. Thus the weights are ordered by
Mi>---
Now	/jLt — fxi+i = at and at = JTA^op by Proposition 10.17. Hence
aj = 2oj, — a>2
ai = ~Mi-i +	— <W/+i for 2 < i < I — 1

It follows that
Mi = "i
p2 — —«1 +<U2
= —0^2 T
Pl— —
Pl+l — Ы1-
By Proposition 13.6 the highest weight of AkV is --------------hp.k = ык. for
1<£</. Thus AkV contains the irreducible module L (шк~) as one of its
irreducible direct summands. However,
dim L (wj) = dim Ak V =
I +1
к
by Proposition 13.2. Hence L (шк~) = AkV.
□
274
Fundamental modules for simple Lie algebras
13.4 Fundamental modules for /> and /А
The fundamental weights oj,,... , a>l for a simple Lie algebra of type P> or
if will be numbered according to the labelling of the Dynkin diagrams:
l-l
O-
* * • —В > О В/
1	2	3
O-
We again use Theorem 13.1 to calculate dimL (w;).
We suppose first that we have an algebra of type
Section 8.3 that the roots have the following form. Let
B;. We know from
Then the fundamental roots are
ai (/i) = A; — A;+1 for 1 < i < I — 1
CKz(^) = Az
The full set of positive roots is given by
A,. — A; for i < i
1	J	J
A, + A; for i < i
1	J	J

where i, j e {1,... , /}. These positive roots can be expressed as combinations
of fundamental roots as follows:
a; 4-----h o,z_1	for i < j
at 4-----+ 2aj 4-----------h 2al	for i < j
-----ha;.
13.4 Fundamental modules for Bt and I)
275
The first two families are long roots and the third family are short roots. Thus
the weights w; are given by
Wj = • • • = w;_j =2	m,= l.
According to Theorem 13.1 we have
dimL (a>j) = ]~[ da
аеФ+
where a = 'ffkiai and
Y!i=ikiwi + kiwi
d —----------------—-
a	1
We have da = 1 if a does not involve aj. We first suppose j e {1,... , I — 1}.
Then the positive roots involving j are:
«,4----ho,4------1 <i<j, j <k<l-l
at 4---h aj 4----h a, l<i<j
«,4----ho, 4-----1аы + 2а44--------h2o; 1 < i < j, j < к - 1 < I
a,.4---hOj, 14-2ai,4-----h2a;4-----h2a, l<i<k, k< j <1—1.
The values of da in these four cases are
k — z'4-2	2/ —214-3	'll — k — z'4-2	ll — k — i + 3
к — z'4-1’	ll — 2z4-l’	ll — к — z'4-1’	ll — к — i4-l
respectively. The product of all possible da in these four cases is
(j + 1)(j + 2)---Z	2/4-1	(2/ —j)(2/ — j —1) •••(/+1)
1-2---Z-J	’	2/-2У+Г	(2/— 2j)(2/— 2j — 1) • • • (/4-1 — j) ’
2/(2/— 1)(2/— 2) ••• (2/4-2 —j)
(2/ - j) (2/ - j - 1) • • • (ll - Ij + 3) (ll - Ij + 2)
respectively. Finally the total product П, ф ^, is C^1)- ^'c now ta'<c J = k
Then the positive roots involving / are
o; 4---h al 1<1<1
o;4----hoJfc_14-2oJfc4---h2o; l<i<k<l.
276
Fundamental modules for simple Lie algebras
The values of d„ in these cases are 2\	‘ J: respectively. The prod-
a	2i—2i+l '	21—i—j+l r	J	r
uct of all possible da in the two cases is
2Z(2Z —2)(2Z —4) • •-2
(2Z— 1)(2Z — 3) • • - 3 • 1 ’
(2Z — 1)(2Z —2) • • • (Z+1)
(2Z — 2)(2Z — 4) • • • 2
respectively. Finally the total product fL, ® da is 2'.
Thus we have shown:
Proposition 13.8 The dimensions of the fundamental modules for the simple
Lie algebra of type If are
2,„ (2,.,) (2,.,)
О---------о------о—
The dimensions of the modules L for 1 < j < I — 1 suggest that these
modules are exterior powers of the (21 + 1)-dimensional natural module. This
is indeed the case.
Theorem 13.9 Let V be the (21 + Ifdimensional natural module for the
simple Lie algebra If (described in Section 8.3). Then the fundamental module
L (a>j) is isomorphic to Л-'T for 1 < j < I — 1.
Proof. Let
Then the weights of V are 0, fir,... , /z;, — fir,... , —p,l where [+(h) = A;.
Since /X; — p.i+1 = at for 1 < i < I — 1 and /z; = al we have
> Mz> •••> M/>‘ o.
13.4 Fundamental modules for Bt and I)
277
Thus the highest weight of Л-'V for 1 <j<l is Mi + м2H--hM7• Expressing
the fis in terms of the as gives
Mi = “id----Ha;
Mi =	-----h a;
Mz = «z-
We also have a; = £ Ajfif’ which in type gives
aq = 2(i)1 — w2
a; = — W;_i +2<w; — <w;+j	2<i<l—2
“/-i = -w;_2 + 2«;_i-2w;
“/=	-W/-i+2«;.
It follows that
Mi = «i
Mi = — и, +oj2
Mz-i — ~ы1-г + o)l_1
М/ = — w/ i +2w;.
Hence Mi H---d M7 = w7 f°r 1 < J < I ~ 1
Mid--------hM/ — 2cu;-
Thus the highest weight of Л-'T is a>j for 1 < j < I — 1. Since
dim L (wj = dim Л-' V =
2/+1
. j
j<l~l
we deduce that L (a>j) is isomorphic to Л-' V. This argument fails when
j = l since the highest weight of A.lV is 2<y; rather than <y;. We shall see
subsequently how to find the remaining fundamental module L (<u;).	□
278
Fundamental modules for simple Lie algebras
We now consider the simple Lie algebra of type /),. This algebra was
described in Section 8.2. Its roots have the following form. Let
/Ai	\
\
Then the fundamental roots are
ai (/i) = A; — A;+1 for 1 < i < I — 1
“/W = Vi + A/-
The full set of positive roots is given by
h^- A; — A; i< i
I	J	J
h^- A; + A; i < i.
1	J	J
These are expressed as combinations of the fundamental roots by
----^aj-i forl<z<j</
----hOy-i+2oyH----l-2al_2 + al_1 + al for l<i<j<l-l
at 4---h a;_2 + o'/	for 1 < i < I — 2.
We take a fixed j with 1 < j < I — 2 and consider dim L (op). By Theorem 13.1
this is given by
dimL(wJ= П da
аеФ+
where a = 'fflkiai and
. _eLa+^
“ eLa
13.4 Fundamental modules for I/ and I)
279
(All weights w; are equal to 1 in type Z>;.) As usual da = 1 if a does not
involve a.j. The positive roots involving ctj are
«Н----------------1 <i<j, j <k<l-l
ai + --- + aj + --- + ak + 2ak+1 + --- + 2al_2 + al_1 + al l<i<j, j<k<l-2
oii + --- + ak + 2ak+1 + • • • + 2aj + • • • + 2a;_2 +	+ at l<i<k<j
at q----h Uj q----h a;_2 + al 1 < i < j.
The values of da in these four cases are
к — i q- 2	21 —i —к 21 — i — k+l l — i+1
к — i q-1 ’	21 — i — к — 1 ’ 21 — i — к — 1 ’ l — i
respectively. The product of all possible da in these four cases is
(j+l)(j + 2) • • • /	(2/— j — 1)(2/— j — 2) • • • (/+1)
1-2----(Z-j) ’	(2/— 2j — 1)(2/— 2j — 2) • • • (/q-1 — j) ’
(2/— 1)(2/— 2) • • • (2/— j q-1)	/
(2/— j — 1)(2/— j — 2) • • • (2/— 2j q-1) ’ ~j
respectively. Finally the total product is .
We next suppose j = l—l. The positive roots involving are
a; q----h	1 < i:< I — 1
at q----h ak_L q- 2ak q--h 2at_2 + al_1 + al 1 < i < к < I - 1.
The values of da in these two cases are	respectively. The
product of all possible da in those two cases is /, 2'1 / / respectively, and so
the total product is 2'
Finally suppose j = /. The positive roots involving a; are
ai
at q----h a;_2 q- a; 1 < i < I — 2
at q----h ak_L q- 2ak q--h 2at_2 + al_1 + al 1 < i < к < I - 1.
The values of da in these three cases are 2,	respectively.
The product of all possible da in these three cases is 2,1/2, 2' 1 /I respectively.
Thus the total product is 2'
280
Fundamental modules for simple Lie algebras
Thus we have shown
Proposition 13.10 The dimensions of the fundamental modules for the simple
Lie algebra of type if are
2'-1
2'-1
□
Again the dimensions of these modules for 1 <j <1 — 2 suggest that they
are given by exterior powers of the natural module.
Theorem 13.11 Let V be the 21-dimensional natural module for the simple
Lie algebra if (described in Section 8.2). Then the fundamental module
L (a>j) is isomorphic to Л-'T for 1 <j <1 — 2.
Proof. Let
Then the weights of V are ... , /х;, —.
have
, —p./ where р./(Ь) = Х/. We


Pi-i+Pi = ai
and
=2.6»! — w2
a2 = — a>1 +2(i>2 — a>3
al-3~ W/-4 + 2.W;_3 W/ 2
13.5 Clifford algebras and spin modules
281
al-l — W/3+2W; 2 w/ I
a/-i = — W/ 2 + 2«/-l
«/ = -"/ 2+ 2w/
using the Cartan matrix of type I).. It follows that
Mi = "i
М2 = — Wl 2” W2
M/-2 — -W/-3 + W/-2
M/-1 — OJI2 + W/ I + W/
М/“	W/-l+W/-
Since > p2 > > /х;_х > fjL[ the highest weight of Л7 V for 1 < j < I — 2 is
px-\----hpj. Also we have px-\------\-pj = (i)j for j <1 — 2. Thus the highest
weight of Л7 V is op when j <1 — 2. Since
I 21
dimL (w;) = dimA7V/ = I .
J</-2
we deduce that L (a>j) is isomorphic to Л7 V for j < I — 2.
□
13.5 Clifford algebras and spin modules
There remain one fundamental module for of dimension 2l and two funda-
mental modules for D; of dimension 2' 1 which cannot be obtained as exterior
powers of the natural module. These are called spin modules and give rise
to spin representations of and D;. We shall now show how these modules
may be obtained in terms of the Clifford algebra.
Let V be a vector space of dimension n over C and suppose we are given
a symmetric bilinear map V x V C under which the pair v, v' maps to
(v, if) e C. Thus we have
(1/, v) = (v, v').
Let T(V) be the tensor algebra of V and J be the two-sided ideal of T(V)
generated by elements
V® v— (v, v) 1
for all v e V.
282
Fundamental modules for simple Lie algebras
Since
(v + i/) ® (v + i/) — (v + v', r + i/) 1 = (v® v— (v, г) I) + (l/® v' — (1/, i/) 1)
+ (v® v' + v' ® v — 2 (v, i/) 1)
we see that
v ® v' + v' ® v — 2 (v, i/) 1 e J for all v, v' e V.
Let C(V) = T(V)/J. Then C(V) is an associative algebra called the Clif-
ford algebra of V.
Now let vr,... , vn be a basis of V. Then the elements
L ® vt — (n;, Vj) 1
L ® Vj + Vj ® vt — 2 (v;, Vj) 1 i<j
lie in J and it is evident that these elements generate J as a 2-sided ideal. We
observe also that
(Cl®V)nj=(r°(V)®r1(V))nj = C>
and so the natural map T( V) C(V) is injective when restricted to Cl ® V.
We shall regard С1ф V as a subspace of C(V). Thus C(V) is generated, as
associative algebra with 1, by elements , vn subject to relations
VfVj = (u;, Vj) 1
VjVj = — VjVj + 2 (vj, Vj) 1 i<j-
By using these relations any polynomial in , vn can be written as a poly-
nomial in which each monomial has form vt vt ... vt where < i2 < • • • < ik
and 0 < к < n. Moreover an element of C( V) in this standard form cannot be
simplified further by the use of the above relations. Thus we have shown
Proposition 13.12 (i) dim C( V) = 2".
(ii) The elements vti vi2... vit for z\ < i2 < • • • < ik with 0<k<n form a basis
for C(V). (The empty product is l.j	□
We note that all generators of J lie in Г°УфГ2У. We define
T(V)+, T(V)- by
T(V)+ = ф TV
i even
T(V)- = фГУ.
i odd
13.5 Clifford algebras and spin modules
283
Then we have
Г( V) = Г( V)+ ® T(V)~
J = (J n Г( V)+) © (J n r( V)-) .
This follows from the fact that J is generated by elements of '/'(V7). Hence
T(y)+ T(VY
CfVl = v 7 ф v 7
V 7 jnr(V)+ Jnr(V)-*
We write C(V)+ = /yff- and C(V)~ = 7^^. Then
C(V) = C(V)+©C(V)-.
In terms of our basis for C( V), C( V7) has basis vt v^... vit for z\ < i2 < • • • <
ik with k even and CIV7) has basis consisting of these elements with k odd.
Thus
dim C( V)+ = dim C( V)" = 2" 1.
Now the associative algebra CIV7) can be made into a Lie algebra [C(V)]
in the usual way by defining [xy] = xy — yx. Let L be the subspace of [C( V)]
spanned by the elements [W] for all v, v' e V. Then L can be spanned by
elements for i<j, and since
[v;v7]=2v;v7-2(v;,v;)1
these elements are linearly independent. Thus dimL = n(n — l)/2. We shall
show that L is a Lie subalgebra of [C(V)].
Lemma 13.13 (i) Let x, y, z e V. Then [[xy]z] = 4(y, z)x — 4(x, z)y.
(ii) Let x, y, z,we V. Then
[[xy], [Zw]] = 4(y, z)[xw]-4(y, w)[xz]+4(x, w)[yz]-4(x, z)[yw],
Proo/. (i) [[xy]z] = (xy - yx)z - z(xy - yx)
= xyz — yxz — zxy + zyx
= -xzy + 2(y, z)x + yzx-2(x, z)y
+xzy - 2(x, z)y - yzx + 2(y, z)x
= 4(y, z)x-4(x, z)y.
284
Fundamental modules for simple Lie algebras
(ii) [[xy], [zw]] = [xy]zw - [xy]wz - zw[xy] + wz[xy]
= [[xy]z]w + z[xy]w — [[xy]w]z — w[xy]z — zw[xy] + wz[xy]
= [ [xy] z] W - [ [xy] w] Z + z [ [xy] w] - w [ [xy] z]
= [[[xy]z]w]-[[[xy]w]z]
= [4(y, z)x-4(x, z)y, w] - [4(y, w)x-4(x, w)y, z]
= 4(y, z)[xw] -4(x, z)[yw] -4(y, w)[xz]+4(x, w)[yz]. □
Corollary 13.14 L is a Lie subalgebra of [C(V)].	□
Now C(V) is a [C(V)]-module giving the adjoint representation so is in
particular an L-module. Lemma 13.13 (i) shows that its subspace V is an
L-submodule.
Proposition 13.15 Suppose the symmetrix scalar product on V is non-
degenerate. Then V is a faithful L-module.
Proof. Let xeL and suppose [хи] = 0 for all v e V. We must show x = 0. Let
x='f2i<j ctj We may define a skew-symmetrix n x n matrix C = (cy)
by c;; = 0 and Cjj = —Cij for i < j. We have
Е4Ы”]=°-
By Lemma 13.13 (i) we have
4 E cij ((vpv) vi - v)vj) =°-
The coefficient of vt in this expression is
n
=4Ecy(^,^).
7=1
It follows that
Ecv(npn)=o
7=1
13.5 Clifford algebras and spin modules
285
for all i and all v e V. Let (v^, v^) = m.jk. Then we have
Ec/'7/'=° for all iff
7=1
that is CM = О where M = (m^. If the scalar product on V is non-degenerate
then M is a non-singular matrix. Then C = О and so x = 0.	□
Lemma 13.16 LetxeL and v, v' e V. Then
([xv], v') + (v, [xv']) = 0.
Proof. It is sufficient to prove this when x = [yz] for y, z e V. Now
([[yz]v], v') = 4(z, v) (y, v') —4(y, v) (z, v')
(v, [[yz]v']) = 4 (z, v') (v, y) —4 (y, v') (v, z)
by Lemma 13.13 (i). The result follows.	□
Thus we have a Lie algebra L of dimension n(n — l)/2, an L-module V of
dimension n, and a symmetric bilinear scalar product on V invariant under L
in the sense of Lemma 13.16.
We now consider some special cases of the above situation. First let V be a
vector space with dim V = 11 + 1 and let v0, vp ... , v;, v_p ... , v_; be abasis
of V. Consider the symmetric bilinear scalar product V x V -+ C determined by
(v0, v0) = 2
(v;, v_;) =1 i = 1,... , I
and all other scalar products of basis elements 0. The matrix of this scalar
product is
/2 0 • • • • 0\
0 О
\o A o'
The condition
([xv], v') + (v, [xv']) = 0
286
Fundamental modules for simple Lie algebras
of Lemma 13.16 tells us that if the element x e L is represented by the matrix
X on V then
X'M + MX=O.
Now L has dimension 1(21 + 1) and, by Proposition 13.15, acts faithfully
on V. However, the Lie algebra of all (21 + 1) x (21 + 1) matrices X satis-
fying X! M + MX = О is the simple Lie algebra (cf. Section 8.3) and has
dimension 1(21+ 1). Since L is contained in this set of matrixes we must have
L = Bl.
We aim to find the spin module for inside the Clifford algebra C(V).
Recall that V has basis
v0, Uj,... ,u;,u_1,... ,v_t.
We define и; = vovt and u_t = vov_t for i = 1,... , /. Let U be the subspace of
C(V) spanned by elements
U ; U ; ... И ; И, . . . U,
-Jl ~J2 ~Jt 1 2	1
for all subsets < j2 < • • • < jt of {1,... , /} with 0 < t < I. Bearing in mind
the natural basis of C(V) we see that these elements are linearly independent.
Thus dim U = 2l. We have U c C(V)+.
Lemma 13.17 C(V)+U c U. Thus U is a left ideal of C(V)+.
Proof We first observe that C(V)+ is generated by the elements и; and и_;
for i= 1,... , /. This follows from the fact that v0 anticommutes with vt and
v_t for all i = 1,... , /.
We note next that u2 = 0 and u2_t =0 for i= 1,... , /, that, и;, Uj anticom-
mute and и_;, u_j anticommute when i^£j and both lie in {1,... , /}, that
и;, u_j anticommute for all i, j e {1,... , /}, and that
UiU_i + u_iUi = -4 -1.
For utu_t + И_;И; = иои;иои_; + V0V_iV0Vi = — Vq (vtv_t + U_;U;) = — 2- 2(Vj, u_;)l
= -4-1.
It follows from these relations that
И; • U ; ... И ; Ml ... И,
1 -Jl -Jt 1	1
_ ±4u_ji...u_i...u_ju1...ul	if ге{Л, ... ,jt]
0	otherwise
13.5 Clifford algebras and spin modules
287
where и_; means the term и_; is omitted.
if i e {ji,  , jt}
u_jui...ul ifi^,... ,jt}.
This shows that UfUcU and u_JJ c U, so C(V)+I7 c U.
□
This lemma shows that we may regard U as a C(V)+-module under left
multiplication. U is therefore a [C( V)+]-module under the same action. Since
L is a Lie subalgebra of [C(V)+] we may regard U as an L-module under
left multiplication.
Warning note Whereas the action of L on U is given by left multiplication
the action of L on C(V) considered earlier in this section was given by Lie
multiplication.
We consider the weights of the L-module U. In order to do this we identify
the diagonal Cartan subalgebra H of L.
Lemma 13.18 Under the above isomorphism L = Bl the element [v;v_;] eL
corresponds to the diagonal matrix

Proof. The matrix representation of L comes from the L-module V with basis
v0, vx,... , vh ... , v_[. Now
[hLihoM
[[v;v_;], vj =4 (y_t, Vj) vt—4 (yt, v?) v_t = 8ij-4vi
[[v;v_;],	=4 (y_t, v ) vt — 4 (yt, v_j) v_t = — 8tj -4u_;
by Lemma 13.13 (i).	□
288
Fundamental modules for simple Lie algebras
It follows from this lemma that the element
, J
4[u;u_;]
is represented by the diagonal matrix
/0	\
А/
-Ax
We recall from Section 8.3 that such matrices form a Cartan subalgebra
H of L.
We consider the action of h on the L-module U. We have
[U;U_;] =	| (l?QVqV~ V^VqVqV^
= —^uiu_i+^u_iui.
Thus
. . U_j ux. . .Щ = —^UiU-i • и
.. u.
.. u_
; U
..U,
.. u
— 2.U;
2u_j ...u
if i e {j'l, • • • , jt}
if Jt}-
Thus
hu ; ... и ; и, ... и, = Т
—Ji ~Jt 1	1	2
.. u.
where £; =
1 if i e i.}
. Let Li: e H* be given by u, (/i) = A;. Then
1 if i^{f,...Jt}
the weights of L coming from the L-module U are
13.5 Clifford algebras and spin modules
289
for all possible choices of the signs £; = ±1. In particular the highest weight
is yE,=lM,-
We recall from the proof of Theorem 13.9 that а>1 = | (/хх 4---h /z;). Thus
U has highest weight <y;. It follows that U contains the spin module L (tof)
as an irreducible direct summand. But
dim U = dim L (a>l~) = 2l.
Thus we have proved
Theorem 13.19 Let L be the simple Lie algebra of type Lf. Then the L-module
U constructed as above in the Clifford algebra is the spin module L (tof) of
dimension 2l.	□
We now consider a second special case. This time let V be a vector space
with dim V = 21 and let , % v_r,... ,u_(bea basis of V. Consider the
symmetric bilinear scalar product V x V C determined by
(v;, v_t) = 1 i = 1,... , I
and all other scalar products of basis elements are 0. The matrix of this scalar
product is
The condition of Lemma 13.16 implies that if xeL is represented by the
matrix X with respect to this basis of V then
X'M + MX = 0.
Now L has dimension 1(21 — 1) and acts faithfully on V. The Lie algebra of
all 21 x 21 matrices X satisfying XlM+MX = О is the simple Lie algebra If.
by Section 8.2. Since dim ЛТ = /(2/ — 1) we have L = Dt.
We again aim to find the two spin modules for /) inside the Clifford
algebra C(V). Let U be the subspace of C(V) spanned by all elements of
form
for all subsets < • • • < jt of {1,... , /}. These elements are linearly indepen-
dent, so form a basis for U. We have dim U = 2l.
Lemma 13.20 C(V)U c U. Thus U is a left ideal of C(V).
290
Fundamental modules for simple Lie algebras
Proof. C(V) is generated by elements vh v_t and we have
±2v_; ... v_t... v4 v^.v, if i e {f,jt}
Vt-V: ...V: V-, ...V} = { Ji	J!
~11	[o	if i^{h,
0	if ге{Л,... ,jt}
V]...V,= <
[±v_ji...v_i...v_jy1...vl	if г'00'i,... ,jt}
Thus vflcU and v_tU<zU, so C(V)UcU.	□
Let U+ = t/nC(V)+ and (/ =Or'C(l/)  Then we have
C(y)+U+cUnC(yf= u+
C(V)+U~ c unc(y)- = IT.
Since L c C(V)+ it follows that
LV+cU+, LU- tlT.
Thus U and U are L-modules under left multiplication, with
dimt/+ = dim tr =2'4
We shall show that these are the two spin modules for L.
We consider the weights of the L-modules U+, U~ by identifying the
diagonal Cartan subalgebra H of L.
Lemma 13.21 Under the above isomorphism L = I) the element [v;v_;] eL
corresponds to the diagonal matrix
(°
0
4
‘	0
0
—4
0
\ °/
13.5 Clifford algebras and spin modules
291
Proof. The proof is the same as that for Lemma 13.18 with the first row and
column omitted.	□
Thus the element
J
^4[u;u_;]
is represented by the diagonal matrix
/ax	\
J
-Ax
We consider the action of h on the L-modules U+ and U . We have
[иги_;]и_71.. .
= V;V ;V ; . . .V ; Vi ... V; ~ V ;V;V ; . . .V ; Vi . . .V;
1 ~l -Jl ~Jt 1	1 ~l 1 -Jl ~Jt 1	1
... v_jt v1...vl if i e {f,... , jt}
since v_ivi + viv_i=21. Thus
where я, =
if Jt}
if i £ {ji, , jt}-
As before let /х; e H* be defined by
\
292
Fundamental modules for simple Lie algebras
Then the weights of the L-module U are
i=l
for all possible choices of the signs £; = ±1.
If / is even, the basis elements with t even lie in U and those with t
odd in U . Thus the weights of U+ have an even number of £; negative
and those of U have an odd number negative. Since >- /z2 >---------М/ the
highest weight of U+ is | E=1 pt and that of U is | (J2;=i Pt ~ Pi)-
If / is odd we have the reverse situation in which | E=1 p-i is the highest
weight of IJ and | AL — Pi) is the highest weight of U+.
Now by the proof of Theorem 13.11 we have
I	(Mi 3	\~Pi-i +pi) = 0Ji
2	(Ml 3---\~Pl-l ~Pl) = Ml-1-
Thus we have proved
Theorem 13.22 Let L be the simple Lie algebra of type If. Then the
L-modules U+, U are the spin modules of dimension 2;-1. If I is even we
have U+= L(p)f), U = L(a>l_1). If I is odd we have U+ = L(p)l_f), U =
LM-
13.6 Fundamental modules for Cz
The fundamental weights oj,,... , <y; for a simple Lie algebra of type C; will
be numbered according to the labelling of the Dynkin diagram
1	2	3
О------О-----о
l-l
I
. . -------() г f>
As before we shall use Theorem 13.1 to calculate dimL (wj. We knows from
Section 8.4 that the roots of C; have the following form. Let
13.6 Fundamental modules for Ct
293
Then the fundamental roots are
a,(h) = A; — A;+1 for 1 < i < I — 1
a;(/i) = 2A;.
The full set of positive roots is given by
h A,. — A; for i < j
1	J	J
h A; + A; for i < j
1	J	J
h^2Xt
where i, j e{l,... , I}. These positive roots can be expressed as combinations
of fundamental roots as follows:
at d----h ci'y_1 1 < i < j < I
«, d----t-aJ_1+2aJ-l----l-2al_1 + al l<i<j<l
2at-\---\-2al_1 + al l<i<l.
The first two families are short roots and the third family are long roots. The
weights w; are given by
= • • • = w;_j = 1 w; = 2.
According to Theorem 13.1 we have
dimL (a>j) = ]~[ da
аеФ+
where a = 'ffkiai and
E;=i kiwi + kiwi
d —-----------------—-
a T!i=1kiWi
We have da = 1 if a does not involve aj.
We first suppose that je{l,... ,1 — 1}. Then the positive roots involv-
ing j are:
a;d-----hOyd-----hajt-1 ^<i<j<k<l
«Н------hOyd-----\-ak_1+2ak-\-------\-2al_1 + al l<i<j<k<l
at d----haJfc_1d-2aJfcd--h2aiyd-----h2a;_1d-a; i<i<k<j
2a;d----h2a7d-------h2a;_1d-a;	1<г</
294
Fundamental modules for simple Lie algebras
The values of da in these four cases are
k — г 4-1	2Z — i — £4-3	2l — i — k + 4	l — i + 2
k — i ’	21 — i — к 4- 2 ’	21 — i — к 4- 2 ’	I — i 4- 1
respectively. The product of all possible da in these four cases is
(j +l)(j + 2) • • • /	(2/— j+1)(2/— j) • • • (/ + 2)
1-2- ---l-j ’	(2/— 2j4-1)(2/— 2j) • • • (/— j'4-2) ’
(2/+l)2/(2/-l)---(2/-j + 2)	l+l
(2/ - j + 2)(2/-j Tl)---(2/— 2j'4-3)’	l-j+1
respectively. The total product fL, ® da is
(2/)!
(2,Jj4:2),j!P'+* 1)(2>-2J + 2).
This expression may be written in a more suggestive form by using the
identity
(2l\ ( 21 \	(2/)! z
СИ-2)=(Йтад(2,+1)(2'-2'+2)'
We now suppose that j = /. The positive roots involving al are
2a;4-----h2a;_14-a; l<i<l
at 4----h etj_x 4- 2aj 4-h 2a;_14- a;
1
The first family are long roots and the second short roots. The values of lda
in these two cases are
I — i + 2	2l — i—j + 4
I — г 4-1 ’ 21 — i — j 4- 2
respectively. The product of all possible da in these cases is
(2Z+l)(2Z)---(Z + 2)
(Z 4-2)(Z 4-1) • • - 3
respectively, and the total product ГТ, ф da is 
13.7 Contraction maps
295
By using the identity
we see that
(2Z+1)!2
(Z + 2)!Z!
' 2/ '
Z-2
Thus we have shown
Proposition 13.23 The dimensions of the fundamental modules for the simple
Lie algebra of type are
o-----о----о— • • • —t)	< (>
13.7 Contraction maps
We shall now identify the fundamental modules whose dimensions we have
obtained. We begin with
Proposition 13.24 The natural 21-dimensional C.-module is isomorplic to
L(wi)-
Proof Let V be the natural C;-module. Let
/л,	\
Then the weights of V are fir,... , /z;, —pt1,, —p,l where pcfh) = A;. Since
Pi~Pi+i = ai
2p.i = ai
we have
----> M/>‘ o.
296
Fundamental modules for simple Lie algebras
Thus the highest weight of V is We have
Mi = “iH------l~al_1 + ^al
M2 =	-----l~ai-i + ^ai
М/ = |“/-
We also have at = JT Ajfitj which in type C; gives
aj = 2(i)1 — a>2
a2 = — <y! + 2w2 —
«/ = -2"/ ! +2w/-
It follows that
Ml="l
M2 = —<«i + <«2
М/— w/-i + w/-
Thus V is a C;-module with highest weight <>j. It therefore contains L (wj
as an irreducible component. However,
dim V = dim Л (oj| ) = 21
thus V is irreducible and isomorphic to I. (op).
□
We now consider the fundamental modules L for j > 2. We have
, , f2l\ ( 21 \
dimL (co;) =	—
V \j) U-2/
This suggests that we should look for L as a submodule of the exterior
power Л-'T. The key idea is to find a homomorphism of C;-modules from
Л-'T into A.j~2V, called a contraction map.
13.7 Contraction maps
297
Proposition 13.25 Let v, v' (v, if) be the skew-symmetric bilinear map
V x V C given by the matrix
H-t o)-
Then there is a unique homomorphism of C [-modules
в : AjV^Aj-2V
satisfying the condition
в (их A - • • Л Иу) = 22(— I )'11 (ur, us)u1A- • • Лиг A • • • A
iis A • • • A Uj forall «j, ... , Uj eV.	(t)
Here as usual the notation ur, us means that those terms are omitted.
Proof. It is clear that if such a map в exists it will be unique. To prove the
existence let vr,... , i>2; be a basis of V. Then there is a unique linear map 0
satisfying
0 A-'-ADiJ = 22(— l)r+s-1 (yt , vt ) Vti A • • • A vt A - • - AU; A - • - Al>;.
for all q,... , ij e {1,... , 21} with q < • • • < f. We show this map has the
required properties. Since both sides of equation (j) are linear in их,... , Uj it
will be sufficient to prove it when each uk is one of the basis elements of V.
If the same basis element appears twice both sides of (j) are 0. Thus we may
assume the basis elements are all distinct. They may not occur in increasing
order, thus we must show that the above formula defining 0 remains valid if
the factors vt ,... , vt. are permuted. In fact it is sufficient to see this if we
transpose two consecutive terms vit,	. When we carry out such a transpo-
sition the expression vt A • • • A vt. changes in sign. We show that each term
(—l)r+s-1 (u; , vt ) Vti A- • - AU; A • • • A A- • - Al>;.
changes in sign also. If neither of r, s lie in {к, k+ 1} the term
l>; A • • • A U; A • • • A U; A • • • A U;
И	lr	ls	lj
will change in sign when we make the transposition. If just one of r, s lies
in {к, £-1-1} the term (—I )''11 will change in sign when the transposition
is made. Finally if r = k, ,y = £-|-l the term (v;	) changes in sign, since
the bilinear map is skew-symmetric. This shows that the linear map 0 we
have defined satisfies (j).
298
Fundamental modules for simple Lie algebras
It remains to show that в is a homomorphism of C;-modules. Let x lie in
the Lie algebra C;. Then
xO (u3 A • • • A Uj) = X If ’1 (ur, uf) Mj A • • • Л Иг A • • • AusA- • • AUj
= 22 У? (— I)'11 (ur, Us) MjA-• • AxukA-• • ЛигЛ-• •
r<s к
k^r,k^s
/\й./\- • • ЛИ;.
d	J
On the other hand we have
0x(hjA • • • A«p
= в JAM"' AxukA-   Auf= хв (и} Л-   Л И;)
к
+	(xuk,
к s
k<s
+ 222ZX- l)iZI (ur, xuf) u1A-• • ЛигЛ-• • ЛикЛ-• • AUj.
к г
r<k
Renaming the suffixes we see that the last two sums cancel since
(хи^ uf) + (uk, xuf) = Q.
This condition is equivalent to
X'M + MX=O
where X is the matrix representing x on V, and we recall from Section 8.4
that the simple Lie algebra C; satisfies this condition. It follows that
вх (их Л • • • A Uj) = хв (их Л • • • Л Uj)
and so в is a homomorphism of C;-modules.	□
This homomorphism в : Л7 V	A7 2 V will be called a contraction map.
Now the weights of Л7 V are sums of j distinct weights of V. By the proof
of Proposition 13.24 the weights of V are
a>, >--a>1 + (i>2>—a>2 + a>3>->—+ <y;
>- COZ_x — <Dl >->- O>2 — fc>3 >- fc>i — U>2 >-«1.
Thus if j < I the highest weight of Л7 V is op. Similarly the highest weight of
A7 2 V is Since op >	2 we see that op is not a weight of A7 2 V. Since
op is the highest weight of Л7 V the module L must be an irreducible
13.7 Contraction maps
299
direct summand of Л-' V. On the other hand L (w;) cannot be a submodule of
A7 ;V, as a>j is not a weight of this module. Thus L (w j must lie in the
kernel of the contraction map 0.
We shall show subsequently that when j < I the contraction map в : Л7 V
X' 2V is surjective. It will follow that
dim (ker ff) =
' 2/ A	, .
= dim L (a>;)
J-2/	V
and therefore that L (wj =ker 0. This will identify the irreducible module
L as the submodule of Л7 V which is the kernel of the contraction map 0.
Let Up ... , % v_p ... , v_j be the natural basis of V with respect to which
the skew-symmetric bilinear form is given by
(vt, v_j) =1	1 < i < I
(v_;, Vj) = — 1
and all other scalar products zero. Let W be the subspace of V spanned by
Vp... , V[ and W the subspace spanned by v_r,, v_;. Then W, W are
isotropic subspaces of V, i.e. the skew-symmetric form restricted to W and
W is identically zero. Also we have V = ТфИ'’ . It follows that
AjV= ф (A^giA^W).
a+b=j
The contraction map в : Л7 V X' 2V satisfies
d (A" IT0 A"VT ) c A" 1 IT0 A" 1 IT
since a basis element in W has a non-zero scalar product only with a basis
element in W . Thus in order to show that в : Л7 V X' 2V is surjective for
j < I it will be sufficient to show that
0 : A" IT®AftWA" 'IT® A" 1 IT
is surjective whenever a + b < I.
For each subset 7c{l,...,/} we define = vt A• • • Avt where 1 =
{z\,... , ik} with z\ < • • • < ik. We also define v_; = v_ti A • •  Av_it. Then any
basis element of A" 1 IT 0 A" 1 IT can be written in the form
± (vx A VT) ® (v_T A U_y)
for some subsets T, X, У of {1,... , /} with
ТГ\Х = ф, ТГ\У = ф, ХГ\У = ф, |X| + |Г| = а — 1, | У | + | Г| = b - 1.
300
Fundamental modules for simple Lie algebras
We write |Г| = r. Since a+ b < / we have |X| + |У| + 2r + 2 < I, that is / — |X| —
| У | > 2r + 2. Thus it is possible to choose a subset 5 of {1,... , /} such that
|5|=2r+l, 8С\Х = ф, 5П¥ = ф, SdT.
We can now describe an element of A“W®AiW which maps under в to a
non-zero multiple of (wzAwJ.)®(ii_TAii_y).
Proposition 13.26 Suppose subsets T, X,Y, S of {1,... , /} are chosen as
above, and let в : AdV AX2V be the contraction map. Then
E(—1/*!(r —z)!	(vx/\vu)®(v_u/\v_Y)
\UCT\=i
= (r + 1)! (vx A VT) ® (w_TAll_y) .
Consequently the map
в :	!W
is surjective when a + b<l.
Proof We note that Sz>T, |5| =2r+ 1, |Г| = r and that we are summing over
all subsets U of S with | U\ = r + 1 and | UCi Г| = i. Since |X| + | Г| = a — 1 and
|y| -|- |Г| = b — 1 we have |X| + |U\ = a and |У| + |U\ = b. Thus the left-hand
side lies in A“W® A"W .
By definition of в we have
v_M) = (—l)r 52 vR®v_R
where the right-hand side involves a sum over all r-element subsets R of U.
Thus
0 E
\|vnz|=,'	/
= (-i)r E E vr®v-
W\=r+1	|7J|=r
|£/ПГ|=г
= (-i)r E
R
vR ® V_R
13.7 Contraction maps
301
Since |t/A Г | = i and R is obtained from U by omitting one element we
have |RПT| = i or |АПГ| = г — l.We split the sum according to those two
possibilities. Thus
= (-iy E	E
R	U
\R\=r	RgU
\RC\T\=i	M=r+1
\|mr|=i
1	+ l)r у
R
\R\=r
\RC\T\=i—l
E
и
RaU
\U\=r+l
\|mr|=i
1 Vjj V_R
/
= (—i)r E G+ i)”/?&”-/?+(-!)r E	(r+i-i)vR®v_R
R	R
\R\=r	\R\=r
\R(YT\=i	\RnT\=i-1
since in the first case the additional element of U can be chosen in i + 1 ways
and in the second case in r+ 1 — i ways. Thus
/ r \
E(-l);/!(r-/)! E vv®v_v
i=0	U
\	\UCT\=i	/
= (—i)rE(—!)г/!(r—0! E G+1)”/?®v-R
i=0	R
\R\=r
\RC\T\=i
+ (-i)''E(-1);;'!(?--0! E (r+i-i)vR®v_R.
i=0	R
\R\=r
|ЛПТ|=г-1
We rename the variable i in the second sum to give
(- 1УE(-г'!(г-0! E (i+^vR®v_R
i=0	R
\R\=r
\RC\T\=i
+(-iy E(-l);+1G+l)!(r-i-l)! E (r-i)vR®v_R
i=-l	R
\R\=r
\RC\T\=i
302
Fundamental modules for simple Lie algebras
= (-l),'E(-1);G+l)!(?’-0! E G-1)”/?®”-/?
i=0	R
\R\=r
\RC\T\=i
+(r+l)! vR®v_R
R
|S|=r
|ЯПГ|=г
= (r+1)! <8> v_T.
We now consider
£(-iyz!(r-O! £
i=0	U
UaS
\U\=r+l
\	\UC\T\=i
/
Since the vt for i G X and the v_t for i G Y have scalar product 0 with all
factors in the above product they are not involved in any contraction. Thus
0 EH)'‘-(.r-0! E
i=0	U
\	\U(YT\=i
2=0
|£7ПТ|=г
= vx A ((r + 1)!vr ® v_T) A v_Y
= (r T 1)! (vx A vT) ® (v_rAl)_y) .
□
Corollary 13.27 The contraction map в : Л-' V Л-' 2 V is surjective when
j<l-
The surjectivity of в enables us to identify the fundamental modules L (wj.
Theorem 13.28 The fundamental modules L for the simple Lie algebra
Cj are given as follows.
(a)	L (wj) is the natural 21-dimensional (f-module V.
(b)	For 2<j<l,L (wj is the submodule of Л-'T given by the kernel of the
contraction map 0 : Л-' V	Л^2 V.
13.8 Fundamental modules for exceptional algebras
303
Proof, (a) was shown in Proposition 13.24. We also pointed out earlier that
L (wj is a submodule of Л-'T contained in the kernel of в. Since в : A>V
A7 2 V is surjective for j <1 we have
/2Z\ / 'll \
dim ker в =	—
\J / \j~2/
and this is equal to dimL (wj by Proposition 13.23. It follows that L (wj =
kerd.	□
13.8 Fundamental modules for exceptional algebras
By applying Theorem 13.1 to the exceptional simple Lie algebras and making
use of the information about their root systems available in Sections 8.5, 8.6
and 8.7 we can show that the dimensions of the fundamental modules for
these algebras are as shown. We omit the details.
G2
14	7
o-4—о
52 CD’52 C2)-52 26
f4	О-------o > o-----------О
E6
6 899 079 264
248	30380 2450240 146325270	6696000	3875
О--------О--------О-------О-------6-------о--------о
Es
147 250
304
Fundamental modules for simple Lie algebras
We shall show in each case how to obtain the fundamental module of small-
est dimension. We begin by obtaining a 27-dimensional fundamental mod-
ule for E6.
Proposition 13.29 (a) The number of positive roots of E-j not in E6 is 27.
(b) The subspace V of E-j spanned by vectors ea for such roots is a 27-
dimensional fundamental E6-module.
Proof We recall from Section 8.7 that the fundamental roots of E7 are given
by
If-lf If-IL fL-lf	Z A
O------О-------О------------о------о
and that the full set of roots of E7 is
±ft±ft i^j z, j e {2, 3,4,5, 6,7}
±(ft+ft)
|E£ift	П£/=1’ £1=£8-
The positive roots are
ft-ft i^j z, j e {2, 3,4,5, 6, 7}
ft + ft i^j z, je{2,3,4,5,6,7}
-ft-ft
|E£/ft £;е{1,-1}, П£/ = 1’ £i = £8 = -1.
The positive roots of E7 which are not roots of E6 are
ft-ft je {3,4,5, 6,7}
ft + ft	je {3,4,5, 6,7}
-ft-ft
IE£/ft П£/ = 1’ si=s8 = -i, £2=i-
The number of such roots is 27.
Now let V be the subspace of E7 spanned by the root vectors ea for such
roots a. Then dim V = 27.
13.8 Fundamental modules for exceptional algebras
305
Now Е-/ may be regarded as an E7-module giving the adjoint representation.
In particular E7 may be regarded as an E6-module. We observe that V is
an E6-submodule. To see this it is sufficient to show that [еае^ e V for all
a e Ф (E6), /3 e Ф+ (E7) — Ф+ (E6). We have
k/E =
Napea+p if a +/3 e Ф (E7)
otherwise.
Suppose a + /3 e Ф (E-f). Since /3 is not a root of E6,/3 will involve the
fundamental root of E7 not in E6, and since /3 is positive this fundamental root
will have positive coefficient in /3. It will therefore have positive coefficient
in a+/3, and so а + /ЗеФ+ (E7). We claim that a + /3 g<t> (E6). Suppose to
the contrary that a + /3 e Ф (E6). Then — a e Ф (E6) and
[ea+/3e-a] ^a+g,-aeg'
Since Na^Q it follows from Proposition 7.1 that Na+p_a^Q and so
/3 еФ(Е6), a contradiction. Hence a + /3 e Ф+ (E7) — Ф+ (E6) and V is an
E6-module.
In order to determine the highest weight of V it is convenient to use the
linear function
8
h :	IR/3; IR
i=l
determined by the property that h (a;) = 1 for each fundamental root at of
Eg. Thus
^(/3;-/3;+1) = l for ie {1,... , 6}
/z(/36 + /37) = l
h = 1-
\ i=l /
Hence we have
^(/30 = 6, /z(/32) = 5, /z(/33)=4, /z(/34) = 3, /z(/35) = 2,
й(/36) = 1,	/z(/37) = 0,	/z(/38) = —23.
Of our 27 roots the one with the highest /i-value is —/Зх—/38. This must
therefore be a highest weight of V. Now the fundamental roots of E6 are
306
Fundamental modules for simple Lie algebras
If-IL. '4 '5	'5 '6 ft+A?
o------o-—-<j>-------О—
/А~А?
8
4 SA
ю 2 i=i
and —ft — ft is orthogonal to all of them except — |	/3;. Moreover the
scalar product {,} satisfies
-ft-ft
thus —ft — ft is the fundamental weight w8. Hence L (w8) is an irreducible
direct summand of V. Since
dim L (w8) = dim V = 27
we deduce that V = L (w8).	□
In order to obtain the other 27-dimensional fundamental E6-module we
introduce the dual module. We recall that, given any L-module V, the dual
space V* of linear maps from V to C may be made into an L-module by the
rule
(x/)v = —/(xv) xeL, f eV*, veV.
The weights of V* are the negatives of the weights of V. In the case of the
27-dimensional E6-module V above, the highest weight of V* is the negative
of the lowest weight of V. The lowest weight of V is the one with the smallest
value of h, i.e. /32 — /З3. Thus the highest weight of V* is ft — ft. This is
orthogonal to all fundamental roots of E6 except for a3 = /33 — ft. Since
{ft-ft, ft-ft} = |{ft-ft, ft-ft}
we deduce that /З3 — ft = a>3. Hence V* = L (<w3).
Now the weight of V with second highest value of h is | (—ft + /32 + ft-F
ft + ft + ft + ft—ft) and the third highest is j (-ft + ft + ft + ft + ft
—ft —ft — ft). Thus the highest weight of A2V is
(-ft-ft)+|(-ft+ft+ft+ft+ft+ft+ft-ft)
= i(-3ft+ft + ft + ft + ft + ft + ft-3ft).
13.8 Fundamental modules for exceptional algebras
307
By considering the scalar products {,} of this weight with the fundamental
roots of E6 we see that this weight is <y7. Since
/27\
dim L (a>i) = 11 = dim Л2 V
we deduce A2V = L (w7).
Similarly the highest weight of Л3 V is
(-ft-ft) + |(-ft+ft + ft+ft + ft+ft + ft-ft)
+ l(-ft+ft + ft + ft + ft-ft-ft-ft)
= -2ft+ft + ft + ft + ft-2ft.
We check by computing scalar products {,} that this is the weight a>5 of E6.
Since
/27\
dimL(w5)=l l=dimA3V
we deduce A3V = L (<y5).
It may be shown similarly that
A2V*=L(w4) and A3V* = A3V = L(w5).
Finally L (w6) is the adjoint module. Thus the fundamental ft-modules are
V*	A2V* A3V*=A3V A2V	V
О-----о-------о-----о------о
о
L
We now consider the simple Lie algebra E7 and obtain a 56-dimensional
fundamental module. The idea is similar to what we have seen for E6.
Proposition 13.30 (a) The number of positive roots of Es not in E-j is 57.
(b) The subspace V of Es spanned by vectors ea for such roots is a
57-dimensional E--module. V decomposes as the direct sum of a
56-dimensional fundamental module with a 1-dimensional module L(G).
308
Fundamental modules for simple Lie algebras
Proof. We see from Section 8.7 that the positive roots of Ls not in E7 are
Pi-Pj j e {2, 3,4,5, 6,7}
Pi+Pj j e {2, 3,4,5, 6,7}
/3;-/38	ze{2,3,4,5, 6,7}
-/3;-/38 ze {2, 3,4,5, 6,7}
П£/=1’ £s = -l, Si = 1-
i=l
The number of such roots is 57.
Let V be the subspace of A's spanned by the ea for this set of roots. The
argument of Proposition 13.29 shows that V is an ^-module. Now /3, —/38
is orthogonal to all fundamental roots of E7 and it follows that
КеЛ -ft]=° for all a e Ф (E7).
Hence Ce(3i_(3g is a 1-dimensional ^-submodule of V. Let V be the subspace
spanned by the remaining ea. The fact that — /3S is orthogonal to all
a e Ф (E7) implies that /3, —/38 cannot be expressed in the form a + /3 where
аеФ(Е7),/ЗеФ+(Е8). This shows that V is an ^-submodule of V. Its
highest weight is obtained by picking the weight with the highest value of h,
and this is /32 — /3S. In fact the first few highest weights are
/з2-/з8, /З3-/з8, /з4-/з8, /з5-/з8, ...
By calculating scalar products {,} with the fundamental roots of E7 we see
that /32 — /38 = <y2. Thus L (<y2) is an irreducible direct summand of V. Since
dim L (w2) = 56 = dim V'
we have V = L (w2). Thus
V = L(w2)®L(0).	□
We can obtain information about some of the other fundamental ^-modules
by considering exterior powers of V. The highest weight of A2V' is
(/з2-/з8) + (/Зз-/з8) = /з2+/з3-Ж-
A calculation of scalar products {,} shows that
/32 + /З3 — 2/38 = a>3.
13.8 Fundamental modules for exceptional algebras
309
Thus A2!7' contains L(w3) as an irreducible direct summand. But we know
that
dim L (w3) =^2^ — 1-
Thus
A2V' = L(w3)®L(0).
The highest weight of A3!7' is
(/32 -/38) + (/33 -/38) + (/34 -/38) = /32 + /33 + /34 -3/38.
We have
Pi + & + /34 - 3/38 = w4.
Thus L (w4) is an irreducible direct summand of Л3 V. We know that
dim L (w4) =^3^—56.
In fact we have
AV =L(w4)©L(w2).
The highest weight of A4!7' is
(& - /38) + (& - /38) + (/34 - /38) + (& - &) = P2 + & + /34 + P5 - 4/38.
We have
/32 + /3Э + /34 + /35 — 4/38 = w5.
We know that
z ч /56А /56А
dimL(w5) = I I - I I.
In fact it turns out that
Л4 V = L (w5) ф L (w3) ф L (0).
Some of the remaining fundamental E7-modules may be identified by
means of the adjoint module. The highest root of E7 is —— /38 and we have
—— /38 = w8. Thus we see that L (w8) is the adjoint ^-module, since
dim Л (ws) = 133 = dim L.
310
Fundamental modules for simple Lie algebras
The second highest root of ft is | (-ft + ft + ft + ft + ft + ft + ft - ft).
Thus the highest weight of A2L is
(-ft-ft)+|(-ft+ft+ft+ft+ft+ft+ft-ft)
= | (—3ft+ft+ ft+ ft+ ft+ ft+ ft — 3ft).
We have
| (—3ft+ft + ft+ft + ft+ft + ft —3ft) = w7.
Thus L is an irreducible direct summand of Aft. Since
we have
Aft = L(«7)®L(0).
We next consider the simple Lie algebra ft. The smallest dimension of a
fundamental module for ft is
dim Л (oj| ) = 24<S.
The highest root of ft is ft—ft, and we have ft—ft = <yp Since
dim L = 248 we deduce that L (wj = L. Thus the fundamental module L (w, )
is the adjoint module.
The description of the remaining fundamental modules of ft is considerably
more complicated than in the other simple Lie algebras. We shall not discuss
the details.
We now turn to the simple Lie algebra ft and show how to obtain the
26-dimensional fundamental module. This will be done by identifying ft with
a subalgebra of ft. We shall retain our previous numbering of the fundamental
roots of E6 given by
3	4	5	7	8
О-------О------------О------о
6
Let <j be the permutation of the vertices given by
а=(3 8)(4 7)(5)(6).
13.8 Fundamental modules for exceptional algebras
311
Then cr gives a symmetry of the Dynkin diagram of E6 with cr2 = 1. We have
А«РЫР = А1з for all z, j.
Thus by Theorem 7.5 there is an automorphism of E6, which we shall also
call cr, satisfying
O’ («/) = ea(i)
<? (fi) = fa(i)
O'(hi) =
Since et, fi, ht generate the Lie algebra, cr is determined by these conditions,
and we have cr2 = 1.
We may define a linear map on the real vector space spanned by the simple
roots, also denoted by cr, to satisfy
a (ai) = aa(t)-
Then we have <т(Ф) = Ф. All the сг-orbits on Ф have size 1 or 2. Examination
of the root system of E6 shows there are 24 orbits of size 1 and 24 of size 2.
Proposition 13.31 Let L be the simple Lie algebra E6 and cr \ L^ L be the
automorphism of order 2 given above. Then the subalgebra IT of cr-stable
elements of L is isomorphic to F4. The elements
— e6 F2 — £5
Fi=fe F2=f5
T^i = h(, H2 = /i5
are standard generators of F4.
E3 — e2 T e- E3 — e3-\~ e%
F3=f4+fy F4 = f3+fs
H3 — h2 4“ h~ H4 — h3 T hs
Proof. Let (aF) be the Cartan matrix of F4 given by
/2—10 0 \
-1 2 -1 °
0-22-1’
\ 0	0—12/
It is straightforward to check that the elements Ei,Fi, Hi satisfy the relations
M=o
lHiFJ]=AijEJ
312
Fundamental modules for simple Lie algebras
[EiFi] = Hi
[E;F7]=O if i^j
[E;,...[E;Ej]=O	if z#j
[F;,... [F;Fj] =0	if i*j
where the last two relations have 1 — factors F;, F; respectively. By Propo-
sition 7.35 there is a homomorphism
в : F4^L
whose image is the subalgebra generated by the elements Ei,Fi,Hi. Since
0^0 and F4 is simple the image of 0 is isomorphic to F4.
We shall also show that im в = La. Since each E^F^ Hi lies in IF we have
im в C La. On the other hand consider the decomposition
L = H® £ Сеаф £ (Cea + Cea(a)).
аеФ	аеФ
сг(а)=а	сг(а)^а
Each direct summand is сг-stable, thus IF is the direct sum of the cr-stable
subspaces of the components. We have
dim//'7 =4
dim (Ce^)17 <1 if o’(a) = a
dim (Сеа + Сео.(а)У < 1 if a(a)^a.
Thus dim IF <4 + 24 + 24 = 52. But dim(im 0) = 52, thus im0 = L<7. Hence
La is isomorphic to F4.	□
Now let V be the 27-dimensional fundamental module L (w8) for F6 con-
structed in Proposition 13.29. Then V may be regarded as an F4-module using
our embedding of F4 in F6. We label the fundamental roots of F4 by the
diagram
12	3	4
О------и > о-------О
Proposition 13.32 The II-module V decomposes as
V = L(w4)®L(0)
where L (<y4) is the 2,6-dimensional fundamental module.
13.8 Fundamental modules for exceptional algebras
313
Proof. We determine the weights of the F4-module V. We recall that the
weights of V have form
3<j<7
+	3<j<7
П£/=1’ £i = -h £2 = 1, £8 = -l-
z=l
Now the fundamental roots of E8 are
a;=/3;-/3;+1	z=l,...,6
a7 = Рб + Pl
“8 = -|Е/?г
z=l
Also ctj (hi) = Aij i, j e {1,... ,8} where (Ay) is the Cartan matrix of F8.
It follows that the numbers f3j (/i;) i, j e {1,... ,8} are given by
/3.(/i.) = l z=l,...,7
/з;+1(М = -1	;=i,...,6
/Ш) = Ч z = l,...,8
(/ij = 0 otherwise.
Let II,. H2, H3, H4 be the fundamental coroots of F4 defined above
and Wj, <y2, <y3, <y4 the corresponding fundamental weights of F4. Then
<y; (if) =8^. By calculating the values /3, (//J we deduce
/31 =/32=/38 = -iW4
&	=|«4
/З4	=w3-|w4
/З5	=w2-«3-i«4
/?6	=w1-w2 + w3-iw4
/З7	=-ш1 + ш3-^ш4
314
Fundamental modules for simple Lie algebras
when the /3; are regarded as weights for F4. Hence the 27 weights of the
F4-module V are
± {w4, a>, — a>3, a>1 — a>4, a>2 — a>3, a>3 — a>4, a>3 — 2w4, a>1 — a>2 + a>3, a>1 — a>2
+ a>4, a>1 — a>3 + a>4, a>2 — a>3 — a>4, a>2 — 2a>3 + a>4, a>1 — a>2 + a>3 — a>4}
U {0,0,0}.
The only dominant weight among these, excluding 0, is a>4. Thus V has
highest weight a>4 and so L (w4) is an irreducible direct summand of V. Since
dimV = 27, dimL(w4) = 26
we have
V = L(«4)®L(0).	□
Using the relation at = JT Ajfitj in F4 we see that
w4 = «। + 2a2 + 3a3 + 2a4.
This is the highest short root of F4. All short roots of F4 are transforms of this
one under elements of the Weyl group W. Thus all 24 short roots of F4 are
weights of L(w4). So the weights of L(w4) are the 24 short roots together
with 0 with multiplicity 2.
We now discuss the other fundamental modules for F4. We first consider
L (toj). The relations at = JT f°r ^4 show that
<j)1 = 2a3 + 3a2 + 4a3+2a4.
We recall from Section 8.6 that
а3=/33—/32 a2 =/32 — (33 a3 = (33 a4=^(—(31 —(32—(33 + (34)
and so
a>1 = 2a3 + 3a2 + 4a3 + 2a4 = + /34.
The long roots of F4 have form ±/3; ± f3j and, since /34 >- Д >- /32 >- /З3,	+ /34
is the highest root. Thus a>, is the highest root of F4 and L (wj is therefore
the adjoint F4-module.
The remaining fundamental modules L (<y2) , L (<y3) for F4 satisfy
/52\
dim L (w2) = ( 2 ) — 52
z s /26\
dim L (a>f) = 11— 52.
13.8 Fundamental modules for exceptional algebras
315
It can be shown that L (w2) , L (w3) appear as irreducible direct summands of
A2L (и, ), A2L (w4) respectively, and that
A2L (wj =L(w1)®L(w2)
A2L (w4) = L (wj ф L (w3) .
Finally we consider the simple Lie algebra G2 and show how to obtain
the 7-dimensional fundamental module. We do this by identifying G2 with a
subalgebra of /)4. The fundamental roots of /)4 will be numbered as in the
diagram
Let a be the permutation of the vertices given by
<т=(1 3 4)(2).
cr gives a symmetry of the Dynkin diagram with cr3 = 1. Since
Дф)^)=Л2 for all/, j
there exists by Theorem 7.5 an automorphism cr of /)4 satisfying
O’ («/) = ea(i)
O' Ш = fa(i)
O'(hi) =
We may also define a linear map cr on the vector space spanned by the simple
roots, satisfying cr (a;) = aa^. We have <т(Ф) = Ф. These are 6 сг-orbits of
size 1 on Ф and 6 orbits of size 3.
Proposition 13.33 Let L be the simple Lie algebra if and <r : L^ I. be the
automorphism of order 3 given above. Then the subalgebra La of cr-stable
elements of L is isomorphic to G2.
The elements
Ei =e2	E2 =	+ e3 + e4
F,=f2	F2=/1+/3+/4
If	~ -J- ^3 -J- ^4
are standard generators of G2.
316
Fundamental modules for simple Lie algebras
Proof. The idea is the same as that for F4 in E6. The Cartan matrix of G2 is
It is again straightforward to check that the elements E2, E2, F2, F2, H1, H2
satisfy the defining relations
M=o
[я;е7]=л7е7
[tf;F;] = -AyF;
[EiFi] = Hi
[F;F7]=0 if i^j
[E;,... [E;Ej] = 0 iff# J
[F;,... [F;Fj] =0 if i*j
where the last two relations have 1 — Atj factors F;, F; respectively.
Thus by Proposition 7.35 there is a homomorphism в : G2 L. The image
im d is isomorphic to G2. We show im0 = L<7. Since F;, F;, Hi lie in La we
have im в C La. Now consider the decomposition
L = H® 22 Сеаф 22 (Ce^ + Ce^j + Ce^^j).
аеФ	аеФ
сг(а)=а	сг(а)^а
Each direct summand is сг-stable, thus IF is the direct sum of the cr-stable
subspaces of the components. We have
dim ^ = 2
dim (Ce^)17 <1 if o’(a) = a
dim[Cea + Cea(a) + Cea2(a)y <1 if cr(a)^a.
Thus dim IF <2 + 6 + 6=14. But dim(im 0) = 14, thus im в = IF. Hence IF
is isomorphic to G2.	□
Proposition 13.34 Let V be the 8-dimensional natural D^-module. Regard
V as a G2-module using the above embedding of G2 in I)4. Then
V = L(w2)®L(0)
where I.(oj2 ) is the 7-dimensional fundamental G2-module.
13.8 Fundamental modules for exceptional algebras
317
Proof. We recall from Section 8.2 that in this 8-dimensional representation
we have
e| =	2 —2 I ’ e2 =-^23 —-^-3-2> e3 =-^34 —-£-4-3’	e4 =-^3-4 —-^4-3
fi = — E_3_2 + E23, f2 =—e_2_3+e32, f3 = ~e_3_4 + e43,
/4 = ~e_34+e_43.
Hence
^i = E31 — E22 — E_1_1 +E_2_2
^2 = E22 ~ E33 — E_2_2 + E_3_3
h3 = E33 E44 E_3_3 T- E_4_4
h4 = E3 3 E 4 —Д E_3 _3 T- £44
and so
Eli = E22 ~E33 — E_2_2 + E_3_3
If = E33 — E22 + 2E33 — E_3_3 +E_2_2 — 2E_3_3.
Let vr, v2, v3, v4, v_r, v_2, v_3, v_4 be the natural basis of V. Let Wj, w2 be
the fundamental weights for G2. Since w, (//J =8tj these basis vectors span
weight spaces with weights
(i>2, w1 — a>2,	—a>3+2a>2,	0,	— ы2,	—a>3 + a>2,	a>3—2a>2,	0
respectively. The highest weight is a>2, thus L (w2) is an irreducible direct
summand of V. We have
dimV = 8, dim£(w2) = 7
and so
V = L(w2)®L(0).
We note that a>2 = a1+2a2 is the highest short root of G2. All short roots
are transforms of this root by elements of the Weyl group, thus all six short
roots are weights of /. (w2). Thus the weights of £(<w2) are the short roots
together with 0.
Now we have
Ei — E23 E_3_2,
E2 — E12 + E34 + £3-4 —£-2-i
£-4-3	£4-3
Ei~ E_2_3-\-E32,
F2— E_3_2 E_3_4 £_34+ £21+£43+ £_43.
318	Fundamental modules for simple Lie algebras
It may be checked that the vector v4 — v_4 is annihilated by Ex, E2, Fx, F2 and
so spans the 1-dimensional submodule L(0).	□
Finally we consider the other fundamental G2-module L (cof). The relations
O'; = £;Агй>; show that
1	*-“4 J1 J
a>, =2a1 + 3a2.
This is the highest root of G2. Therefore the fundamental module Ь(ор) is
the 14-dimensional adjoint module.
14
Generalised Cartan matrices
and Kac-Moody algebras
In 1967 V. G. Kac and R. V. Moody independently initiated the study of
certain Lie algebras ISA) associated with a generalised Cartan matrix A. An
и x и matrix A = (A^) is called a generalised Cartan matrix if it satisfies
the conditions
Au = 2 for i = 1,... , n
A,, e Z and Ati <0 if i j
Atj = Q implies Ajt = Q.
The Cartan matrix of any finite dimensional simple Lie algebra is a generalised
Cartan matrix, as shown in Section 6.4. We shall see that, in the special
case when A is a Cartan matrix, the Lie algebra L(A) constructed by Kac
and Moody coincides with the finite dimensional simple Lie algebra with
Cartan matrix A. However, the Lie algebra ISA) can in general be infinite
dimensional.
The term ‘generalised Cartan matrix’ will be abbreviated to GCM. The
Lie algebra ISA) associated to a GCM A will be called the Kac-Moody
algebra associated to A. We shall explain the definition and some of the basic
properties of L(A) in the present chapter. In fact the introductory ideas do
not use the fact that A is a GCM - we shall assume initially that A is any
и x и matrix over C.
14.1 Realisations of a square matrix
Let A be an и x и matrix over C. A realisation of A is a triple (77, П, IP)
where:
H is a finite dimensional vector space over C
IIV = {/ij,... , hn} is a linearly independent subset of H
319
320
Generalised Cartan matrices and Kac-Moody algebras
П = {aj,... , an} is a linearly independent subset of H*
aj (.hi) = Aij for all bi-
Proposition 14.1 If (II, П, IIV) is a realisation of A then dim H > In —
rank A.
Proof. Let rank A = l and dim H = m. We extend the set IP to give a basis
hr,... ,hm of H and extend П to give a basis ax,... , am of H*. Consider
the mxm matrix	This is non-singular so its rows are linearly
independent. Thus the n x m matrix given by the first n rows has rank n.
This matrix therefore has n linearly independent columns. Now the leading
и x и submatrix is A, so has rank /. Thus the remaining n x (m — n) matrix
has rank at least n — l. It follows that m — n > n — I, that is m > In — I. □
Definition A minimal realisation of A is a realisation in which
dimH = 2n — rank A.
Proposition 14.2 Any и x и matrix over C has a minimal realisation.
Proof. Since rank A = l, A has a non-singular / x / submatrix. By reordering
the rows and columns we obtain a matrix
n — I
I
A22/
П — I
in which An is non-singular. Let
/ Ai
C = I ^21
\ О
^12
A22
L,
О
In—l
о
n — I
n — I
I n — l n — l.
Since det C = ±det An ^0 we see that C is a non-singular (2n — /) x (2n — /)
matrix. Let H be the vector space of all (2n — /)-tuples over C. Define
aj,... ,anEH* to be the first n coordinate functions
(Ai,... ,Л2л_;)^Л;	/=1,...,и.
Define hr,... , hn e H to be the first n row vectors of C. Then аъ... , an
and /ix,... , hn are linearly independent and we obtain a realisation of
^11 ^12
^21 A22
14.1 Realisations of a square matrix
321
with dim /7 = 2н — /. By reordering ar,... ,an and hr,... ,hn appropriately
we obtain a minimal realisation of A.	□
Now let (H, П, IIV) and (!/', IT, (П')У) be two realisations of A. We say
the realisations are isomorphic if there is an isomorphism of vector spaces
ф :
such that ф (h) = hl and ф* (a) = a; where
ф* :
is the isomorphism induced by ф.
Proposition 14.3 Any two minimal realisations of an n,<n matrix A over C
are isomorphic.
Proof. Let (II. П, IIV) be the minimal realisation of A constructed in Proposi-
tion 14.2 and (IP. П', (П')У) be another minimal realisation. We reorder the
rows and columns of A as before to obtain
^11 ^12
^21 ^22
where An is non-singular.
We complete h),... ,h'n to a basis hf ... ,	of H'. Then the matrix
(al (/ij)) for i = 1,... , 2n — I ; j = 1,... , n has form
^11 ^12
^21 ^22
B2
Since af ... , a'n are linearly independent this matrix has rank n. Thus it has
n linearly independent rows. Since rows 1+ 1,... , n are linear combinations
of rows 1,... , / the matrix
Bi
I
A12A I
B2 J n — I
n — I
must have linearly independent rows, so is non-singular.
322
Generalised Cartan matrices and Kac-Moody algebras
We now extend a(,... , a'n to a(,... , so that the (2n — /) x (2n — /)
matrix (al (/i.)) is
Л1
^21
Bl
I
^22
B2
n — I
° \
In—l I
О /
n — l.
n — I
n — I
This matrix is non-singular, thus a[,... , a'^^ are a basis for (//')*•
Since An is non-singular, by adding suitable linear combinations of the
first / rows to the last n — I rows we may achieve B, = O. Thus it is possible
to choose h'n+1,., h2n_l so that h(, ... , h2n_l are a basis of H' and
Мп A12 О \
(aj Mi)) = I ^21 ^22 Ifi—l I •
\ О B’2 О /
The matrix B2 must be non-singular since the whole matrix is non-singular.
We now make a further change to h'n+1,... , equivalent to left multi-
plying the above matrix by
h О О \
o In-i О
О О (B'2yl)
Then we obtain
/Ац A12 О \
(aj Mi)) = I ^21 ''M In-l I •
\ О In_t о }
This is equal to the matrix C above. Thus the map /i; hl gives an iso-
morphism H H' which induces the isomorphism (Я')* H* given by
al aj. This shows that the realisations (H, П, IIV) and (H', П', (П')У) are
isomorphic.
14.2 The Lie algebra L(A) associated with a complex matrix
Let A be an и x и matrix over C with rank /. Let (II. П, IP) be a minimal
realisation of A. Then we have
dim H = 2n —I
nv = {/ii, • • • ,hn]cH, П={а1,... ,s„|c»
“;Mi) = Ay
14.2 The Lie algebra L(A) associated with a complex matrix 323
We define a Lie algebra L(A) by generators and relations.
Let X={er,... , en, fx,... , fn,x for all x e H} and let R be the following
set of Lie words in X:
x — Ay — pz for all x, y, z e H, Л, p e C with x = Ay + pz
[xy] for all x,y eH
[e;/;] — ht for i = 1,... , n
[e;/J for all i ф j
[xe;] — a;(x)e;	for all x e H and i = 1,... , n
[x/;] + a; (x)/;	for all x e H and i = 1,... , n.
We define L(A) = L(X ; R) to be the Lie algebra generated by the elements
X subject to relations R.
Lemma 14.4 If a different minimal realisation of A is chosen the Lie algebra
L(A) is the same up to isomorphism.
Proof. This follows from Proposition 14.3.	□
We note that if A is a Cartan matrix then /.(A) is the Lie algebra investigated
earlier in Section 7.4 and Example 9.13. For in this case A is non-singular
and H is the vector space with basis ht = [e;/;].
Proposition 14.5 There is an automorphism a> of L(A) uniquely deter-
mined by
^(ei~) = -fc ^Ш = ~ео ш(х) = -х
for allxeH. Also a>2 = 1.
Proof. There is a map ы : X FLfX) given by the above formulae. By
Proposition 9.9 there is a unique Lie algebra homomorphism FLfX) FLfX)
extending this map. We shall denote this map also by a>. It satisfies w2 = I.
Let (R) be the ideal of I'L(X) generated by the above set R of Lie words.
By applying a> to the elements of R we see that &>((/?)) c (R). Thus we may
define the induced map
S> : FL(X)/(7?)^FL(X)/(7?).
Since w2= 1, <b is an automorphism of L(A).
□
324
Generalised Cartan matrices and Kac-Moody algebras
Let H be the subalgebra of L(A) generated by the elements x for all x e H.
Let N be the subalgebra generated by ex,... ,en and N the subalgebra
generated by , fn. Then we have
ю(Я) = Я, &>(#) = AT, w(N~)=N.
Now let V be an и-dimensional vector space over C with basis , vn
and let
Г(У) = фГ(У)
s>0
be the tensor algebra of V. Thus T'(V) has basis
Vti ®	® Vt = Vti ... vt
for all , is e {1,... , и}. For each linear map A eH* we define a map
0A : X End T(V).
It is sufficient to define the effect of these endomorphisms on the basis
elements of T(V). Г°(Е) has basis 1. We define
0A(i) - l = A(x)l
0A(i)•	. vj = (A - air--aj (x)v;i ...v^
for xeH.
МЛИ = ^
We define 0A (ej by induction on \ as follows
M^H=o
(^)-v; = 8yA(^)l
(^)-(v;i...vJ = v;i (0A (e7) (l2---Ls))
+5y(A-a;2------aj(^)v;2...v;s	S>1.
Proposition 14.6 The above map : X^- End T(V) can be extended to a
Lie algebra homomorphism L(A)^ [End Г(У)].
Proof. The idea of the proof is essentially the same as in Proposition 7.9.
0A can first be extended to a homomorphism
0A : FL(X)^ [End Г(У)]
14.2 The Lie algebra L(A) associated with a complex matrix 325
by Proposition 9.9. We have
L(A) = FL(X)/(7?)
and so in order to show that 0A induces a homomorphism /.(A) [End Г( V)]
we must verify that 0A(r) = 0 for all r eR.
The elements of R have form
X — Xy — fJLZ
[*y]
Kfil - К
[eifj] i^j
[xe;] — a;(x)e;
[xf^ + a^xyfi.
The relation 0A(r) = 0 may be checked for each such r e R in a straightforward
manner, just as in the proof of Proposition 7.9	□
Corollary 14.7 The map x^x is an isomorphism of vector spaces H H.
Proof. H is the subalgebra of L(A) generated by x for all xeII. However,
these elements form a Lie algebra since
i1 + x2 = Xj + x2
Ax = Ax
[iii2] = 0.
Thus^ = {x; xeH}.
Consider the map H H given by x x. This is a homomorphism of Lie
algebras. It is suijective. To show it is an isomorphism we must show it is
also injective. Thus supposexeH and x = 0. Then 0A(x) = O. Thus A(x) = 0.
Since this holds for all A eH* we may deduce that x = 0.	□
We next consider the restriction of 0A to N . It is clear from the definition
that this is independent of A. We call it
в : N~^ [End T(V)].
326
Generalised Cartan matrices and Kac-Moody algebras
Now 0 (/;) is left multiplication by vt. Thus, for any Lie word w (/j, ... , /„)
in /1, • • • ,	0 (w (/i, • • • , /„)) is left multiplication by w (v^ ... , v„).
Proposition 14.8	. , fn generate N~ freely, and so N~ is isomorphic to
FL (/
Proof. Define ф : N [F(V)] by cb(ii') = 0(w)  1. Thus
^(w(/i,... ,f„)) = w(v1,... ,v„).
Then ф is a Lie algebra homomorphism, since
ф [w(/j,	w' (/j,	= [w (Uj, ... , u„) , w' (Uj, ... , v„)]
= [Ф («ЧА, • • • , A)A (A, • • • , A))] •
Now T(V) = F (vp ... , vn), the free associative algebra on vr,... , vn. Thus
the free Lie algebra FL(u1,... , ил) lies in [F(V)] and consists of all Lie
words in Wj,... , vn. Thus FL(u1,... , ил) is the image of ф. Hence the
homomorphism
ф : N~ -^FLtv^... ,vn)
is surjective. But there is a Lie algebra homomorphism
ф' : FL(vj,... ,vn)^N-
with </>'(v;)=A Moreover we have фоф' = 1 on FL (vp ... , vn) and
ф' оф=1 on N . Thus ф, ф' are inverse isomorphisms and N is isomorphic
to FL	□
Corollary 14.9	,... , en generate N freely.
Proof. Apply the automorphism w of Proposition 14.5. We have w (A~) = N
and w (/;) = —e;. Thus the result follows from Proposition 14.8.	□
Proposition 14.10 L(A) = N~®H®N, a direct sum of subspaces.
Proof. The proof is similar to that of Proposition 7.12. We show that
I = К +H + N is an ideal of L(A). It is sufficient to show that
ade;-Zc7, ad/;-Zc7, adi-ZcL
14.2 The Lie algebra L(A) associated with a complex matrix 327
Since the defining relations show that
ad e; • H c N, ade^NcN
ad ft-HcN-, ad fi-N~cN~
adi-H=O,	ad A • /VcN,	ad x• N~ cN~
it is sufficient to check that
ad f-N CH+N
ade,-,V CH+N2
We have
ad fi-ej = 8ijhiEH+N.
Suppose w2eN satisfy
ad/;•	eH + N,	ad f-w2EH + N.
Then
ad fj [wj w2] = [ad f-u^, w2] + [ , ad f • w2] eH + N.
Thus adf-N CH + N.
The relation ad e; • N с H + N follows similarly. Thus I is an ideal of
L(A) containing all the generators, and so L(A) = N + H + N.
In order to show the sum is direct we verify that if w_ E N~, xeH,weN
satisfy
w_+i+w = 0
then we have w_ =0, i = 0, w = 0. Thus suppose w_+i+w = 0. Then
0A(w_+i + w) is the zero endomorphism of '/(V). In particular 0A(w_ +
x + w)-l=0. Now 0A (w_) • 1 = ф (w_) , 0A(i) • 1 = A(x)l and 0A(w)-l=O.
Hence
ф (w_) + A(x)l = 0.
Now ф (w_) E ф5>1Г5(У) and A(x)l e T°(V). It follows that ф (w_) = 0 and
A(x)l=0, that is A(x) = 0. Since this holds for all Х.ЕН* we have x = 0.
Hence £ = 0.
Now ф : N FL (vp ... , v„) is an isomorphism, and so <^(w_) = 0
implies w_=0. Finally w_+i+w = 0 implies w = 0. Thus
L(A~) = N-®H®N.	□
328
Generalised Cartan matrices and Kac-Moody algebras
Let Q be the subgroup of H* given by Q = {a = k1a1-\---\~k„an ; kr,... ,
kn e Z}. Let Q+ = {a 0 e Q ;	> 0 for all i} and Q = {a 0 e Q ; kt < 0
for all i}. For each a e Q let
La = {yeL(A); [iy] = a(x)y for all x e //[.
Proposition 14.11 (i) L(A) = ®aEQLa
(ii)	dim La is finite for all aeQ.
(iii)	L0 = H.
(iv)	If of I) then La = 0 unless a e Q+ or a e Q~.
(v)	[LaLp\(zla+g fOr all Ct, /3&Q.
Proof. To show L(A) = J)aegLa it is sufficient to show H <z^fa^QLa, N c.
К C E„y Е,- It is clear that H c Lo. To show that N c E„y La
we observe that each Lie monomial w in ex,... , en satisfies [iw] = a(x)w
for all x e H and some a e Q+. For
[ie;] = afx)et
and if
[iw1] = /3(x)w1,	[iw2] = y(x)w2
we have
[i К w2]] = (/3 + y)(x) [wi w2].
This shows NG^2aeQ+La and similarly we have К cE«eg- La. Thus
L(A) = E«eg La.
In order to show that the sum is direct we show that
vi 3----И14 = 0
for vt with /3j,... , /Зк distinct implies each vt = 0. Suppose this is false.
Choose the minimal value of к for which it is false. Suppose vx 4-h vk = 0
for this value of k but that not each vt = 0. Then
[i, Uj 4---h vft] = 0 for all x e H.
Thus
We also have
ftWfi + -+ftWft = 0.
14.2 The Lie algebra L(A) associated with a complex matrix 329
Hence
(/3i (x) - /3k (x)) Vj + • • • + (ft (x) - /Зк (x))	= 0.
By the minimality of k we have
(/3;(x) — ft (x)) vt = 0	for i= 1,... ,k — 1.
Since /3^/3к there exists xeH with /3;(х)^/3k(x). Hence vt = 0 for
i=l,...,k— 1. It follows that vk = 0. This contradicts our assumption.
Hence
L(A) = ®Lff.
Since L(A~) = N~ ® H ® N by Proposition 14.10 and
,v c E 7.,,, Hc'L0,
aeQ~	aeQ+
it follows that
H = L0, N=^La, N~=^La.
aeQ+	<xeQ~
Also we have La = O if a^O, a$.Q+, a$.Q~. The Jacobi identity shows
that c La+p for all a,/3eQ.
Finally we show dim La is finite. We have dim Lo = 2n — /. So let a e Q+.
Then La cN. Now N is spanned by Lie monomials in ex,... , en and each
Lie monomial lies in some La. Let a = k1a1-\----\~knan with kteZ and
к > 0. A Lie monomial lies in La if and only if e; appears к times in it for
each i. But there are only finitely many Lie monomials in which e; appears к
times for each i. Thus dim La is finite. A similar argument proves this when
a e Q~. We note in particular that
dim La. = 1, dim L_a. = 1
dim=0, dimL_fa. =0 if A:>1.	□
The following lemma will be needed in the proof of the next proposition.
Lemma 14.12 Let H be a finite dimensional abelian Lie algebra and V be
an H-module such that
V=
Ле//*
330
Generalised Cartan matrices and Kac-Moody algebras
where VA = {v e V ; xv = A(x)u for all x e H}. Let U be a submodule of V.
Then
u= ф (t/nvA).
Лей*
Proof. Let и e U. Then u = u1-\--------\-um where ut e VA. and Ax,... , Am are
distinct elements of H*. Let
Hij = {xeH ; А;(х) = АДх)} for i^j.
Htj is a subspace of H of codimension 1. Now H	since a finite
dimensional vector space over C cannot be the union of finitely many proper
subspaces. So we can find x e H with Ax(x),... , Am(x) all distinct.
Let 0(x) : V -+ V be the linear map given by D(x)v = xv. Then we have
и — u} T • • • T um
0(x~)u = Aj (х)И1 + • • • + Ат(х)ит
0(х)2и = Aj (х)2И14--h Am(x)2«m
0(х)'"-1и = A, Cvf 4 + • • • + A.H(x)'" 1 /7jH.
We have here m equations in , um whose coefficients have non-zero
determinant. Thus , ... , um may be expressed as linear combinations of
u, 0(x)u, 0(х)2и,... , d(x)'" 1 /7. These vectors all lie in U. Thus ut e UCi VA,.
Thus we have shown that U = J2AeAf* (U П VA) and the sum is direct because
^2лей* Уд is a direct sum.	□
Proposition 14.13 The algebra /.(A) contains a unique ideal I maximal with
respect to IC\H=O.
Proof Let J be any ideal of L(A) with J Г11 =0. We have
L(A)= ф/.„
«ей*
by Proposition 14.11, and we consider L(A) as an //-module. By
Lemma 14.12 we have
J= Ф (ьапт).
aeH*
14.3 The Kac-Moody algebra L(A)
331
Now each La with а ф 0 lies in N or in N . Thus
J = (2V-nj)®(2Vnj).
In particular J c N~ ф N.
Now consider the ideal I of L(A) generated by all ideals J with J O il = 0.
All such ideals J lie in N фN, thus I lies in К фN. Hence IOH = O.
Thus I is the unique ideal of L(A) maximal with respect to 1ПН= О. □
14.3 The Kac-Moody algebra L(A)
We now suppose that A is a GCM. Let L(A) be the Lie algebra associated
with A defined in Section 14.2 and I be the unique maximal ideal of L(A)
with IOH=O. Let L(A) be defined by
L(A) = L(A)/L
The Lie algebra L(A) is called the Kac-Moody algebra with GCM A. We
have a natural homomorphism 0 : L(A) L(A). We define N = 0(N) and
N~ = 0(N~).
Proposition 14.14 L(A) = N ® 0(H)® N. Moreover the map 0 : H 0(H)
is an isomorphism.
Proof. We know from the proof of Proposition 14.13 that
I=(N-OI)®(NOI).
Since L(A) = N~ ®H®N it follows that
L(A) = N ®0(H)®N
and that 0 : H 0(H) is an isomorphism.	□
We recall from Corollary 14.7 that there is a natural isomorphism H H.
Combining this with 0 we obtain an isomorphism H 0(H). We shall subse-
quently use this isomorphism to identify 0(H) with H, and we shall write
L(A) = N~®H®N.
In order to show that a given Lie algebra is isomorphic to L(A) the
following result is often useful.
Proposition 14.15 Suppose we are given an и x и GCM A = (Ay). Let L be
a Lie algebra over C and H be a finite dimensional abelian subalgebra of L
332
Generalised Cartan matrices and Kac-Moody algebras
with dimH = In — rank A. Suppose П = {aq,... , an} is a linearly indepen-
dent subset of H* and IP = {lq,... , hn} a linearly independent subset of H
satisfying oij (hi) = Aij.
Suppose also that ex,... , en, fy,... ,fn are elements of L satisfying
[eifi] =
[«;/;]= 0 if i^J
[xe;] = afxfyi for xeH
[*/;] =-“/(Ж forx^H
Suppose that ex,... , en, fy,... ,fn and H generate L and that L has no
non-zero ideal J with .1П11 =(). Then L is isomorphic to the Kac-Moody
algebra I.(A).
Proof The elements ,... ,en,fy,... ,fn and x e H generate L and satisfy all
the defining relations of I.(A) given in Section 14.2. Thus there is a surjective
Lie algebra homomorphism в : I.(A) -+ L and L is isomorphic to I.(A)/ ker 0.
The restriction map в : II II is an isomorphism by Corollary 14.7,
thus ker dn/7 = 0. It follows that ker 0 c I, the largest ideal of L(A) with
IC\H=0. In fact we have ker0 = Z since L has no non-zero ideal J with
jr\II = (). Hence
L = Z(A)// = L(A).
Corollary 14.16 If A is a Cartan matrix then 1ЛЛ) is the finite dimensional
semisimple Lie algebra with Cartan matrix A.
Proof In this case we have rank A = n, so dim H = n. The finite dimensional
semisimple Lie algebra satisfies all the hypotheses of Proposition 14.15, so
is isomorphic to the Kac-Moody algebra 1ЛЛ).	□
This result shows that the theory of Kac-Moody algebras is an extension
of the theory of finite dimensional semisimple Lie algebras, which we have
already described.
We shall now describe some further basic properties of the Kac-Moody
algebra 1ЛЛ). We shall denote the images of e;, ht, ft e L(A) under the
natural homomorphism L(A)^L(A) by e;, ht, ft eL(A). This should not
lead to confusion as we shall subsequently be concentrating on L(A) rather
than L(A).
14.3 The Kac-Moody algebra L(A)
333
Proposition 14.17 There is an automorphism a> of L(A) satisfying co2 = 1
determined by
а(еЗ) = -^, co(fi) = -ei
<y(x) = —x for all xeH.
Proof By Proposition 14.5 L(A) has an automorphism ы with w:= I. Thus
w(/J is the unique maximal ideal with
й(/)Пй(Я) = О.
But a>(H) = H so a>(I) = I. Thus ы induces an automorphism a> of L(A)/I =
L(A) satisfying the stated conditions.	□
There is also an analogue of Proposition 14.11. For each aeQ define
by
La = {yeL(A); [xy] = a(x)y for all хеH}.
Proposition 14.18 (i) L(A) = фaeQLa
(ii)	dim La is finite for all aeQ.
(iii)	L0 = H
(iv)	If aft) then La = O unless a e Q+ or a e Q~.
(v)	[LaLp\ C La+p for all a, fie Q.
Proof Let в : L(A) L(A) = L(A)/I be the natural homomorphism. We
have
(|'/1)=ф(,, by Proposition 14.11.
«eg
Also
/ = (J) (/П Z,b) by Lemma 14.12.
«eg
It follows that
L(A) = e0(ZJ.
«eg
Now we clearly have в (La)cLa, thus L{A') = 'ffaeQLa. This sum is direct,
just as in the proof of Proposition 14.11. It follows that L(A) = ®a^QLa and
that La = 0 (La). Now
А(/1) = ;У	by Proposition 14.14
334
Generalised Cartan matrices and Kac-Moody algebras
La, hence we have
N~=Q)La, H = L0, N = ф La.
aeQ~
dim Л,ь is finite because La = 0 (La) and dimLff is finite. Finally [LaL^ C
La+p follows from the Jacobi identity.	□
Definitions H will be called a Cartan subalgebra of L (A). This fits in with
our previous terminology when A was a Cartan matrix. An element ucJI ' is
called a root of L(A) if a A® and La A- O. Every root lies in Q+ or Q~. The
roots in Q+ are called positive roots and those in Q~ negative roots. If a is
a root then La is called the root space of a. The dimension of La is called
the multiplicity of a. When A is a Cartan matrix we recall that all roots have
multiplicity 1. However, we shall see that this is not always the case when A
is a GCM.
Proposition 14.19 (i) dim La. = 1 and dim L _a. = 1.
(ii) If k > 1 then dim Lka. = 0, dim L_ka. = 0.
Proof. Since L„ =0 (L„ ) and dim L„ = 1 we have dim L„ < 1. If dim L„ = 0
j	a,	\ a,/	a,	a, —	a,
we would have etel = kerв. This would imply [eifi] = hiel, contrary to
IC\H=O. Thus dimLa. = 1. A similar argument gives dimL _a. = 1.
Since Lka. = O and L_ka. = O for k>l it follows that Lka. = O and
L~kai = 0.	□
a1,a1,... ,an are called the fundamental roots of 1ЛЛ), again in agree-
ment with the earlier terminology when A is a Cartan matrix.
Remark 14.20 For a general n x n matrix A over C we constructed a
minimal realisation (II, П, IIV) where H is a vector space over C of
dimension In— rank Л, I Г' =	, hn} is a linearly independent subset
of H and П = {cij,... , an} is a linearly independent subset of II such that
aj (^i) — A-ij-
In the case when A is a GCM the matrix A is real and so we can find
a real vector space T/R, of dimension In — rank A over R, contained in II
such that /ij,... , hn lie in T/R and are linearly independent and ar,... , an,
when restricted to T/R, remain linearly independent. In the construction of
H, described in Proposition 14.2 as the vector space of all (2n — /)-tuples
over C, we define T/R as the subset of all (2и— /)-tuples over R. The triple
(//,11,11) with li e// and IJcT/R is called a real minimal realisa-
tion of A.
14.3 The Kac-Moody algebra I.(A)
335
We denote by I.(Ay the subalgebra of L(A) generated by e^... ,en,
/1,
Proposition 14.21 (i) La lies in L(A)' for each root a of L(A).
(ii) £(А)' = (НПЦА)')фЕа/0£а.
(iii) L(A)' = [L(A)L(A)].
Proof We know from Proposition 14.18 that LafO implies a e Q+ or a e
Q~. If a^Qf then LacN and if aeQ then !.„cP . Since N is the
subalgebra generated by ,... ,en and N is the subalgebra generated by
/),... , fn we have La c L(A)' for each a.
Since L(A) = H® and La a L(A)' we have
L(Af = (»П/.|Л))Ф£/.,
a^O
It follows that 1АЛ) = L(A~)' + H. We also have [H, L(A)'] c L(A~)' and so
L(A)' is an ideal of L(A). We have
L(A)/L(A)' = H/H П L(A)'
and so L(A)/L(A)' is abelian. Hence [L(A)L(A)] c L(A)'. On the other hand
we have [eifi\ = hi,\hiei\ = 2ei,\hifi\ = —2fi and so e;, ft e [L(A)L(A)].
Thus L(A)' c [L(A)L(A)] and we have equality.
15
The classification of generalised
Cartan matrices
The structure of the Kac-Moody algebra ISA) depends crucially on the
GCM A. In the present chapter we shall discuss various possible types of
GCM A which can occur.
15.1 A trichotomy for indecomposable GCMs
Two GCMs A, A' are called equivalent if they have the same degree n and
there is a permutation cr of 1, n such that
&Г alii, j.
A GCM A is called indecomposable if it is not equivalent to a diagonal sum
Aj О
О A2.
of smaller GCMs Ab A2. If A is a GCM so is its transpose A1. Moreover A
is indecomposable if and only if A1 is indecomposable.
We shall now define three particular types of GCM. Let v= (vp ... , un)
be a vector in R". We write v > 0 if vt > 0 for each i, and v > 0 if vt > 0 for
each i.
Definitions A GCM A has finite type if
(i)	detA^O
(ii)	there exists и > 0 with Au > 0
(iii)	Au > 0 implies u>Q or u = Q.
336
15.1 A trichotomy for indecomposable GCMs	331
The GCM A has affine type if
(i)	corank A=1	(i.e. rank A = n —1)
(ii)	there exists и > 0 such that Au = O
(iii)	Аи>0 implies Au = O.
The GCM A has indefinite type if
(i) there exists и > 0 such that Au < 0
(ii) Au > 0 and и > 0 imply u = 0.
All vectors и in these definitions are assumed to lie in IR", and are column
vectors.
We aim to prove the following theorem.
Theorem 15.1 Let A be an indecomposable GCM. Then exactly one of the
following three possibilities holds:
(a)	A has finite type
(b)	A has affine type
(c)	A has indefinite type.
Moreover the type of A1 is the same as the type of A.
This section will be devoted to the proof of Theorem 15.1, which gives a
trichotomy on the set of indecomposable GCMs.
We begin with a lemma on inequalities.
Lemma 15.2 Let vl = (u;1,... , vin) e IR" for i = 1,... , m. Then there exist
Xp ... , xn e IR with vtjXj > 0 for i = 1,... , m if and only if Xp)14-----h
Xmvm = 0, A; > 0 implies A; = 0 for i = 1,... , m.
Proof. Suppose there exists a column vector x = (xb ... , л„)' satisfying vlx >
0 for all i. Suppose A1u1H----hAmum = 0 with all A; >0. Then AjiAxH--------h
Amif"x = 0. But vlx > 0 and A; > 0, thus we have A; = 0 for all i.
Conversely suppose A1u1H-------hAmum = 0, A; >0 implies A; = 0 for all i.
Let
; A;>0,
i=l
m
EA; = 1
2 = 1
Define f : .S'^IR by /(y)=||y|| where ||y||=/y(4---------hy^. Then .S' is a
compact subset of IR" and f is a continuous function from 5 to IR. Thus f(S)
is a compact subset of IR. Hence there exists .ve.S' with ||x|| < ||x'|| for all
x' eS. Clearly xf() since the zero vector does not lie in S. We shall show
338
The classification of generalised Cartan matrices
vlx > 0 for all i as required. In fact we shall show that (у, л ) > 0 for all у e .S',
where (y, x) = 22 У;Х;. This implies the required result since each vl lies in S.
Now 5 is a convex subset of R". We assume у fix, then ty + (1 — t)x e 5
for all t with 0 < t < 1. By the choice of x we have
(ty+(l— t)x, ty + (1 — t)x) > (x, x)
that is
y-x) + 2(y-x, x) >0
for 0 < t < 1. This implies (y — x, x) > 0, that is (y, x) > (x, x) > 0.	□
We make use of this lemma in the following proposition.
Proposition 15.3 Let M be an m^n matrix over R. Suppose
и >0 and Mlu >0 imply u = Q.
Then there exists v > 0 with Mv < 0.
Proof. Let M= (wiy) and consider the following system of inequalities:
n
— У' m1;x; > 0 i = 1, ... , m
j=l
X; > 0 Z = 1, . . . , П.
J	J '	'
We shall use Lemma 15.2 to show that these inequalities have a solution.
Thus we consider an equation of form
-,-«O+Em/o, ... , 1,... ,0)=0
i=i	y=i	i
with A; > 0,	> 0 for all i, j. Then
£A;m;> = ^.
i=l
Let и = (Ль ... , Am)1. Then Mfi= (p^,... , /хЛ)1. Thus we have и > 0 and
Mlu>Q. This implies that и = 0. We also have Mlu = Q. Thus A; = 0 and
fij =0 for all i, j. Hence Lemma 15.2 shows that the above inequalities have
a solution. Thus there exists v > 0 with Mv <0.	□
15.1 A trichotomy for indecomposable GCMs
339
We now consider our three classes of GCM A. Let
5F = {A; A has finite type}
SA = {A; A has affine type}
S} = {A; A has indefinite type}.
It is easy to see that no GCM can lie in more than one of these classes.
Lemma 15.4 8рП8А = ф, 8рП81 = ф, .S'A П .S', = </>.
Proof. If A e .S'r n ,S'A then det A 7^ 0 and corank A = 1, a contradiction.
If A e .S'j, П 5j there exists и > 0 with Au > 0. But Au > 0 and и > 0 imply
и = 0, a contradiction.
If A e SA П 5j there exists и > 0 with Au = 0. But Au > 0 and и > 0 imply
и = 0, a contradiction.	□
We must therefore show that each indecomposable GCM lies in one of the
three classes.
Lemma 15.5 Let A be an indecomposable GCM. Then и > 0 and Au > 0
imply that и > 0 or u = 0.
Proof. Suppose и 0 and и 0. Then we can reorder 1,... , и so that ut = 0
for i = 1,... , s and ut > 0 for z = s + 1,... , n. Let
A=(P *
\ W S J n — s
s n — s
Now all entries of the block Q are < 0 and if Q has an entry < 0 then Au has
a negative coefficient, which is impossible. Thus Q = 0. This implies R = O
by the definition of a GCM, thus A is decomposable, a contradiction. □
Now let A be an indecomposable GCM and define KA by
KA = {u ; Au > 0}.
KA is a convex cone. We consider its intersection with the convex cone
{и; и>0}. We shall distinguish between two cases:
{и; и > 0, Au > 0} {0}
{и; и > 0, Au > 0} = {0}.
340
The classification of generalised Cartan matrices
The first of these cases splits into two subcases, as is shown by the next
lemma.
Lemma 15.6 Suppose {и; и > 0, Au > 0} fi- {0}. Then just one of the follow-
ing cases occurs:
Kag{u; u > 0} U {0}
KA = {w, Au = 0} and KA is a 1 -dimensional subspace of IR".
Proof We know there exists и^0 with и>0 and Аи>0. By Lemma 15.5
this implies that и > 0. Suppose the first case does not hold. Then there exists
vfiO with Au>0 such that some coordinate of v is < 0. If v > 0 then r>0
by Lemma 15.5, thus some coordinate of v is < 0.
We have Au > 0 and Av > 0, hence A(tu + (1 — tjvj > 0 for 0 < t < 1. Since
all coordinates of и are positive and some coordinate of v is negative there
exists t with 0< t< 1 such that tu+(l — t)v>0 and some coordinate of
tu + (1 — tjv is 0. But then tu + (1 — tjv= 0 by a further use of Lemma 15.5.
Thus и is a scalar multiple of u. We also have
0 = A(t«+(1 — t)w) = M«+(l — t)Av.
Since Au > 0, Av > 0 this implies that Au = 0, Av = 0.
Now let w e KA. Then Aw > 0. Either w > 0 or some coordinate of w is
negative. If w > 0 then w > 0 or w = 0 by Lemma 15.5. Suppose w > 0. Then
by the above argument with и replaced by w, v is a scalar multiple of w. Hence
w is a scalar multiple of u. Now suppose some coordinate of w is negative.
Then by the above argument with v replaced by w, w is a scalar multiple of u.
Thus in all cases w is a scalar multiple of u. Hence KA is the 1-dimensional
subspace IR«. Thus we have shown that KA is a 1-dimensional subspace of IR".
We have also shown that if w e KA then Aw = 0. Thus KA = {w; Aw = 0}.
Thus if the first case does not hold the second case must hold. We note
finally that the two cases cannot hold together since in the first case KA
cannot contain a 1-dimensional subspace of IR".	□
We now identify the first case in Lemma 15.6 with the case of matrices of
finite type.
Proposition 15.7 Let A be an indecomposable GCM. Then the following
conditions are equivalent:
A has finite type
{и; и > 0 and Au > 0} fi {0} and KA с {и; и > 0} U {0}.
15.1 A trichotomy for indecomposable GCMs
341
Proof. Suppose A has finite type. Then there exists и > 0 with Au > 0.
Hence {и; и>0 and Аи>0}^{0}. Also detA^O. Thus {и; Аи = 0}
is not a 1-dimensional subspace of R". Hence /САс{и; и>0}и{0} by
Lemma 15.6.
Conversely suppose {и; и>0 and Аи>0}^{0} and /САс{и; «>0}U
{0}. Then there cannot exist и^О with Аи = 0. For this would give a
1-dimensional subspace contained in KA. Thus detA^O. Now there exists
и 0 with и > 0 and Au > 0. By Lemma 15.5 we have и > 0. If Au > 0, A has
finite type. So suppose to the contrary that some coordinates of Au are zero
and some are non-zero. We choose the numbering 1,... , и so that the first s
components of Au are 0 and the last n — s are positive. Let
P
R
A =
s)
s
n — s
s n — s
Now the block Q satisfies QfO since A is indecomposable. We choose the
numbering so that the first row of Q is not the zero vector. We have
fl’u'+Qifs
Au= I II I= I	I.
5J \«2/ yRu1 + Su2 J
Hence Pu1 + Qu2 = 0 and Ru1 + Su2 > 0. We also have и1 > 0, и2 > 0. Thus
Qu2 < 0 and the first coordinate of Qu2 is < 0. Hence Pu1 > 0 and the first
coordinate of Pu1 is > 0. Since Ru1 + Su2 > 0 we can choose s > 0 such that
A(1 -f-s)!/1 + Su2 > 0.
We now consider, instead of our original vector и = (“2), the vector )•
We have
/(l + s)^ >o
\ u2 )
/(l-Fs)^1^	fPux+ Qu2 + ePux\	/ sPu1 \
\ u2 J	yR^ + S^ + ERu1/	\A(1 -f-s)!/1 +Su2J
The first coordinate and the last n — s coordinates of this vector are positive
and the remaining coordinates are > 0. Thus
/ (1 + £W\
а Г / >o
\ u2 f
and the number of non-zero coordinates in this vector is greater than that in Au.
We may now iterate this process, obtaining at each stage at least one more
non-zero coordinate than we had before. We eventually obtain a vector v > 0
such that Av > 0. Thus A has finite type.	□
342
The classification of generalised Cartan matrices
We next identify the second case in Lemma 15.6 with that of an
affine GCM.
Proposition 15.8 Let A be an indecomposable GCM. Then the following
conditions are equivalent:
(i) A has affine type
(ii) {и; и>0 and Au > 0} {0}, KA = {и; Аи = 0}, and KA is a 1-dimen-
sional subspace of IR".
Proof Suppose A has affine type. Then there exists и>0 with Аи = 0. It
follows that {и; и>0 and Аи>0}^{0}. Also XueKa for all AelR. By
Lemma 15.6 we see that KA = {w, Aw = 0} and that KA is a 1-dimensional
subspace of IR".
Conversely suppose the three conditions of (ii) are satisfied.
Then corank A = 1.
Also there exists и 0 with и > 0 and Au > 0. By Lemma 15.5 we have и > 0.
So there exists и > 0 with Au > 0. But KA = {u; Au = 0}. Hence there exists
u>0 with Аи = 0. Finally Аи>0 implies Аи = 0. Thus A has affine type.
□
Proposition 15.9 Let A be an indecomposable GCM. Then:
if A has finite type A1 has finite type
if A has affine type A1 has affine type.
Proof. To prove these results we shall make use of Proposition 15.3.
Suppose A has finite type. We show there does not exist v > 0 with Av < 0.
For if Av < 0 then A(—v) > 0 and so — v > 0 or — v = 0. Hence v < 0 or v = 0.
This contradicts r>0. We may now apply Proposition 15.3 to show there
exists ufif) with и>0 and А‘и>0. So {и; и>0 and А‘и>0}^{0}. By
Lemma 15.6 either
KA, с {и; и > 0} U {0}
or KAi = {и; А‘и = 0} and this is a 1-dimensional subspace. Now det A 0 so
det А1 0. Thus the latter case cannot occur. The former case must therefore
occur, so by Proposition 15.7 A1 has finite type.
Now suppose A has affine type. We again show there does not
exist r>0 with Au<0. For A(—u) > 0 is impossible in the affine case.
15.1 A trichotomy for indecomposable GCMs
343
By Proposition 15.3 there exists и^О with и>0 and А‘и>0. So {и; и>0
and А‘и > 0} {0}. By Lemma 15.6 we may again conclude that either
Кл, с {и; и>0}и{0} or
KAi = {и; Аи = 0} and this is a 1-dimensional subspace.
Now corank A = 1 so corank A1 = 1. This shows that we cannot have the first
possibility. Thus the second possibility holds, and then by Proposition 15.8
we see that A1 has affine type.	□
We may now identify the case not appearing in Lemma 15.6 with that of
an indefinite GCM.
Proposition 15.10 Let A be an indecomposable GCM. Then the following
conditions are equivalent:
A has indefinite type
{и; и > 0 and Au > 0} = {0}.
Proof. Suppose A has indefinite type. Then и > 0 and Au > 0 imply и = 0.
Conversely suppose {и; и > 0 and Au > 0} = {0}. Then the same condition
holds for A1, i.e. {и; и > 0 and А‘и > 0} = {0}. This follows from Lemma 15.6
and Propositions 15.7, 15.8 and 15.9. But then Proposition 15.3 shows that
there exists v > 0 with Av < 0. Thus A has indefinite type.	□
We are now able to achieve our aim of proving Theorem 15.1. For each
indecomposable GCM A Lemma 15.6 shows that exactly one of the following
conditions holds:
(а)	{и; и > 0 and Au > 0} {0} and KA с {и; и > 0} U {0}.
(b)	{и; и>0 and Au > 0} {0}, KA = {и; Аи = 0}, and KA is a 1-dimen-
sional subspace.
(с)	{и; и > 0 and Au > 0} = {0}.
By Proposition 15.7 A satisfies (a) if and only if A has finite type. By
Proposition 15.8 A satisfies (b) if and only if A has affine type. By Propo-
sition 15.10 A satisfies (c) if and only if A has indefinite type. Thus we
have the required trichotomy for GCMs. Moreover Proposition 15.9 shows
344
The classification of generalised Cartan matrices
that the type of A1 is the same as the type of A. This completes the proof of
Theorem 15.1
Corollary 15.11 Let A be an indecomposable GCM. Then:
(a)	A has finite type if and only if there exists и > 0 with Au > 0.
(b)	A has affine type if and only if there exists и > 0 with Au = 0.
(c)	A has indefinite type if and only if there exists и > 0 with Au < 0.
Proof (a) Suppose и > 0 and Au > 0. A cannot have affine type as then Au > 0
would imply Au = 0. A cannot have indefinite type as then и > 0 and Au > 0
would imply и = 0. Thus A has finite type.
(b) Suppose и>0 and Аи = 0. A cannot have finite type since detA = 0.
A cannot have indefinite type since then и > 0 and Au > 0 would imply
и = 0. Thus A has affine type.
(c) Suppose u>0 and Аи<0. Then A(—и)>0. A cannot have finite type
as this would imply — u>0 or — u = 0. A cannot have affine type since
A(—u) > 0 would then imply A(—u) = 0. Thus A has indefinite type.
Remark 15.12 In proving the results of Section 15.1 we have assumed that
A is a GCM. However, we have not used the full force of this assumption.
Inspection of the proofs shows that we have nowhere assumed that A;; = 2 or
that Atj e Z. This remark will be useful in some subsequent applications.
15.2 Symmetrisable generalised Cartan matrices
In this section we shall consider a special type of GCM which plays a key role
in the theory of Kac-Moody algebras. These are the symmetrisable GCMs.
Before giving the definition we obtain some preliminary results.
Let A= (Ay) be a GCM with i, j e {1,... , n} and let J be a subset of
{1,... , n}. Let A7 = (Ay) , i, j e J. Then A7 is also a GCM, called a principal
minor of A.
Lemma 15.13 (i) Suppose A is an indecomposable GCM of finite type and
A7 is an indecomposable principal minor of A. Then A} also has finite type.
(ii) Suppose A is an indecomposable GCM of affine type and A, is a proper
indecomposable principal minor of A. Then A} has finite type.
15.2 Symmetrisable generalised Cartan matrices
345
Proof, (i) By passing to an equivalent GCM we may choose the numbering
so that J = {1,... , m} for some m < n. Let K={m+1,... , n}. Let
P
R
A =
Q\
s )
m
n — m
m n — m
Now there exists и > 0 with Au > 0. Let и = (“J). Then
\uKJ
p
R SJ \uK)
Puj + QuK
Ruj + SuK
Since Au > 0 we have Pu7 + QuK > 0. However, QuK < 0 so Pu7 > 0. Thus
there exists и7 > 0 with Apt, > 0. By Corollary 15.11 Л, has finite type.
(ii)	As before we may assume J = {1,... , m}. This time we have m < n. Let
A =
P
R
m
n — m
s)
where P = A,
m n — m
Since A has affine type there exists и > 0 with Аи = 0. We have
p
R S J \uK)
Puj + QuK
Ruj + SuK
Hence Ри7 + QuK = 0. Now QuK < 0 so Pu7 > 0.
Suppose if possible that Ри7 = 0. Then QuK = 0, and since uK>0
this implies that Q=O. But then R = O also and A is decomposable, a
contradiction. Hence we have и7 > 0, Pu} > 0, Pu} 0. This implies that
P = Aj cannot have affine type or indefinite type. Thus Л7 has finite type.
□
We next describe our trichotomy in the special case in which the indecom-
posable GCM is symmetric.
Proposition 15.14 Suppose A is a symmetric indecomposable GCM. Them.
(a)	A has finite type if and only if A is positive definite.
(b)	A has affine type if and only if A is positive semidefinite of corank 1.
(c)	A has indefinite type if and only if A satisfies neither of these conditions.
Proof, (a) Let A have finite type. Then there exists и > 0 with Au > 0.
Hence for all A>0 we have (А + А7)и>0. Thus A + XI has finite type by
Corollary 15.11. (Note that A + A7 need not be a GCM, but the results of
Section 15.1 can be applied to it by Remark 15.12.) Thus det(A + A7)^0
346
The classification of generalised Cartan matrices
when A > 0, that is det (A — A/) fi 0 when A < 0. Now the eigenvalues of the
real symmetric matrix A are all real. Thus all the eigenvalues of A must be
positive. Hence A is positive definite.
Conversely suppose A is positive definite. Then det.A^O so A has finite
or indefinite type. If A has indefinite type there exists и > 0 with Au < 0.
But then и* Au < 0, contradicting the fact that A is positive definite. Thus A
must have finite type.
(b) Let A have affine type. Then there exists и>0 with Аи = 0. Hence for
all A>0 we have (А + А/)и>0. Thus by Corollary 15.11 А + А/ has
finite type when A > 0. (We are again using Remark 15.12 here.) Thus
det(A + XT) fi 0 when A > 0, that is det(A — A/) fi 0 when A < 0. Thus all
eigenvalues of A are non-negative. But A has corank 1 so 0 occurs as
an eigenvalue with multiplicity 1, and the remaining eigenvalues are all
positive. Hence A is positive semi-definite of corank 1.
Conversely suppose A is positive semi-definite of corank 1. Then
det .4 = 0 so A cannot have finite type. Suppose A has indefinite type.
Then there exists и > 0 with Au < 0. Thus u'Au < 0, which contradicts
the fact that A is positive semi-definite. Thus A must have affine type.
(c) This follows from (a) and (b).	□
In general a GCM need not be symmetric, but it may nevertheless satisfy
the weaker condition of being symmetrisable.
Definition A GCM A is symmetrisable if there exists a non-singular diagonal
matrix D and a symmetric matrix В such that A = DB.
Lemma 15.15 Let A be a GCM. Then A is symmetrisable if and only if
A: : A: : ... A: : = A: : A: : ... A: :
ZP2 г2г3	lkll l2ll l3l2	lllk
for all f, i2,... , ik e {1,... , n}.
Proof. Suppose A is symmetrisable. Then A = DB with £> = diag (dx,... , </„)
and B= (Bfi. Thus Afi = diBij. Hence
X lJ /	lJ 1 lJ
A; ; . . . A; : = d; ... </; B, ; . . . B, ;
ZP2	llfil	ll	lk	lll2	lkll
A; ; . . . A; : = d; ... </; B, ; . . . B, ;
Z2Z1	г1гк	и	lk	г2г1	г1гк
and these are equal since В is symmetric.
Conversely suppose
A: : ... A: : = A: : ... A: :
ZP2	Ikh г2г1	г1гк
15.2 Symmetrisable generalised Cartan matrices
347
for all q,... , ik. We may suppose A is indecomposable since the result in this
case implies it for all A. Thus for each i e {1,... , n} there exists a sequence
1=71,72,••• Jt = i
with
Aj j ^0, Aj j ^0,	... , Aj j
J1J2 ' y J2J3 ' У	y Jt-lJt '
We choose a number in IR. We wish to define dj by
A;j ...Ajj
d	-----ы d
Aj j ...Aj j 1
JlJ2 Jt-lJt
However, we must check that this definition of dt depends only upon i and
not on the sequence chosen from 1 to i. So let
1=^Д2,... ,ku = i
be a second such sequence from 1 to i. We claim that
Ajiii-l • • • ^1211 _	• • • ^k2kj
Ajlh' ' ' Aji-lii ^kjk2 • • • ^ku_jku
that is А1к2Ак^кз... A^jAjj^ ... А^—Ак21Акзк2... Aj^Aj^^ ... A^. This
is in fact one of the given conditions on the matrix A. Thus dt E IR is well
defined and dtj^0. Let D = diag (dx,... , <7„). Define by Ay = djBy. We
show that Bjj = Bjj, that is ф- = If A;; = 0 then Ai; = 0 also and the con-
J1 lJ	dj dj	v	7‘
dition is satisfied. So suppose Ay ^0. Let 1 =jk,j2,... , jt = i be a sequence
from 1 to i of the type described above. Then 1=^, j2,... ,jt,j is such a
sequence from 1 to j. These sequences may be used to obtain dt and dj
respectively, and we have
Thus Bji = Bij. Hence A = DB where D is diagonal and non-singular, and В
is symmetric. Thus A is symmetrisable.
Corollary 15.16 Let A be a symmetrisable indecomposable GCM. Then
A can be expressed in the form A = DB where D = diag (cf,... , df), В is
symmetric, with dr,... , dn>0 in Z and By E Q. Also D is determined by
these conditions up to a scalar multiple.
Proof. We choose any dx gQ with dx >0. Then Lemma 15.15 shows that
dj E Q and dj > 0 for each i. Thus by multiplying by a positive scalar
348
The classification of generalised Cartan matrices
we may assume each with Also Bij = Aifidi lies in Q. The
proof of Lemma 15.15 also shows that D is determined up to a scalar
multiple.	□
The following important result shows that indecomposable GCMs in the
first two classes of our trichotomy are symmetrisable.
Theorem 15.17 Let A be an indecomposable GCM of finite or affine type.
Then A is symmetrisable.
Proof. First suppose there is no set of integers z\, z2,... ,ik with к > 3 such
that z\ i2, i2^fi • • • > ik~i 7- г'ь fi fi aad
A;  ф 0, A;  ф 0, ... , A; 	0, A; 	0.
!1!2'	’	!2!3 '	’	’ lk-llk' ’ Vl '
Then Lemma 15.15 shows that A is symmetrisable.
Thus we suppose there is such a sequence z\,... , ik with к > 3 and we
choose such a sequence with minimal possible value of k. We thus have
А;л #0 if (r, s) e{(1,2), (2, 3),... , (k, 1), (2,1), (3, 2),... , (1Д)}.
The minimality of к shows that A;; =0 if (r, s) does not lie in the above
set. Otherwise there would be such a sequence with a smaller value of k.
Let J = {f,... ,ik}. Then the principal minor A7 of A has form
/2 -rj 0
-^i	2	-r2
0	— s2	2
~sk \
0
0
\~rk 0
2	•	0
2 -rk_x
0 -Su	2 /
with eZ satisfying z',> 0, \,> 0. In particular we see that A7 is inde-
composable. Now Aj must have finite or affine type by Lemma 15.13. Thus
there exists и > 0 with AjU > 0. Let u=(u1,... , uk~). We define the к x к
matrix M by
M = diag (zzf \ ... , ИдГ1) A7 diag (u1,... , uk) .
15.2 Symmetrisable generalised Cartan matrices
349
Then Mtj = и; 1 (Aj)^ Uj. Thus
j	j
In particular we have	j Mt - > 0. Now we have
/2	-r(	0
—4	2 — r2
0	-s'2 2
0	-4 \
о
M =
о
• 2 -<-1
o -4-1	2 /
with r. = и; 1riui+1, s[ = u^SjUj. (We define uk+1 = u1.)
We note that r( > 0, s[ > 0 and r'^ = r:s e Z. We also have
E Ma=2k - (4+4)--(4+4) •
Now > y/r'is'i =	> 1 hence r( + s'i>2. Since JT j My > 0 we deduce
that r( + \' = 2 and г[$[ = 1. Hence r.s, = 1 and, since rt, st are positive integers,
we have r; = 1, st = 1. It follows that
/2-10
-1 2 -1
0-12
\-l 0
-1\
0
•	•	0
•	2 -1
0-12,
Let v= (1,... , 1). Then v>0 and A7v = 0. Thus A} has affine type by
Corollary 15.11. Lemma 15.13 shows that this can only happen when A} = A.
Thus A is symmetric, in particular symmetrisable as required.	□
We are now able to prove the following basic description of our trichotomy.
It generalises the description previously obtained in Proposition 15.14 for
symmetric indecomposable GCMs.
350
The classification of generalised Cartan matrices
Theorem 15.18 Let A be an indecomposable GCM. Them.
(a)	A has finite type if and only if all its principal minors have positive
determinant.
(b)	A has affine type if and only if det A = 0 and all proper principal minors
have positive determinant.
(c)	A has indefinite type if and only if A satisfies neither of these two
conditions.
Proof, (a) Suppose A has finite type. Then A is symmetrisable by Theo-
rem 15.17, hence A = DB where D = diag (dx,... , df) with dt>0 and В
is symmetric, by Corollary 15.16. The matrix В need not necessarily be a
GCM, but Remark 15.12 shows that we can nevertheless define the type of B.
Moreover Corollary 15.11 shows that A and В have the same type. Thus В
is a symmetric indecomposable matrix of finite type, and so det В >0 by
Proposition 15.14. It follows that detA>0 also. Now all principal minors
of A also have finite type by Lemma 15.13. Thus these also have positive
determinant.
Conversely suppose that all principal minors of A have positive deter-
minant. Suppose there is a set of integers z\,... , ik with k>3 such that
h 7^г2’	’ Lh with
Aj j Aj j ... Aj j ф 0.
*1*2	*2*3	lkll '
Choose such a sequence with minimal possible к and let J = {f,... , z^}.
Then the principal minor A} of A has form
/2-10	• • • 0 -1\
-12-1-	0
0 -1 2	•	•
•	• • • 0
0	• • 2 -1
rl 0 	•	• 0 -1 2 )
by the proof of Theorem 15.17. But then detA7 = 0, a contradiction. Thus
there is no such sequence q,... , ik. By Lemma 15.15 A is symmetrisable.
Hence A = DB where D = diag (cf,... , d„) with dj > 0 and В is symmetric.
Again В need not be a GCM but we can nevertheless define its type using
Remark 15.12 and, by Corollary 15.11, A and В have the same type. Now
the principal minors of the symmetric matrix В all have positive determinant
15.3 The classification of affine generalised Cartan matrices 351
and so В is positive definite. Thus В has finite type by Proposition 15.14, and
so A has finite type also.
(b) Now suppose A has affine type. Then detA = 0. All proper principal
minors of A have finite type by Lemma 15.13 and so have positive
determinant by (a).
Suppose conversely that det A = 0 and that all proper principal minors
of A have positive determinant. Suppose there is a set of integers q,... , ik
with к > 3 such that f ^i2,... , ik ф q with
А,- t At ; ... At t ф 0.
*1*2	*2*3	lkll '
Choose such a sequence with minimal k. and let J = {f,... , z^}. Then
the principal minor Aj has form
/2—10
-1 2 -1
0-12
• 0 -1\
0
\-l 0
•	• 0
•	2 -1
0-12,
as above. Since det A, = 0 we have A, = A. But then A is affine since
Аи = 0 with и=(1,...,1). Thus suppose there is no such sequence
q,... ,ik. Then A is symmetrisable by Lemma 15.15, and has form
A = DB where D = diag (dx,... , df) with dt > 0 and В is symmetric. Now
det В = 0 and all proper principal minors of В have positive determinant.
This implies that the symmetric matrix В is positive semidefinite of corank
1. Hence В is of affine type by Proposition 15.14. Thus A has affine type
also, by Corollary 15.11.
(c) This follows directly from (a) and (b).	□
15.3 The classification of affine generalised Cartan matrices
In this section we shall determine explicitly which indecomposable GCMs
lie in each class of our trichotomy. We begin with indecomposable GCMs of
finite type.
352
The classification of generalised Cartan matrices
Theorem 15.19 Let A be an indecomposable GCM. Then A has finite type
if and only if A is a Cartan matrix. Thus the indecomposable GCMs of finite
type are those on the standard list 6.12.
Proof. We recall from Sections 6.1 and 6.2 that a GCM is a Cartan matrix if
and only if it satisfies the conditions:
(a)	A4 e {0, -1, —2, -3} for all i^j
(b)	Atj = — 2 or —3 implies Л ., = — 1
(c)	the quadratic form
Q(*i,... , x„) = 2J2^- ^s/ntjXiXj
i=l
is positive definite, where ntj = A^A^.
Suppose A is a Cartan matrix. Then Л,, = 2^—Л Let D= diag (dx,... , df)
where d= = J (a,, a,). Then (DAI) ') = 2——r and so DAD1 is
the matrix of the quadratic form Q (xp ... , xn). Since Q is positive definite
det (DAD'} > 0 and so det A > 0.
Now any principal minor A, of the Cartan matrix A is also a Cartan matrix.
Hence detA7>0 for all principal minors of A. Thus A has finite type by
Theorem 15.18 (a).
Now suppose conversely that A has finite type. Suppose i^j and consider
the 2x2 principal minor
(2 M
2/
By Theorem 15.18 (a) the determinant of this minor is positive, hence
A,.A< 4. Since Atj and Ajt are both non-positive integers such that Atj = 0
if and only if Ajt = Q we deduce that Atj e {0, —1, —2, —3} and that Atj e
{—2, —3} implies Ajt = — 1.
Since A has finite type A is symmetrisable by Theorem 15.17. Thus A =
DB where D = diag (d,,... , dn), dt>0, and В is symmetric. Although В
need not be a GCM we may define the type of В by using Remark 15.12.
Thus В is indecomposable of finite type, and so В is positive definite by
Proposition 15.14 (a). Let yi = s/dixi. Then
Q(xi,... ,x„) = 2J2%2-
i i¥=j
15.3 The classification of affine generalised Cartan matrices 353
2	1	1
i
Since В is positive definite we see that Q(xx,... , xn) is positive definite.
Thus A is a Cartan matrix.	□
Having determined the indecomposable GCMs of finite type we next deter-
mine those of affine type. This will also determine those of indefinite type,
as those remaining.
To each GCM A we define an associated diagram A(A) called the Dynkin
diagram of A. This extends the definition of the Dynkin diagram of a Cartan
matrix given in Section 6.2. The vertices of A(A) are labelled 1,... , n where
A is an и x и matrix. Suppose i, j are distinct vertices of A(A). We explain
how i, j are joined in A(A). This depends on the pair (A^, Ajt). We recall
that A;; and A;; lie in Z, A;; < 0, A;; < 0 and A;; = 0 if and only if A;; = 0. The
rules are as follows.
(a)	If AyA^; = 0 vertices i,j are not joined.
(b)	If Atj Ац = 1 vertices i, j are joined by a single edge.
(c)	If AjjAjj = 2, Ay = — 1, Ajj = —2 vertices i, j are joined by a double
edge with an arrow pointing towards j.
(d)	If AyAj; = 3, Ay = —1, Ajj = — 3 vertices i,j are joined by a triple
edge with an arrow pointing towards j.
(e)	If AjjAjj =4, Ay = — 1,	A= —4 vertices i, j are joined by a quadru-
ple edge with an arrow pointing towards j.
(f)	If AtjAji =4, Ay = —2, Aji = —2 vertices i, j are joined by a double
edge with two arrows pointing away from i, j.
a> <o
i j
(g)	If AtjAji >5 vertices i,j are joined by an edge with the numbers |Ay |, |A^;|
shown on it.
i	j
It is clear that the GCM A is determined by its Dynkin diagram A (A).
Moreover A is indecomposable if and only if A(A) is connected.
We now consider a set of connected Dynkin diagrams called the affine list.
354
The classification of generalised Cartan matrices
15.20 The affine list of Dynkin diagrams
15.3 The classification of affine generalised Cartan matrices 355
o---o----о > о—о
^4
о---О
g2
о---о----п < о—о
~t
^4
О---0—^-0
~t
g2
We note that many, but not all, of the Dynkin diagrams on the affine list
appeared in Lemma 6.8. We also note that every proper connected subdiagram
of a Dynkin diagram on the affine list appears on list 6.11 of Dynkin diagrams
of finite type. We shall call this the finite list.
Proposition 15.21 Let Abe a GCM whose Dynkin diagram lies on the affine
list. Then det A = 0.
Proof. First suppose that A(A) has 2 vertices. Then either Д(А) = А1 and
/ 2 -2\	-	/2 -4\
A = I I	or Д (A) = A\	and A = I I.	In either case det A = 0.
\-2	2 ) v 7	1	\-l	2 )
Next suppose that Д(А) = At for / > 2. Then the sum of all the rows of A
is zero, and so det/1 = 0.
In all other cases Д(А) has a vertex, say 1, joined to just one other vertex,
say 2. Moreover we can choose these vertices so that they are joined by a
single or a double edge. In the case of a single edge we have
det A = 2 det В — det C
where В is obtained from A by removing row and column 1, and C is obtained
from В by removing row and column 2. This relation between determinants
is obtained as in the proof of Theorem 6.7. The connected components of
В and C are Cartan matrices of finite type, so their determinants are known
from the proof of Theorem 6.7. In all cases this gives det/1 = 0.
In the case when vertices 1, 2 are joined by a double edge we obtain
det A = 2 det В — 2 det C
again as in the proof of Theorem 6.7. Again В, C have connected components
of finite type so we know their determinants, and in each case we obtain
det/1 = 0.
Proposition 15.22 Let Abe a GCM whose Dynkin diagram lies on the affine
list. Then A has affine type.
356
The classification of generalised Cartan matrices
Proof. By Proposition 15.21 we have det.4 =0. Also the Dynkin diagram of
any proper principal minor has connected components on the finite list. Thus
all proper principal minors have positive determinant. It follows that A has
affine type by Theorem 15.18 (b).	□
We shall now prove the converse.
Theorem 15.23 Let A be an indecomposable GCM. Then A has affine type
if and only if its Dynkin diagram A (A) lies on the affine list.
Proof. Suppose A has affine type. Then every proper indecomposable principal
minor of A has finite type, by Lemma 15.13 (ii). Thus all proper connected
subdiagrams of A (A) lie on the finite list, by Theorem 15.19.
If A (A) has only one vertex A has finite type, so there is no possible
affine A.
If A (A) has two vertices then
where a, b are positive integers. Since det A = 0 we have ab = 4. The possi-
bilities are (a, b) = (1,4)(4, 1)(2,2). Thus A(A) = Ax or A'r
Now suppose A(A) has at least three vertices. If A(A) contains a cycle
then the proof of Theorem 15.17 shows that A(A) = A; for some / > 2. Thus
we suppose that A (A) contains no cycle. Since all the connected subdiagrams
with two vertices lie on the finite list all edges of A (A) have one of the forms
О---------О n n (1 > ()
Suppose A (A) has a triple edge м > n Then A (A) must have exactly
three vertices, otherwise A (A) would have a proper connected subdiagram
with three vertices containing a triple edge, whereas there is no such diagram
on the finite list. Thus we have
/2 -1 0 \	/ 2 -3 0 \
A= I —3 2 —a I or 1—12 —a I
\ 0 —b 2/	\0 —b 2 /
where a, b are positive integers. Thus det A = 2(1 — ab). However, det A = 0
and so a = 1, b= 1. Thus A(A) = G2 or G\.
So we now suppose that A(A) has no triple edge. Now A(A) has at
most two double edges, as every proper connected subdiagram appears on
the finite list so has at most one. Suppose A(A) has two double edges.
15.3 The classification of affine generalised Cartan matrices 357
Then every edge which can be removed to give a connected subdiagram must
be a double edge. This implies that A(A) must be one of C;, C\, Cf
Thus we suppose that A(A) has just one double edge. If A (A) has a branch
point then no proper connected subdiagram can contain both a double edge
and a branch point, since the subdiagram lies on the finite list. This implies
that A (A) is Bl or B\.
Now suppose that A(A) has one double edge but no branch point. Then
A (A) has form
О---------о > <>--------------О---------О
------------ b --------------►
with a + fo + 2 vertices. We have a>0 and b>Q since A (A) is not on the
finite list. Also b < 2, otherwise there would be a proper subdiagram
o----о > о-----о----о
and a < 2, otherwise there would be a proper subdiagram
О----О---О > о----о
Thus the possibilities are
(a,b) = (l, 1), (2, 1), (1,2), (2, 2).
The case (a,b) = (l, 1) appears on the finite list so is not affine. The case
(a, b) = (2, 2) is impossible, since it would give proper subdiagrams as above.
Thus (a, b) = (2, 1) or (1, 2) and A(A) is F4 or Ff
Thus we may now assume that A(A) has only single edges. Consider the
branch points of A (A). Each branch point has at most four branches, otherwise
there would be a proper subdiagram
which does not appear on the finite list. If there is a branch point with
four branches then A(A) = Z)4, as otherwise there would again be a proper
subdiagram Z)4.
Thus we may assume that all branch points in A(A) have three branches.
There cannot be more than two branch points, as otherwise there would be a
proper connected subdiagram with two branch points which could not be on
the finite list.
358
The classification of generalised Cartan matrices
Suppose A(A) has 2 branch points. Then any proper connected subdiagram
has only one branch point, and this implies that A (A) = Dl for some I > 5.
So suppose A (A) has just one branch point. Let the branch lengths be
f,l2,l3 with h — h — h so that there are l3 + l2 + l3 + \ vertices. We must
have Zx < 2, otherwise there would be a proper subdiagram
which is not on the finite list. Suppose f = 2. Then we must have l2 = 2 and
Z3 = 2, otherwise there would again be a proper subdiagram as above. Thus
A(A) = E6.
Thus we may assume f = 1. Since l2= 1 would give a diagram of finite
type we must have l2 > 2. However, l2 < 3 as otherwise there would be a
proper subdiagram
О------О------О------<j>----о------о------о------о
which is not on the finite list. Thus Z2 = 2 or 3.
Suppose Zx = 1, l2 = 3. Then we must have l3 = 3, otherwise there would be
a proper subdiagram
О------О------О------<j>----о------о------о
which is again not on the finite list. Thus A (A) = E-1.
We may now suppose that f = 1, l2 = 2. Since the diagrams with Z3 =2, 3,4
are of finite type we must have Z3>5. But Z3<5 also, as otherwise there
would be a proper subdiagram
О------О------<j)----О------о------о------о------о
which is not on the finite list. Hence Z3 = 5 and A(A) = E8.
Finally if A (A) has only single edges and no branch points then it lies on
the finite list so A cannot be affine.
Thus we have shown that whenever A is affine A (A) must appear on the
affine list. This, together with proposition 15.22, completes the proof. □
A GCM A such that A (A) is on the affine list will be called an affine
Cartan matrix.
15.3 The classification of affine generalised Cartan matrices 359
Corollary 15.24 Let A be an indecomposable GCM. Then A has indefinite
type if and only if its Dynkin diagram A (A) does not appear on the finite list
or the affine list.
Proof. This follows from Theorems 15.1, 15.19 and 15.23	□
16
The invariant form, Weyl group,
and root system
We now turn to the study of the Kac-Moody algebra L(A) associated with a
GCM A.
16.1 The invariant bilinear form
We recall from Section 4.2 that when A is a Cartan matrix the corresponding
finite dimensional Lie algebra L(A) has a non-degenerate symmetric bilinear
form
(,) : ШиШнС
which is invariant in the sense that
([xy],z) = (x, [yz])
for x, y, z e L(A). The Killing form has these properties.
In the case of a GCM A we cannot define the Killing form on L(A) as in
the finite dimensional case. We can nevertheless ask whether there is a non-
degenerate, symmetric, invariant bilinear form on L(A). This is not always the
case, but we shall show that such a form does exist when A is symmetrisable.
Thus suppose A is a symmetrisable GCM. Then A = DB where D is diago-
nal and В is symmetric. Let D = diag (dx,... , dn). Let (H, П, IP) be a mini-
mal realisation of A, where 11' = {/?,,... , hn} is a linearly independent subset
of H, П = {ap ... , an} is a linearly independent subset of H*, Uj(hi) = Aij
and dim H = 2n — l where / = rank A.
Let H' be the subspace of H spanned by hr,... ,hn and let H" be a
complementary subspace of H' in H. Then we have
H =	dimH' = n, dimH" = n-l.
360
16.1 The invariant bilinear form
361
We define a bilinear form (,) : H < W C by the rules:
{ht, hj} = djdjBjj i,j=l,...,n
{hi,x) = {x,hi) = diai(x) forxeH"
(x, y)=0 for x, ye H".
This is evidently a symmetric bilinear form on H.
Proposition 16.1 This form on H is non-degenerate.
Proof. We have A = DB where D is diagonal and non-singular and В is
symmetric. We have rank B = l. We observe that the symmetric matrix В of
rank / has a non-singular / x / principal minor. If I = n we can take В itself
as the principal minor, so suppose / < n. Then, for some i, the zth row of В
is a linear combination of the remaining rows of B. Since В is symmetric
the zth column of В is a linear combination of the remaining columns of B.
Let B' be the (и — 1) x и matrix obtained from В by removing the zth row.
Then rank B' = /. Let B" be the (и — 1) x (и—1) matrix obtained from B' by
removing the zth column. Then rank В" = I. Now B" is symmetric of degree
n — 1 and rank /. Thus by induction B" has a non-singular / x / principal
minor, and this is the required principal minor of B.
It follows that the symmetrisable matrix A has a non-singular / x / principal
minor. For let B, be non-singular where J is a subset of {1,... , n} with
| J\ = I. Then Aj = DjBj where D} = diag {dj, j e./} with each dj Ф 0. Since
Dj is non-singular A2 is also non-singular.
We now consider the special case in which J = {1,... ,/}. Then A has
form
n — I
An non-singular.
By Proposition 14.2 we may extend the linearly independent sets hr,... ,hne
H, aq,... , an e H*, to bases hx,... , h2n_j ; ar,... , a2n-i such that a.j (h) =
Cjj where
/ ^11
C= a21
\ О
^12
^22
I
O\
I
Of
n — I
n — I
362
The invariant form, Weyl group, and root system
Let
n — I
Then
£>iBn
/L/L|
О
DyBn
ITJf,
I
The symmetric matrix M of the bilinear form (ht, i, j e {1,... , 2n — /} is
/L/L|/)|
О
M =
DrBuD2
D2B22D2
B>2
O\
B>2 •
o /
This matrix is non-singular since
det/W = ± (detDj)2 (det D2)2det Bn ^0.
Now suppose A is any n x n symmetrisable GCM of rank /. Then A has
a non-singular / x / principal minor A, for some J c {1,... , n}. Let К be
the complementary subset of J in {1,... ,n} and L = {n+1,... ,2n — /}.
Then there exists a realisation hr,... , h2n_i ; ar,... , a2n-i whose matrix
aj (h) = Cjj may be written symbolically in the form
/ Aj	AJK	O\	J
C=	AK	I p
\ О	I	Of	L
J	К	L
Let
(Dj O \ J
\О DK) К
J К
Then
/В}В} DJBJK O\
C= I DKBK] DKBK I I .
\ О I Of
16.1 The invariant bilinear form
363
This time the symmetric matrix M of the bilinear form (h^h^ for i, j e
{1,... , 2и — /} is
M =
DjBjDj
dkbkjd.
О
D.
0\
Dk I •
о /
DKBKDk
Since det/И = ± (det/)л)2 (det/)A ): det Л’7 0 the bilinear form is non-
degenerate on H.	□
Theorem 16.2 Suppose A is a symmetrisable GCM. Then the Kac-Moody
algebra L(A) has a non-degenerate symmetric invariant bilinear form.
Proof. We have L(A) = ф La. For a = m1a1 d--------h mnan eQwe define the
n'Q
height of a by ht a = mx d----h mn. Then
L(A) = ®L;
where L; is the direct sum of all La with hta = z. Since	we
have [TjTj] <^Li+j. Thus L(A) may be considered in this way as a Z-graded
Lie algebra.
We define, for each integer r > 0,
L(r) = Ф Li-
Then we have
fl = L(0)cL(l)c£(2)c-
and Ur>oL(r) = L(X)-
We have already defined a symmetric bilinear form on H = L(G). We shall
extend this definition to give a symmetric bilinear form on L(r) for r =
1,2,3,... thus eventually defining such a form on L(A). We shall define
the form on L(r) by induction on r, assuming it is already defined on
L(r-l).
We begin with the case r= 1. We have
(n	\	tn
z=l	/	\z=l
364
The invariant form, Weyl group, and root system
We define a bilinear form {,) on L(l) which is uniquely determined by the
following rules:
{,) agrees with the form already defined on H
(Lt, L^ = 0 unless i + j = Q
(ei,fi) = (fi,ei) = di
if^j-
This bilinear form on L(l) is clearly symmetric. We show
<[xy],z) = (x, [yz]>
for all x, y, z e L(l). In showing this we may assume x e Lt, у e Lj, z e Lk for
some i, j, zeZ with |i|, |j|, |£| < 1. We may assume i + j + k = O as otherwise
both sides of our required equality are zero. The relation is known already
when i, j, k are all 0. Thus we may assume i, j, k are 1, —1,0 in some order.
There are six possible orders, but it is only necessary to check three of them
as the other three follow from them. Thus we show
Н-//НМ
= [ejj]}
for heH. Both sides are zero in these relations if i^j. If i = j the relations
are valid because
lfii,h}=diai(h') for all he H.
This follows from the definition of the form (,) on H. Thus we have
([xy], z) = (x, [yz]) for all x, y, z e L(l).
Now suppose inductively that a symmetric bilinear form has already been
defined on L(r— 1) and satisfies:
(L;, Tj) = 0 unless i + j = 0 for |i|, \j\ < r — 1
<[xy],z) = (x, [yz]) for all xeLi,yeLj,zeLk with |i|, |j|, |£| <r-l
and i + j + k = Q.
16.1 The invariant bilinear form
365
We shall show this form can be extended to one on L(r) with analogous
properties. We extend the form to L(r) by defining
(L;, Lj} = 0 unless i + j = 0 for |i|, \j\ < r.
We must also define (x, y) = (y, x) for x e Lr, у e L_r. We assume r > 2.
Now we have L(A) = N~ ®H®N with H = L0,N~ = @i<0Li,N =
The algebra N is generated by /x,... , fn, thus each element of
/V can be written as a Lie word in /x,... , fn, so is a linear combination
of Lie monomials in /x,... , fn. An element of L_r is a linear combination
of Lie monomials in ,... , fn such that the number of factors in each
Lie monomial is r. If r > 2 each Lie monomial is the Lie product of Lie
monomials of degree s,t say with s+t = r. It follows that each element
у e L_r can be written in the form
j
where Cj e L_u., dj e L_„. with Uj > 0, Vj > 0 and Uj + Vj = r. The expression
of у in this form need not be unique.
Given x e Lr, у e L_r we write у = JT [cjdj] as above and wish to define
J
The right-hand side is known since [xcj and dj lie in L(r — 1), so if there
is a form of the required type on L(r) it must satisfy the above relation in
order to be invariant. However, the right-hand side appears to depend on the
particular expression у = JT [cjdj] 'or У which need not be unique. We must
therefore show that the right-hand side remains the same if a different such
expression for у is chosen.
In a similar way we can write x^Lr in the form
x = E laibil
i
where a; e Ls, bt e Lt. and st > 0, t; > 0 with si + ti = r. We shall show
^{ai,[biy]) = ^{[xcj],dj}.
i	j
This will imply that the right-hand side is independent of the given expres-
sion for y, and also that the left-hand side is independent of the given
expression for x. In fact it is sufficient to show
366
The invariant form, Weyl group, and root system
Now
([[аД] cj , d}] = {[[a;9] b;], d}] - {[[M a;], d}}
= (hc,] ’ [VJHLM ’
= k lbi M]]|
using the invariance of the form on L(r— 1). Hence our form (x, y) is now
well defined on L(r), where it is bilinear and symmetric.
We must now check that
<[xy],z) = (*> [yz])
when xef, у &Lj, z&Lk with |i|, |j|, |£| < r and i + j + k = O. This is known
already by induction unless at least one of |i|, |j|, |£| is equal to r.
It is impossible for all of |i|, |j|, |£| to be equal to r since i + j + k = O. We
suppose first that just one of |i|, |j|, |£| is r. Then the other two are non-zero.
If | i | = r then
(x, [yz]) = ([xy], z)
by definition of the form on L(r). Similarly if |£| = r this relation also holds
by definition. So suppose \ j\ = r. We may assume that у has the form у = [ab]
where a eLs, beLt,s + t = j and 0 < |sj < \ j\, 0 < |t| < |j|. Then
<[xy],z) = <[x[ab]],z)
= ([[bx]a], z) + {[[xa]b], z)
= ([Z?x], [az]) + {[xa], [bz])
= {[xb], [za]) + ([xa], [bz])
= (x, [b[za]]) + (x, [a[£>z]]>
= (x, [[ab]z])
= (x, [yz])
using the invariance of the form on L(r — 1).
Now suppose that two of |i|, |j|, |£| are equal to r. Then i, j, k are r, —r, 0
in some order. Thus one of x, y, z lies in H.
16.1 The invariant bilinear form
367
Suppose x e H. We may again assume у = [ab] where a e Ls, b e Lt, s +1 =
j, o<и<\j\, o<\t\<\j\.
Then
([xy], z) = ([x[afe]], z) = ([[xa]fe], z) - ([[xfe]a], z)
= ([xa], [fez]) — ([xfe], [az])	by definition of(, )on L(r)
= (x, [a[fez]]) — (x, [fe[az]])	by invariance on L(r— 1)
= (x, [[afe]z]) = (x, [yz]).
If z G 77 the result also holds by using the symmetry of the form.
Finally suppose yeH. Then we may assume z=[afe] where
b eLt, s + t = k,0 < l^l < \k\, 0 < |t| < |£|. Then
(x, [yz]) = (x, [y[afe]]) = (x, [a[yfe]]> + (x, [[ya]fe])
= ([xa], [yfe]) + ([x[ya]], fe)	by definition of(,) on L(r)
= ([[xa]y], fe) + ([x[ya]], fe)	by invariance on L(r — 1)
= (ItoHfe)
= ([xy], [afe]) by definition of(, )on L(r)
= (ky],z).
We have therefore proved invariance when xeLt,y eLj,zELk with
|i|, |j|, |£| < r and i + j + k = O. It follows that invariance holds for all
x, y, z G L(r). By induction the form is therefore invariant on 1.(Л).
Thus we have now defined a symmetric invariant bilinear form on 1.(Л).
We show it is non-degenerate. Let I be the kernel of (,), i.e. the set of
xeL(A) such that (x, y)=0 for all yeL(A). Since the form is invariant I
is an ideal of 1.(Л). Since by Proposition 16.1 the form is non-degenerate on
restriction to 77 we have I Г\Н = 0. But the Kac-Moody algebra I.(A) has no
non-zero ideal 7 with IП 77 =0. Hence 1 = 0 and the form is non-degenerate
on L(A).	□
Note The proof of this theorem shows that any symmetric invariant bilinear
form on I.(A) is uniquely determined by its restriction to 77.
Definition The form constructed in Theorem 16.2 will be called the standard
invariant form on L(A).
Corollary 16.3 For each i e Z the pairing Lt x C given by x, у
(x, y) is non-degenerate.
368
The invariant form, Weyl group, and root system
Proof. Suppose x&Li satisfies (x, y)=0 for all yeL_;. Since (L;,L^ = 0
unless i + j = 0 we have (x, у) = 0 for all у e L(A). Hence x = 0.	□
Corollary 16.4 (La, L^ = 0 unless a + /3 = 0.
Proof Suppose a + /3f; 0 and let x e La, у e Lp. Choose h e H with
(a + /3)(/0#O.
Then
([x/i],y) = (x, [/ry]>
implies
-a(li){x, y)=/3(h)(x, y)
that is
(a + /3)(/i)(x, y)=0.
Hence (x, y) =0.	□
Since the form {,) is non-degenerate on H it determines a bijection H*	H
given by a h'a where
{h'a, h) = a(fi) for all
Corollary 16.5 (i) Suppose xeLa, у eL_a, Then [xy] = (x, y)h'a.
(ii) The pairing La x L_a C given by x, у (x, y) is non-degenerate.
(iii) For each x eLa with x 0 there exists у e I. ,, with [xy] 0.
Proof, (i) Consider the element [xy] — (x, y)h'a e H. For all h e H we have
([xy] - (x, y)h'a, h) = ([xy], h) - (x, y) (h'a, h)
= {x, [y/i])-a(/i)(x, y)
= 0.
Since the form is non-degenerate on H we deduce that [xy] — (x, y)h'a = 0.
(ii) Since the form is non-degenerate on L(A) and if.rt,Lf = () unless
/3=—a the pairing La x L_a C must be non-degenerate.
(iii) For each xeLa with x^O there exists yEl. ,, with (x, y)^0. Hence
[xy]#0by(i).	□
We now consider to what extent a non-degenerate symmetric invariant
bilinear form on L(A) is unique. The following proposition deals with this
question.
16.1 The invariant bilinear form
369
Proposition 16.6 Suppose A is an indecomposable symmetrisable GCM and
{,} is a non-degenerate symmetric invariant bilinear form on the Kac-Moody
algebra L(Af Then there exists a non-zero £gC such that
{*, у} = £(x, У) for al1 x,ye L(Af.
Thus such a form is determined on the subalgebra I.(A)' up to a non-zero
constant.
Proof. The argument of Corollary 16.4 shows that {La, L^} = 0 whenever a +
/3^0. In particular we have {H, La} = 0 whenever a 0. Since L(A) = H®
La it follows that {,} is non-degenerate on restriction to H. The form {,}
on L(A) is determined by its restriction to H and by the map La x L_a C
given by x, у {x, y} for each a e Ф. The argument of Corollary 16.5 shows
that, for x G La, у G L_a, we have
[xy] = {x, y}k'a
where k'a is the unique element of H satisfying {k'a, h} = a(h) for all h G H.
We therefore have
[LaL_J = C^ = Cra
for each u G(l>. Thus there exists a non-zero f,. G'C with h'a = ^ak'a. This
implies that
{h'a,h} = £a{h'a,h} for all he H
since both sides are equal to t;aofh). Let a;, a, be simple roots. Then we
have
| h' h' I = £ lh' h' \
['%•’ uj I Sai\fLai’’ fLa.j
and so by the symmetry of the forms
f lh' h' \ = f lh' h' \
Sai\fLai’’ fLa.j b<Xj \fLai’’ fLa.j '
If Atj then (hf, h'a.) ^0 and we have £a. = £a.. If the GCM A is inde-
composable this shows that there exists ffO in C such that fr/=f for all
simple roots a;. Thus
{h'a. ,h] = g(h'a., h} for all h G H.
Now for any a G Ф h'a is a linear combination of the h'a,. Hence
{h'a,h} = g(h'a,h) for all heH.
370
The invariant form, Weyl group, and root system
Thus <:„ = <: for all a e Ф. Using the equations
[xy] = {u )K = U y)h'a
for x e La, у e L_a we deduce that
=	for xeLa,yeL_a.
Now L(Af was defined as the subalgebra of L(A) generated by eb ... , en,
,f„. We recall from Proposition 14.21 that L(Af = [L(A)L(A)]. It
follows that L(Af Г\Н is generated by [LaL_a] = Ch'a for all a e Ф. It follows
that
{h', h} = g{h', li) for all/1,/1'е£(А)'ПЯ
{*, ?} = £<*, У) for all xeLa, y&L_a
But Л(Л)'= (Л(Л)'П//) Ф also by Proposition 14.21. Thus we see
that
{x, y} = £{x, y) for all x, у e L(Af.	□
Corollary 16.7 Т(А)'Г\Н is the subspace of H spanned by hr,... , hn.
Proof. We saw in the proof of Proposition 16.6 that L(Af rd I is the subspace
generated by the elements h'a for all a e Ф. Each h'a is a linear combination
of hx,... , hn and so the result follows.	□
Corollary 16.8 Any non-degenerate symmetric invariant bilinear form on a
finite dimensional simple Lie algebra is a constant multiple of the Killing
form.
Proof. Since L(A) is simple we have L(Af = [L(A)L(A)] = L(A). Thus the
given form is a constant multiple of the Killing form on the whole of IfA).
□
Important comment on notation. In the case when IfA) has finite type
the standard invariant form is not the same as the Killing form. It is a constant
multiple of the Killing form.
In our development of the theory of finite dimensional simple Lie algebras
we have used the notation {,) to denote the Killing form. In the theory of Kac-
Moody algebras the Killing form does not exist in general, but the standard
invariant form exists whenever the Kac-Moody algebra is symmetrisable.
In the subsequent development the notation {,) will denote the standard
16.2 The Weyl group of a Kac-Moody algebra
371
invariant form of a symmetrisable Kac-Moody algebra. This will be so even
in the case of finite dimensional simple Lie algebras, i.e. (,) will subsequently
denote the standard invariant form rather than the Killing form.
16.2 The Weyl group of a Kac-Moody algebra
Lemma 16.9 Let xeL(A) and J be the ideal of L(A) generated by x. Then
J = 11 (L(A))x.
Proof The adjoint representation of L(A) gives a Lie algebra homomor-
phism L(A) [End L(A)]. By Proposition 9.3 there is an associative algebra
homomorphism
H(L(A))—> End L(A).
A subspace К of L(A) satisfies [L(A),K]cK if and only if 1I(L(A))K c
K. Now we have [L(A), J] c J. Hence 1I(L(A)) J c J. Since x e J we have
H(L(A))xC J.
On the other hand H(L(A))(lI(L(A))x) = H(L(A))x, thus
[L(A), U(L(A))x] c H(L(A))x.
Hence H(L(A))x is an ideal of L(A) containing x. Hence 1I(L(A))xd J.
Thus we must have equality.	□
Proposition 16.10 In L(A) we have, for ij^j, (ade;)1-Ay e^ = 0 and
(ad/’;)1-Ay/j = O.
Proof We shall show (ad /', )1'" fj = 0. The other relation holds similarly.
Let x= (&&ff)l~Aii fj e N . We shall show [ek, x| = 0 for all k= 1,... , n.
Suppose this is so. Then the set of all yeL(A) with |vx| = 0 is a subalge-
bra containing ex,... , en, so contains N. Thus |/V, x] = О and so H(A)x =
Cx. Since L(A) = N ®H®N we have П(Л(Л)) = II(/V)II(/7)II(/V) by the
PBW basis theorem. Hence
П(£(А))х = П(А~)П(Я)П(ЛО* = П(А~)П(Я)х.
Since [H, /V | c N we have H(/7)/V c N and ll(/7)x c N . Thus
H(L(A))x C U(N~)N~ C N~.
Let J = H(L(A))x. This is the ideal of L(A) generated by x, by Lemma 16.9.
We have J C;V so jr\II=(). This implies J=O by definition of L(A).
Thus x = 0.
372
The invariant form, Weyl group, and root system
Thus in order to obtain the required result x = 0 it is sufficient to show
[^, Aid/)1 л"/;] =0 when i^j.
We first suppose k^i and k^j. Then
[ek, (ad/)'/] = [eb [f, (ad/)'-1/]]
= [^, f], (ad/)'"1 /] + [/, [£(fc, (ad/)'-1 /]]
= [fi,[ek, (ad/)'"1/;]].
Repeating we obtain, for each t,
7,(ad/)'/] = (ad/)'7,/]=0.
Next suppose k = j. Then
[er (ad/)'/] = (ad/)' [ep /] = (ad/)' /
as above. If 1—A;;>2 then this shows that [g;, (ad/)1-Ay/] = 0. If
1-Av = 1 then Aij=° so lfi’hj]=Ajifi = Q- Thus I/’	in
this case also.
Finally we suppose k = i. Then
[e;,(ad/)'/;]=[e;,[/,(ad/)'-7;]]
= [fe;,/] , (ad/;)'-7;] + [/;, [£;, (ad/;)'"'/;]]
= [ht, (ad/)'-7;] + [/, [£;, (ad/)'-1/;]]
= - ((t- l)a; + a;) (/) (ad/;)'-7;+ [/, [£;, (ad/;)'"1/;]]
= (-2(t- 1) - A;;) (ad/)'-7;+ [/, [£;, (ad/;)'"1/;]] .
Repeating, we obtain
(-2(1 - 1) - A;.) (ad/)'-1 /; + (-2/ - 2) - A;;) (ad/)'-1 /; + •••
+ (-A;;) (ad/)'-1 /; = ~t (t - 1 + A;;) (ad/)'-1 /; .
We now put t = 1 — A;;. Then we have
[^(ad/;)1-^/;]^.
This completes the proof in all cases.	□
16.2 The Weyl group of a Kac-Moody algebra	313
Using this result of Proposition 16.10 we may deduce, as in Proposi-
tion 7.17, that the maps ade; and ad/; are locally nilpotent. Then the proof
of Proposition 3.4 shows that expade; and exp ad (—/;) are automorphisms
of L(A). Let
и; = exp ad e; • exp ad (—/;) • exp ad e; e Aut L(A).
Proposition 16.11 nfH) = H. For x eH we have
nfx) = x — afxflp.
Proof. Let x e H. Then
exp ad e; • x= (1 +ad e;) x = .v+ [е,л'| = x — a;(x)e;
1 — ad/; + -—— I (x — a;(x)e;)
= x-ai(x)ei - [/;x] + afx) [/;e;] +1 ad/; ([/;x] + afxfhj)
= x — ai(x)ei — afx)ft — ai(x)hi + |a;(x) • 2/;
= x — afx)et — afxflp
exp ad e; (x — afx)e! — afx^hj) = (1 +ad e;) (x — a;(x)e; — a;(x)/i;)
= x — a;(x)e; — a;(x)/i; + [e;x] — a;(x)
= x — a;(x)e; — afx')hi — a;(x)e; + 2a;(x)e;
= x — a;(x)/i;.
This gives the required result.	□
Proposition 16.12 The map st : H H induced by nt satisfies sj = 1,
si(hj) = —hi, st(x) = x when (Ip, x) =0.
Proof. This follows from sfx) = x — ср(х)1р together with ai(hj) = 2 and
(ht, x) = diCpfx).	□
The maps st : H H are called fundamental reflections. The group W
of non-singular linear transformations of H generated by	is called
the Weyl group W of L(A).
Proposition 16.13 The bilinear form (,) on H is invariant under W.
Ъ1Ь
The invariant form, Weyl group, and root system
Proof. Let x,yeH. Then
(stx, sty) = (x — Uifxfhi, у - afy)hi)
= (x, y) - afx) If^, y) - afy) (x, ht) + afxjafy) (ht, ht)
= (x, y) — ai(x)diai(y) — afy^diafx) + а;(х)а;(у) -2</;
= (x,y).	□
We may also define an action of W on II by
(wA)x = A (w! 'л) for we W, XeH*, xeH.
This action is compatible with the isomorphism H* -e H given by A -e h'x
where (h'x, x) = A(x) for all xeH. For suppose w(A) = /z for A, jxeH*.
Then
(w (/i'A) , x) = (/i'A, w-1(x)} = A (uT^x))
= (wX)x = p.(x) = [h'll,x}
for all xeH. Thus w (/iA) = h'^.
Proposition 16.14 The action of st on H* is given by
.v,(A) = A — A (/i;) at.
Proof. Let xeH. Then
(^A) x = A (.s^x) = A (six) = A (x — a;(x)/i;)
= A(x) — A (/i;) a;(x) = (A — A (ht) at) x.	□
In fact the Weyl group acts on the root system Ф of L(A).
Proposition 16.15 If аеФ, weW then т(а)еФ. Moreover dimLff =
dimLw(a).
Proof. The proof of Proposition 7.21 also applies in our present situation.
□
We shall now determine the order of the product stSj of two distinct
fundamental reflections.
16.2 The Weyl group of a Kac-Moody algebra
375
Theorem 16.16 Suppose i ф j. Then the order of stSj e W is:
2	if	AijAfi	= 0
3	if	АаАа	= 1
4	if	AaAji	= 2
6	if	АаАл	= 3
00	if	Ai.iAji	>4.
Proof. The Weyl group W acts faithfully on H*. Let К be the 2-dimensional
subspace of H* given by К = Ca; + Cay. We have
s. (a;) = — a;, S; (a^ =	— A^a,
sj(ai) = ai~Ajiaj, Sj(aj) = -aj.
Thus the subgroup s^ of W acts on K. We obtain a 2-dimensional repre-
sentation of (si, Sj) given by
4	\ ~АЛ -4
Consider the order of this 2x2 matrix representing stSj. Its characteristic
polynomial is
Л+ 1 - AjjAji
Ar
~A6
Л+1
— A2 4- (2 — AfjAj^ Л+1.
The discriminant of this polynomial is
D = (2 - A^AA2 - 4 = А,Ап (А,А. - 4).
Thus there are two equal eigenvalues if A^A^ = 0 or 4, two distinct complex
eigenvalues if .f .A^ = 1,2 or 3, and two distinct real eigenvalues if A^A^ > 4.
Suppose AjjAjj = 0. Then stSj	) and the matrix has order 2.
Suppose AtjAjt = 1. Then the characteristic polynomial is A2 4- A 4- 1 so the
eigenvalues are <y, <y2 where <y = e2m'/3. Thus the matrix is similar to (“ J,)
and so has order 3.
Suppose А^А^ = 2. The characteristic polynomial is then A2 + l =
(A — i) (A + i). Thus the matrix is similar to (q _°i) and so has order 4.
Suppose А^А^ = 3. The characteristic polynomial is A2 — A 4-1 =
(A-|-<u) (A4-<u2). Thus the matrix is similar to (f _°2) and so has order 6.
376
The invariant form, Weyl group, and root system
Suppose AtjАЦ = 4. The characteristic polynomial is A2 — 2A + 1 = (A — I)2.
The eigenvalues are 1, 1, so the matrix is similar to with This
matrix has infinite order.
Now suppose А^А^ > 4. Then the eigenvalues are real and their product is
1. They are also positive and unequal, so have form A, 1: ' where A> I. Thus
the matrix is similar to (j ) so has infinite order.
We have so far considered the action of stSj on the 2-dimensional subspace
К of H*. We now consider the action of stSj on the whole of H*. Let
K' = {XeH* ; A(7i;) = 0, A(/i7)=0}.
Then dim K' = dimH* — 2. Let A e КПК'. Then A = ^ai + pa, and
А(й;) = 2^ + 7/Ау = 0
X(hj) = ^Aji + 2V = 0.
Now I 42 I = 4 —A;iA... Thus if A,-;A;> #4 we have A = 0, 17 = 0 so К П К' =
О. Then H = K®K'. Now stSj acts trivially on K' since, for A~K', we have
siSj(X') = st (A — A	a J = si(W) = A — A (/i;) at = A.
Thus the order of stSj on II is equal to the order of s^j on К provided
A,;A;, ^4. If A,;A;, =4 the order of SjS, on К is infinite, so SjS, has infinite
order on H*.	□
We now define /(w) and n(w) for w e W in the same way as when L(A) is
finite dimensional, /(w) is the minimal length of w as a product of generators
$!,... , sn, and n(w) is the number of аеФ+ with iri'ujtO . Then the
proof of Theorem 5.15 also applies in our present situation and shows that W
satisfies the deletion condition. Also the proof of Corollary 5.16 applies in
our situation and shows that /(w) = n(w). Finally the proof of Theorem 5.18
applies and shows that W is generated by Sp ... , sn as a Coxeter group. Thus
we have:
Theorem 16.17 The Weyl group W of the Kac-Moody algebra L(A) is a
Coxeter group generated by sx,... , sn with relations
sj= 1
= 1
W3 = 1
W4 = i
W6 = i
if АчАЛ = °
if AyAj; = l
if АИАЛ = 2
if AijAji = 3-
16.3 The roots of a Kac-Moody algebra
ЪТ1
16.3 The roots of a Kac-Moody algebra
Let A be a GCM and L(A) the corresponding Kac-Moody algebra. Then
/.(Л) = //©£/.„
аеФ
where Ф = {а^0 ; L,fO}- Ф is the set of roots of L(A). We recall that
ф = ф+ и Ф where Ф+ = Ф П Q+ and Ф = Ф П Q . These are the positive
and negative roots. П = {а1,... , an} is a subset of Ф+ called the set of
fundamental roots. The multiplicity of the root a is defined as dim/.,,. We
know from Proposition 14.19 that the fundamental roots aq,... , an have
multiplicity 1. We also know from Proposition 16.15 that the Weyl group W
acts on Ф and preserves multiplicities.
Definition a e Ф is called a real root if there exist if t II and w e IT such
that a = w (a;).
a e Ф is called an imaginary root if a is not real.
We note that if a is a real root so is —a. For let a = w(a;). Then — a =
wsi (a;). It follows that if a is an imaginary root so is —a.
Proposition 16.18 Let a be a real root. Then a has multiplicity 1. Also, for
k eZ, ka is a root if and only if k = ±l.
Proof. Since a = w(a;) and at has multiplicity 1, Proposition 16.15 implies
that a has multiplicity 1. We also know from Proposition 14.19 that if k > 1
then kai is not a root. Since ka = w (kctj) , ka is also not a root.	□
We now consider the imaginary roots. Let Ф||И be the set of positive
imaginary roots.
Proposition 16.19 If a e Ф^ and w e W then w(a) e Ф^.
Proof. We know that W acts both on Ф and on the set ФКе of real roots.
Hence W acts on the set Ф^ of imaginary roots. We must show that an
element w e W cannot change the sign of an imaginary root. Let
a =	£;>0.
i=l
Now at least two coefficients kt must be positive. Otherwise a would be
a multiple of some at and hence equal to a;. But then a would be real,
a contradiction. Now si(a) = a — a(hj)ai, thus sfa) contains at least one
378
The invariant form, Weyl group, and root system
fundamental root with positive coefficient. Hence G Ф^. Since w G W
is a product of fundamental reflections st we have w(a) G Ф^.	□
We now introduce the fundamental chamber in the context of Kac-Moody
algebras. We recall that in Section 12.3 the fundamental chamber was defined
for finite dimensional semisimple Lie algebras. In the present context we
begin with a GCM A and take a real minimal realisation (Як, П, IP) as in
Remark 14.20. We then define the fundamental chamber as
C={Ag77* ; A(7i;)>0 for i= 1,... , n}.
Its closure is
C = {Ag77* ; A(/i;)>0 for i= 1,... , n}.
Proposition 16.20 Suppose a G Ф^. Then there exists w G W with w(a)
e-C.
Proof. Consider the set of all elements w(a) for w G W. These are all positive
imaginary roots by Proposition 16.19. Let /3 be such a root for which ht/3
is as small as possible. Let /3 = ^f,kiai. Then sf/3) = /3 — /3 (/i;) at. Since
ht st(/3) > ht /3 we have /3 (/i;) <0. This holds for all i, thus /3 G —C.	□
Proposition 16.21 Let аеФ, ct = 'fff=1kiai and supp a = {i ; A;^0}. Then
supp a is connected.
Proof. We may assume асФ+. Let supp a = Jc{l,... , n}. Suppose if
possible that J is disconnected, that is J = JyJ.f with J2 non-empty and
A^ = 0 for all i G , j G J2.
We shall show that [e;	= 0 for all i G , j G J2. We first show the weaker
condition
[[eiey]A]=° for z'e7i> 7еЛ, A=l,... ,n.
We have
ЫЛ] = Ь [еЛ]] + [[е;Л1еу]-
If k^{i,j} then [e;/J=0 and [е,Л]=0.
If k = i then [[е;е7]Л] = [/1;е7]=ЛЛ = О.
If k = j then [[е;е;]Д] = [е;й;] = -А;;е; = 0.
Thus in all cases Д] = 0 for all k.
16.3 The roots of a Kac-Moody algebra
379
Write x=[e;ej. The ideal of L(A) generated by x is U(L(A))x
as in Lemma 16.9. Since L(A~) = N®H®N~ we have 1I(L(A)) =
11 (/V) U (//) U (/V). Now N is generated by /i,... , fn and [x/;] = 0 for
each i, thus II (/V).v = C.v. Since |//;V|c;V we have lI(7/)/Vc/V and so
ni7/)lI(;V ).rC;V since xeN. Finally II(,V)II(//)II (A~) x c U(N)N c N.
Thus lI(L(A))x is an ideal of L(A) intersecting H in O. By definition of
L(A) this ideal must be O. In particular we have x = 0. Thus [e;	=0 for
all i G Ji , j G J2 •
We use this fact to obtain the required contradiction. Since a G Ф+ we
have LafO and La G N. The elements of La are Lie words in ... , en
of weight a, and so are linear combinations of Lie monomials in
ex,...,en of weight a. Thus there exists a non-zero Lie monomial m in
ex,... ,en of weight a. We show that any such Lie monomial must be 0
since it contains factors e; both with 7 and with iG./;. We can write
m = [m1 m2] where m1,m2 are shorter Lie monomials. If either m1 or m2
involves factors e; both with i G and with i G J2 we have m1 = 0 or m2 = 0
by induction. Otherwise all factors e; of m1 have i G and all factors e; of
m2 have IeJ2, or vice versa. But then [m1 m2]=0 since [e; eJ=O for all
i G j G J2. Thus m = 0 and we have the required contradiction.	□
In order to understand the imaginary roots it will by Proposition 16.20 be
sufficient to understand the positive imaginary roots which lie in — C, the
negative of the closure of the fundamental chamber. Such roots satisfy the
conditions:
a G Q+, a 0, supp a is connected, a G —C.
It is a remarkable fact that, conversely, any element a satisfying these condi-
tions is a positive imaginary root. Before being able to prove this we need a
lemma.
Lemma 16.22 (i) Suppose a G Ф, a ±a;, satisfies a — at 0 Ф and a + at 0
Ф. Then a (hf) = 0.
(ii) Suppose a G Ф, af- —at, satisfies a + at 0 Ф. Then a(hi)>0.
Proof, (i) Since a G Ф we have La O. Let x G La with x 0. Let
и; = exp ad e; • exp ad (—/;) • exp ad e; G Aut L(A).
We show that nyx G Ls.(ay For [hx] = afh)x for all h G H, hence [w;/i, n;x] =
aftynpc. Now nfH) = H by Proposition 16.11 and so
\h', ntx\ = a (n^h'^ ntx for all h' G H.
380
The invariant form, Weyl group, and root system
We have n; 1h' = si 1h' = sih' also by Proposition 16.11. Thus [h', пре] =
a(sih')nix=(sia) (h'ynpc for all h' eH. Hence ntx e L^y
Now ad e; • x ELa+a. and so ade;-x = 0 since а + а^Ф and a + a^O.
Thus exp ad e; • x = x. Also ad/'(-.ve„ so ad/;-x = 0 since a — a; Ф
and a — ai^0. Thus expad (-fi)-x = x. Hence ntx = x. Since xeLa and
Л|Хб1,.(а) we deduce sfa) = a. But sfa) = a — a (/i;) at and so a (/i;) = 0.
(ii) Again let x be a non-zero element of La. As before exp ad e; • x = x since
a + at 0 Ф and a + at 0. We have
where (ad x = 0, since ad(—/;) is locally nilpotent. Thus
Now (ade;)<+1 (ad/;)( x = 0 for each positive integer t, since a + a; Ф
and a + at Hence (ad et~)k (ad/;)(x = 0 for all к > t+ 1. It follows by
considering the above expression for ntx that
ntx eLa®La_a. Ф • • • ФLa_pa..
However,	as in (i). Thus si(a) = a — a(hi)ai = a — kai for
some k with 0 < к < p. Hence a (hj) > 0.
We now define
K=[aeQ+, a^0, suppa is connected, ae- C}.
Proposition 16.23 К с Ф^.
Proof. Let a e К. Then a = J2"=1 kjOij with each kj > 0 and kj > 0 for some i.
Also suppa = {z ; kjf^Q}.
We define a set Ф of roots by
Ф = /3 e Ф+ ; /3 = mjCtj with mt < kt for each i
i=l
Ф is a finite non-empty set of positive roots, since it contains at least one
fundamental root. We choose a root /3 G Ф such that ht/3 is as large as
possible. We aim to show that /3 = a and hence that a e Ф. We shall show
first that supp /3 = supp a.
16.3 The roots of a Kac-Moody algebra
381
Suppose if possible that supp /3 supp a. Since supp a is connected there
exist j e supp a — supp /3 and j' e supp /3 such that Ajj, 0. Let /3 = J2"=1 пцо^.
Then rrij = 0. Now /3 — aj 0 Ф since rrij = 0 and /3 + aj 0 Ф by the maximality
of ht/3. By Lemma 16.22 (i) we have /3 (hj) =0. But
/3(/гу)= E miai(hj)= E miAji<Q
iesupp/3	iesupp/3
since mi > 0, Ац < 0 and Ajj, < 0. This contradicts /3 (hj) = 0. Thus supp /3 =
supp a.
Now we have a = J2"=1 ktah /3 = J2"=1 m,tt, with mj<kj. Let
J = {i e supp a ; к = mJ .
We aim to show that J = supp a and so that /3 = a.
Suppose if possible that J supp a. Let i e supp a — J. Then mi<ki. Hence
/3 + at 0 Ф by the maximality of ht /3. Thus /3 (ht) > 0 by Lemma 16.22 (ii).
Let M be a connected component of supp a — J. Then /3 (/i;) > 0 for all
i e M. Let /3' =	m;a;. Then
0’ (К) = 0 (hi) - E mjaj(hi)
jesuppa—M
= 0(hi)-^ rrijA^.
jesuppa—M
If ieM then /3(h) >0, rrij >0 (since supp/3 = suppa) and Ay<0. Hence
/3' (hj) > 0. Since supp a is connected there exists i' e M and j' e supp a — M
with Aj,j, Then
/3'(M = /3(M-E mjAi'j
jesuppa—M
We have /3'(/i;J>0 as before; however, in fact we have /3'(/if)>0. The
strict inequality holds since rrij, > 0 and A^, < 0.
Let AM be the principal minor (Atj) with i, j e M. Let и be the column
vector with entries m; for j e M. Since
J J
/3' (hj) =	Aijmj	f°r г e м
jeM
we have и > 0, AMu > 0, AMuf 0. Now we recall that if the indecomposable
GCM AM is of affine type then Ами>0 implies Ами = 0. Also if AM is of
indefinite type then Ами>0 and и>0 imply и = 0. Thus AM cannot have
affine type or indefinite type. Hence AM has finite type.
382
The invariant form, Weyl group, and root system
Now let у =	(k, — mt) at. We have kt — mt > 0 for all i e M. We recall
that
a — /3= 22	(^i —тг)аг
zesuppa—J
Thus for i e M we have
(a-/3)(/i;) = 22 (kj~mj)Aij=I2 (kj~mj)Aij = y(hd
jesuppa—J	jeM
since M is a connected component of supp a — J. Thus у (Zi;) = a (Zi;) — /3 (Zi;)
for all i e M. Now a (hi) < 0 since aeK and /3 (Zi;) > 0 since i e M. Thus
7 (hi) < 0 for all i e M.
Now let и be the column vector with entries kt — mt for i e M. Then we
have и > 0 and AMu < 0. Since AM has finite type AM(—u) > 0 implies — и > 0
or — u = 0, that is и<0 or и = 0. This is a contradiction since и>0. This
contradiction shows that J = supp a and hence that /3 = a. Thus a e Ф. Since
aeQ+ we have аеФ+. Thus aeK implies аеФ+. Now aeK implies
laGK, so 2аеФ+. By Proposition 16.18 this implies that аеФ^. This
completes the proof.	□
This remarkable proof, due to V. Kac, enables us to determine the set of
all positive imaginary roots, and hence the set of all imaginary roots.
Theorem 16.24 The set of positive imaginary roots of L(A) is given by
^Im = ^weWW(K)
where
К = [a e Q+ ; a^O, supp a is connected, a e — C}.
The set of all imaginary roots is Ф^и (—Ф^)-
Proof This follows from Propositions 16.19, 16.20, 16.21 and 16.23.	□
Corollary 16.25 Let a e Ф^. Then ка e Ф^ for all positive integers k.
Proof This follows from Theorem 16.24 and the fact that aGK implies
kaeK.	□
We next consider the real and imaginary roots of L(A) when A is sym-
metrisable. Then L(A) has an invariant bilinear form (,) described in Sec-
tion 16.1. This form is non-degenerate on restriction to H, so determines
16.3 The roots of a Kac-Moody algebra
383
an isomorphism H* H under which A hf where А (л ) = (hf x) for all
x e H. We can then transfer the bilinear form to II by defining
In particular we can define (a, a) for a e Ф.
Proposition 16.26 Suppose A is a symmetrisable GCM. Then if a is a real
root of L(A) we have (a, a) >0. If a is an imaginary root then (a, a) < 0.
Proof. The form (,) on H is И'-invariant by Proposition 16.13, so the induced
form on H* is also Vl'-invariant. By definition of the form on II we have
(hf, x) = djttfx) for alive H.
Hence </;a; e H* corresponds to ht e H under our map H* H. Thus
1	2
(acai) = hi^ = ~7'
df	at
In particular (at, af > 0. Now each real root has form w (af for some w e W
and some i. Hence
(w (af , w (a;)) = (ah af > 0.
Now consider the imaginary roots. Let aeK. Then a = ffkiai with each
к > 0 and a (hf < 0 for each i. Thus
(a, a) =	(a, af = sffki- — a (hf < 0
since f >0, dt>0, a (hf < 0. Every positive imaginary root has form w(a)
for some w e W, a e K, thus
(w(a), w(a)) = (a, af < 0.
For the negative imaginary roots we have
(—w(a), — w(a)) = (w(a), w(a)) < 0.	□
We next obtain information about the imaginary roots in the three cases of
our trichotomy.
Theorem 16.27 Let A be an indecomposable GCM.
(i)	If A has finite type then L(A) has no imaginary roots.
384
The invariant form, Weyl group, and root system
(ii)	Suppose A has affine type. Then there exists и > 0 with Au = 0. The
vector и is determined to within a scalar multiple. Thus there is a unique
such и whose entries are positive integers with no common factor. Let
u=(a1,... ,an). Let 8 = a1a1-\---------\-anan. Then the imaginary roots of
L(A) are the elements kS for k eZ,
(iii)	Suppose A has indefinite type. Then there exists a e such that a =
J)"=1 with > 0 and a (/i;) < 0 for all i = 1,... , n.
Proof, (i) If A has finite type L(A) is a finite dimensional simple Lie algebra
by Theorem 15.19. Thus each root of L(A) is real by Proposition 5.12.
(ii)	Suppose A has affine type. We first consider the imaginary roots in K.
Let aeK satisfy a = J2"=1 kj/cq. Let v be the column vector (kr,... , kn).
Then we have v > 0 and Av < 0, since a (hf) < 0 for each i. But in affine
type A(—v) > 0 implies A(—v) = 0. Thus vf() and Av = 0.
We also have и > 0 and Au = 0. Since A has corank 1, и is a multiple
of u. Since the coefficients of и have no common factor v = ku for some
к e Z with к > 0. Thus a = k8.
Now every positive imaginary root has form w(a) for some a e K, by
Theorem 16.24. We have
si (8) = 8 — 8 (ht) at = 8
since 8(hj) = 0 follows from Аи = 0. It follows that w(8) = 8 for each
w e W. Thus the only positive imaginary roots are the elements k8 with
к e Z ,k > 0. Hence the only imaginary roots are the k8 with к e Z, к 0.
(ii	i) Suppose A has indefinite type. Then there exists и > 0 with Au < 0.
Suppose u= (kr,... , kn). Let a = J2"=1 £;a;. Then atK and a (hf) < 0
for all i. Thus a is a positive imaginary root of the required kind.
□
A significant consequence of the last result is as follows.
Corollary 16.28 If A is an indecomposable GCM of affine or indefinite type
then the dimension of L(A) is infinite.
Proof. In both cases L(A) has an imaginary root a. Thus it has infinitely
many imaginary roots ka for IeZ,I^0, by Corollary 16.25. Since L(A) =
//®	dimL(A) must be infinite.	□
16.3 The roots of a Kac-Moody algebra
385
We next consider which of the imaginary roots of I.(A) when A is sym-
metrisable satisfy (a, a) =0.
Proposition 16.29 Let A be symmetrisable and a be an imaginary root of
L(A). Then {a, a) = 0 if and only if there exists w e W such that the support
of w(a) has a diagram of affine type.
Proof. First suppose (a, a) = 0. We may assume without loss of generality
that a e Ф+. Thus there exists w e W with w(a) eK, by Theorem 16.24. Let
/3 = w(a). Then /3 (/i;) < 0 for all i. Let J be the support of /3 and /3 = J);e7 £;a;.
Then J is connected. Now
</3,/3) = E^ (/3,a;) = £^/3(M-
i(=_J	i(=_J ai
Now ki > 0, > 0 and /3 (/i;) < 0 for all i e J. Since we also have (/3, /3) =
{w(a), w(a)) = (a, a) =0 we deduce that /3(/i;) = 0 for all ieJ. Hence
fff/kff, (/i;) = 0, that is У. (.LL h Let и be the column vector with
entries kj for j e J. Then и > 0 and Д,и = 0. Since A} is an indecomposable
GCM this implies that A} has affine type, by Corollary 15.11.
Conversely suppose /3 is a positive imaginary root whose support J has a
diagram of affine type. Then I.ffO and so I.(A) contains a non-zero Lie
monomial in ,... , en of weight /3. The letters e; in this Lie monomial all
have i e J. Thus the Lie monomial lies in L (Aj) and so /3 is a root of L (Aj).
If /3 were a real root of L (Aj) it would have form w (a;) for some w e W (Л7)
and i e J, and so /3 would be a real root of L(A). Thus /3 is an imaginary root
of L(A7). Since L(A7) has affine type, /3 = k8 where 3 is the element for
L (Л7) defined in Theorem 16.27 (ii). Let f> = Z),7«,«,. Then
О=ЕаД«() = £^) = 0
ie7	ie7 ai
since 3 (/i;) = 0 for all i e J. Thus (/3, /3) =0 also. Finally if a is any root of
L(A) satisfying w(a) = /3 for some w e W, we have (a, a) =0 also. □
17
Kac-Moody algebras of affine type
17.1 Properties of the affine Cartan matrix
We now consider the Kac-Moody algebras ISA) where A is a GCM of affine
type. Let A be an и x и matrix of rank /. Then we know that n = I + 1. We
shall number the rows and columns of A by the integers 0, 1,... , I. There
exists a unique vector a ={a0,a1,... ,at) whose coordinates are positive
integers with no common factor such that
The possible Dynkin diagrams of such matrices A were obtained on the affine
list 15.20. We shall choose the numbering of the vertices in such a way that
node 0 is the one in black in the diagram below. We also show in each
diagram the integer a; associated to each vertex.
17.1 The integers a0, ar,... , at.
386
17.1 Properties of the affine Cartan matrix
387
388
Kac-Moody algebras of affine type
1	2	3	4	5	6	4	2
•------О------О-----О--------О----Q---------О------О
1	2	3	ч	4	2	1	2	3	2	1
---------О---------О	>	О----------О	•---------О-----------о	<	<>--------о
There exists also a unique vector (c0, cp ... , c;) whose coordinates are pos-
itive integers with no common factor such that
(c0,Ci,... , c;)A=(0,0,... ,0).
In fact the vector (c0, cp... , c;) for A is the same as the vector
(a0,	, a;) for the transpose A1. Thus the vector (c0, cp ... , c;) may
also be read off from the diagrams in the list 17.1.
Proposition 17.2 (i) c0=l.
(ii) a0 = 1 unless A has type C, or A'v In these cases a0 = 2.
Proof. This is clear from 17.2	□
Let (//,11, II ) be a minimal realisation of A. Then dim H = 2n — I =
1 + 2. IP = {h0, Ip,... , h[} is a linearly independent subset of H and П =
{a0, Op ... , ct[} is a linearly independent subset of H*. These exists an ele-
ment dell such that
a0(t/) = l a;(</) = 0 for i = 1,... , I.
d is called a scaling element.
Proposition 17.3 h0, Ip,... , Ip, d is a basis of H.
17.1 Properties of the affine Cartan matrix
389
Proof. We must show that d is not a linear combination of h0, hr,... , /i;. Sup-
pose if possible that d = '^li=okihi. Then afd) =	(.hi) = 'f2li=okiAij.
Hence
i
(A;o,... , A;;) = (1,0,... , 0).
i=0
In particular, omitting the first column of A,
£Z:;(A;1,...,A;;) = (0,...,0).
i=0
However, we also have
£C;(A;1,...,A;;) = (O,...,O).
i=0
Since the (/ +1) x I matrix (Ay), 0 < i < I, 1 <j <1 has rank /, this implies
that (k0,... ,k[) is a scalar multiple of (c0,... , c;). But this would imply that
, £;)A=(0, 0,... ,0)
a contradiction.	□
We now define an element у e H* determined uniquely by
у (h0) = 1	y(/i;) = 0 for i= 1,... , I y(d)=0.
Proposition 17.4 a0, аъ ... , ahy is a basis of H*.
Proof. The (/ + 2) x (/ + 2) matrix obtained by applying these elements of II
to the basis of H in Proposition 17.3 is
0
1
/
I +1
0 1 • • • I l+l
where A0 is a Cartan matrix of finite type. Thus det A0 ^0 and so the deter-
minant of the above matrix is also non-zero. Hence а0, ar,... ,ahy must be
a basis of H*.	□
390
Kac-Moody algebras of affine type
We know that any indecomposable GCM of affine type is symmetrisable.
We shall now express the affine Cartan matrix A in an explicit way as the
product of a diagonal matrix D with positive diagonal entries and a symmetric
matrix B. The diagonal entries of D are rational, but not necessarily integral.
Proposition 17.5 We have A = DB where D = diag (d0, dx,... ,dj) and В is
symmetric, where d^afc^
Proof. By Theorem 15.17 there exists a diagonal matrix D with positive diago-
nal entries and a symmetric matrix В such that A = DB. Let c = (c0, cx,... , c;)
and at = (a0, ax,... , a;). Then Aa = 0 so DBa = Q, and hence Ba = 0. Thus
aff = 0. Also cA = 0 so (cD)B = 0. Since В has corank 1 cD must be a scalar
multiple of a1. In fact we can choose D so that cD = a1, that is dt = af ct. □
Now we have a non-degenerate bilinear form on H defined as in Proposi-
tion 16.1. This form satisfies
(ht, hj) = didjBij=ajcf1Aij for i, j = 0, 1,... , I
(ho,d) = doao(d) = ao
{ht,d)=Q for/=1,... ,1
(d, d}=0.
This standard invariant form on H defines a bijection H* H given by
A hj where Л (л ) = (hj, x) for all x e H.
Proposition 17.6 Under this bijection between H and H*, h, e H corresponds
to ajcffcti e H* for i = 0, 1,... , / anddtH corresponds to aoy e H*.
Proof. For j = 0, 1,... , / we have
ajcf1ctj (hj) = djA^ = (hj, hj) for i = 0, 1,... , I
ajCflctj(d) = djCtj(d) = (hj, d}
thus ajC^ctj eH* corresponds to hj eH. We also have
aoy (hj) = (d, hj) for i = 0, 1,... , I
aoy(d) = (d, d)
thus aoy eH* corresponds to deH.	□
17.1 Properties of the affine Cartan matrix
391
We may transfer the standard bilinear form from H to H* using this
bijection. The form on H* is then given by
a^ = af1 c^j	i, j = 0, 1,... , I
{a0,y) = aj1
{at, y) =0 i= 1,... , I
y) = 0.
We note in particular that
2(a;, aj
Corollary 17.7 Under the given bijection between H and H*, jell corre-
sponds to f'j e H*.
Proof This follows from Proposition 17.6.	□
We now define an element cel! by c = J2-=oUnder the bijection
H H* c corresponds to 3. For <5 = a;a; and /i; corresponds to ayfcy
by Proposition 17.6.
Proposition 17.8 The element c lies in the centre of I.(A). In fact the centre
is 1-dimensional and consists of all scalar multiples of c.
Proof. For each simple root a) we have aj (c) = Jj.=0 ctaj (hj) = ff!i=o ci&ij= 0-
It follows that a(c) = 0 for all a e Ф. Now L(A) = H ® ^аеФРа. Thus each
element of L(A) has form h + fjx,, where heH, xaeLa and finitely many
xa are non-zero. Thus
с, h + Y,xa
a
E“(cK=°-
Hence c lies in the centre of L(Aj.
Now let h + fjrixr. be any element of the centre of L(Aj. Then we have
x, h + Y,xa
a
for all x e H.
= 0
Thus fjfa a(x)xa = 0 for all x e H. This implies a(x)xa = 0 for all x e H. Now
for each аеФ there exists xeH with ajxj^fQ. Hence xa = 0. This shows
that the centre of L(Aj lies in H.
392
Kac-Moody algebras of affine type
So let h e H lie in the centre of L(A). By Proposition 17.3 we have
i
+ for£;,£eC.
i=0
Let xeLa_. Then \hx\ = af/iff. Hence аД/г) = 0 for each j = 0, 1,... ,/.
Thus
i=0
that is
E£A=o for7=i,...,/
i=0
and
Еело=-е-
i=0
However, we have	= hence Ai0 = — a°1 Aijaj- Thus
Е=о£гДу = О for	implies E=o= 0- Thus we deduce
that £ = 0, and so h = ^li=0^ihi. This in turn gives E=o£/E = O for
j = 0,1,...,/. This implies that (£0, ff,... , £,.) is a scalar multiple of
(c0, cx,... , c;) since A is an (/+ 1) x (/+ 1) matrix of rank /. Thus h is a
multiple of c. Thus the centre of L(A) is the 1-dimensional subspace spanned
by c.	□
c is called the canonical central element of L(A).
Summary
We will find it convenient to summarise in one place the properties of the
various elements discussed in this section:
(a)	h0, h.^... , h[, d are a basis of H.
(b)	c = coho 4--h clhl is the canonical central element.
(c)	a0, aj,... , U[, у are a basis of H*.
(d)	8 = aoao-\--\-alal is the basic imaginary root.
(e)	The standard invariant form on H is given by
{hi’hj) = ajcf1Aij i,j = 0, 1,... ,/
(h0,d)=a0
{hi,d}=0
(d, d}=0.
17.1 Properties of the affine Cartan matrix
393
(f) The standard invariant form on II is given by
	a^ = afx	i, j = 0, 1,... , I (ao,y) = ao1 (a;,y)=0	i=l,...,l t) = o.
(g) The action of H*	on H is given by aj(.hd = Aij /,./ = 0,1,...,/ a0(X) = l afd) = Q	i=l,...,l У (.hf) = 1 7(/i.) = 0	i=l,...,l y(d) = 0.
(h) The properties of the central element c.
	(hi,c)=0	/ = 0,1,..., 1 (d, c) = a0 (c, c) =0 “y(c) = 0 j = 0,l,...,/ y(c)=l.
(i) The properties of the imaginary root 3.
	M = 0	7 = 0,l,...,/ (%5) = i (<5,<5)=O 5(/i.) = 0 z = 0,l,...,/ 8(d) = a0 8(c) = 0.
394
Kac-Moody algebras of affine type
(j) Properties of the standard bijection H	H*.
ht aicf1ai i = 0,l,...,l
d aoy
c—> 8.
17.2 The roots of an affine Kac-Moody algebra
Let A0 be the matrix obtained from the affine Cartan matrix A by removing
the row and the column 0. Then A0 is an I x I Cartan matrix of finite type.
By list 17.1 we see that A0 is given in each case by the following list.
The underlying Cartan matrix A0
A	A0
A;	I > 1	A;
Aj	Ax
Z>3
I > 3	c;
C;	Z>2	C;
c;	i > 2	b;
c;	z>2	c;
Di	l>4	Dl
Et	1 = 6,1,8 El
f4
F4
g2	g2
G‘	G2
Let Ф0 be the set of roots of the finite dimensional Lie algebra L (A"). Ф0
has a fundamental system П0 = {cup ... , a;}. Let IT" be the Weyl group of
Ф0. Then IT" is generated by the fundamental reflections ... ,
Now we know that the imaginary roots of L(A) are the elements kS with
keZ and k^O, by Theorem 16.27 (ii). (However, we do not yet know the
multiplicities of these roots.) Thus we shall now consider the real roots of
L(A). These have the form w (a;) for some w e W and i = 0, 1,... , I. We con-
sider the squared lengths (a, a) of the roots a e ФКе. Since (w (a;) , w (a;)) =
(ai, al) the length of any real root is equal to the length of some fundamental
П.2 The roots of an affine Kac-Moody algebra
395
root. The relative lengths of the fundamental roots may be obtained from list
17.1 using the formulae
Aij = 2±-^-
(ar aj) _ Aa
An'
Proposition 17.9 (a) If A is an affine Cartan matrix of types Ah i2>l,E6,E-1, Es
all the fundamental roots have the same length.
(b)	If A has types Bh B\, Cb C(, Ff there are fundamental roots of two
different lengths. The ratio (/3, /3)/(a, a) where a is short and /3 is
long is 2.
(c)	If A has type G2 or Gf there are fundamental roots of two different
lengths with (/3, /3)/(a, a) = 3.
(d)	If A has type Af there are fundamental roots of two different lengths with
{/3, f3}/(a, a) =4.
(e)	If A has type C[ there are fundamental roots of three different lengths,
say a, /3, y, with (/3, f3}/(a, a) =2 and (% y}/(f3, f3} =2.
Proof. This is clear from list 17.1.	□
We shall denote by ФКе the set of short real roots, by ФКе j the set of long
real roots and by ФКе; the set of real roots of intermediate length. The latter
set is non-empty only when A has type C;' for some /. If all real roots have
the same length we use the convention ФКе = ФКе .
We now aim to characterise the set ФКе s. We consider the possible values
of (a, a) for a e Q. Let a = J2;=o £;a;. Then (a, a) = fT j kff a^. Now
Uj'j e Q for all i, j. Thus there exists d e Z with d > 0 such that e
|Z for all i, j. Thus if (a, a) > 0 then (a, a) >j. Hence there exists m > 0
such that m = min (a, a) for all a e Q with (a, a) > 0.
Proposition 17.10 If aeQ satisfies (a, a) = m then a e Q+ or a e Q~.
Proof. Suppose if possible there exists a^Q with (a, a) = m but agQ+
and uf(Q . Then a = /3 — у where /3, у e Q+, /3^0, у ^0 and supp /3 П
supp у = ф. Hence
(a,a) = (13, /3) + (y,y)—2 (/3, y)
and (/3, y) < 0 since supp /3 П supp у = ф. Hence (a, a} > (f3,	+ (y, y).
396
Kac-Moody algebras of affine type
Now all proper connected principal minors of A have finite type. Thus,
considering the connected components of supp /3, we have /3 = /31-\---\-/3r
with supp /3, connected for each i, Iff, (3fj = <) for i^j, and (/3;, /3;) > 0. Thus
(/3,/3) = (/31,/31) + --- + (/3r,/3r)>0.
Hence (/3, /3) > m. Similarly (y, y) > m. But then (a, a) > 2m, a contradic-
tion. Hence a e Q+ or a e Q~.	□
Proposition 17.11 Let A be an indecomposable GCM of finite or affine type.
Then the set ФКе s of short real roots of L(A ) is given by
фге,8 = {a^Q ; (a,af = m}.
Proof Suppose ueQ satisfies (a, a)=m. We show аеФКе. By Proposi-
tion 17.10 a e Q+ or a e Q~. We may suppose a e Q+. Consider the set
{w(a) ; weW}O2+.
We choose an element	= in this set with ht/3 minimal. Then
(/3, /3) = m, so JT kt {at,	= m. Since ki > 0 and m > 0 there exists i with
(а;, /3) >0. Thus /3 (hf = 2> 0. Now sfff) = /3 — /3 (hf) at so ht st(/3) <
ht/3. By minimality of ht /3 we must have sfff) e Q . But /3eQ+, sfjf) e Q
imply /3 = ra; for some r e Z with r > 0. Since
(ra;, raf = r2 (a;, off > r2m
we have r = 1. Thus /3 = 0^ and (a;, af = m. Hence /3 e ФКе s and so a e ФКе s
also.
Conversely if a e ФКе s then a = w (a;) for some w e W and some i, and
(a, a) = (ai, af. However, we have seen that the short fundamental roots
have (a;, af =m. Thus (a, a) = m also.	□
We aim next to characterise the set ФКе j of long real roots. In order to do
this we compare the roots of L(A) and L (A1). Here A can be any GCM.
Proposition 17.12 If (II, П, IP) is a minimal realisation of the GCM A then
(// , I Г11 ) is a minimal realisation of A'.
Proof Let A be an их и matrix of rank /. Then dim/7 = 2/! — I, IIV =
{/ij,... ,/i„} is a linearly independent subset of H, П = {a^,... , an} is a
linearly independent subset of H*, and ер (Н2) = А^.
П.2 The roots of an affine Kac-Moody algebra
397
We now replace H by its dual space H*. We still have dim77*=2w — I,
(77*)* can be identified with 77 by means of the formula
/i(A) = A(/i) for heH, XeH*.
Since

we see that (77*, IIV, П) is a minimal realisation of A1.
□
Now suppose A is symmetrisable. Then we have an isomorphism between
77 and 77* induced by our standard invariant form. Under this isomorphism
ht corresponds to =diai. For each real root аеФКе we define the
corresponding coroot ha e H to be the element of 77 corresponding to e
H*. The element ha can also be described by using the Weyl group. Since
the H'-actions on 77 and 77* are compatible with the above isomorphism, if
a = w (a;) then ha = w (hf). For ht corresponds to ^a’a and {a, a) = (a;, a;).
Thus the coroots ha for a e ФКе for L(A) may be interpreted as the real roots
for L (A1). Moreover we have
Hence a is a short root for L(A) if and only if ha is a long root for L (A1).
The fact that short roots give long coroots and long roots give short coroots
is very useful. We shall apply this to characterise ФКе j in the case when A is
of finite or affine type.
Proposition 17.13 Let A be an indecomposable GCM of finite or affine type.
Then the set ФКе j of long real roots of L(A) is given by
Ф„ . =	= e Q ; (a, a) = M, k,^-—for all i
’ I	(a, a)
where M = max {(a, a) ; аеФКе}.
Proof. We first show the long real roots satisfy the given conditions. Let
a e ФКе1. Then (a, a) =M. Let a=ff.a.. Then
2a _ (a;, at) 2at
------- = / Л;------------------
(a, a)	(a, a) (at, at)
398
Kac-Moody algebras of affine type
and so
_ (at, af
ha = Hki-,----
(o', a)
This expresses a root for L (A1) as a linear combination of fundamental roots,
thus the coefficients k. 'fyfi'1 lie in Z.
1 {a,a)
Conversely suppose a e Q satisfies the given conditions. Then ha effZh,
and (ha, ha) =4/M. Now 4/M is the minimum possible value of (/3, fif for
all real roots /3 of L(Al). Thus by Proposition 17.11 ha is a short root of
L (A1). Hence a is a long root of L(A).	□
We next wish to characterise the set ФКе; of intermediate roots of L(A)
when A has type C;'. We first need a lemma.
Lemma 17.14 (a) Suppose A is an indecomposable GCM of finite or affine
type. Then the set of all a = 'ffkiai e Q satisfying kt € Z for all i is
invariant under W.
(b) If a = k^i e Q satisfies к e Z for all i then aeQ+ or a e Q .
Proof (a) Suppose a satisfies our condition. It is sufficient to show that
Sj(a) satisfies it also. Now sfa) = a — a (hj) Uj. Thus it is sufficient to show
that
.	. .. (af, af
7 J {a, a)
that is a (hf e Z. Now we have
. . (ah O';)	. . lah O';)
a (hij V—Г = Ykiai (hi) V—Г
' J {a, af t ' J {a, af
— z x (cr, cr) — (cr, cr)
= ЕМЛМу^=ЕДл¥Н^
; J (a, a) J (a, a)
as required.
(b) Suppose the result is false. Then a = /3 — у where /3, у e Q+, /3fi0, у fi0
and supp /3 П supp у = ф. Then
(a, a) = (fi, /3) + (y, y) -2(13, y) > (/3, /3) + (y, y).
П.2 The roots of an affine Kac-Moody algebra
399
Now /3 = E;esuppy3^a; so
and

(a, a)
Hence e Z. Similarly we have 7Ц G Z.
{a,a)	J	{<x,a)
Now all proper connected principal minors of A have finite type. Thus
we have /3 = /31-\--\-{3r with supp/3; connected for each i, Iff, pf = k
for i^j, and (Pt, PP > 0. Thus
(P,P) = Z dis-
similarly we can show (y, y) >0. Thus (a, a) >0 also. But now we
have 7^7 e Z so (P, P) > (a, a), and 7Ц e Z so (y, y) > (a, a). Hence
(a, a) > (P, P) + (у, у) > 2(a, a), a contradiction.	□
We now suppose A is a GCM of affine type C;'. The diagram of A is
О < О-------------О------о------о-------о------о-------о < о
0	12	1-2	1-1	I
Let т' be defined by (ah at) = m' for i = 1,... , I — 1. Thus m' is the squared
length of the intermediate roots.
Lemma 17.15 Suppose A has type Cf Suppose и =ffii=^Ikiai e Q satisfies
(a, a) = m'. Then к £ Z for all i.
Proof The required condition is obvious for all z^O since (ah af =m' for
i = 1,... , I — 1 and (a,, a,) = 2m'. We must therefore show e Z, that
is that k0 is even.
NOW U' = ^oQ'o + ELl^iQ'i’ ^us
(1	1	\
i=l	i=l	I
I	I
= kl “0)+кок1Аю <“1, “1) +E^ ai)+E kikjAij ad 
400
Kac-Moody algebras of affine type
Thus (a, a) ek® (a0, a0) + Zm'. But (a, a) = m' so kf (a0, a0) G Zm'.
Since (a0, a0) = ^m' we have k^/2 G Z and so k0 is even, as required. □
We can now characterise ФКе
Proposition 17.16 Suppose A is a GCM of type C'[. Then
фке,1 = {“е2 ; {a,a)=m'}.
Proof. Let a^Q satisfy {a, a)=m'. By Lemma 17.15 a = U!i=okiai with
kt e ^ог each ВУ Lemma 17.14(b) aeQ+ or acQ . We may
assume a g Q+.
Consider the set
{w(a) ; weW}O2+.
We choose an element /3 = '}f!i=ok'iai in this set with ht /3 minimal. Then
(/3, /3) = m' and so J2-=o k't {at, jf) = m'. Since m' > 0 and k't > 0 there exists
i with (a;,/3)>0. Thus /3 (/гг) = 2	>0. Now sfff) = /3 — /3 (/i;) ai so
ht st (/3) < ht /3. By the minimality of ht/3, sfff fQ. But sf/3)eQ+ or Q
by Lemma 17.14 (a) and (b). Thus /3 G Q and st(J3) G Q  Hence /3 = rat for
some reZ with r>0. Thus (/3, /3) = r2 (at, at) = m'. However, (a;, at) >
|m' thus r = 1. Thus /3 = aie ФКеIt follows that a G ФКед also.	□
We are now able to obtain explicitly the set ФКе of all real roots of each
affine Kac-Moody algebra individually. We recall that Ф0 is the root system
of the Lie algebra L (A0) of finite type obtained by removing vertex 0 from
the diagram of A. We denote by Ф°, Ф;0 the set of short and long roots in Ф0.
If all roots of Ф0 have the same length we write Ф° = Ф°.
Theorem 17.17 The real roots of the affine Kac-Moody algebra L(A) are
as follows.
(a)	If A is one of the types Al,Bl,Cl,Dl,E6,E-l,Ei,Fir,G1 then ФКе =
{a + r3; a G Ф°, rGZ}.
(b)	If A is one of the types B\, C\, then
ФКе s = {a + r<5 ; асФ°, rGZ}
ФКед = {a + 2r3 ; аеФ°,ге%].
(c)	If A is of type G\ then
ФКе8 = {o' + r8 ; асФ°, rGZ}
ФКе1 = {о' + ЗгЗ ; a,GФfl,rGZ}.
П.2 The roots of an affine Kac-Moody algebra
401
(d)	If A is of type С\ then
Фке,з = Н(“ + (2г-ф) :M>gZ)
ФКе1 = {a + rd ; a g Ф°, reZ}
ФКеi = {a + 2r<5 ; аеФ;0, reZ}.
(e)	If A is of type A\ then
Фке,8 = {К« + (2г-1)й); (геФ' .гег|
ФКс! = {a + 2лй ; аеФ0, reZ[.
Proof (i) Suppose first that A is not of type C;' or Af Then Ф° c Let
a G Ф°. Then (a, a) = m. Hence for r G Z we have (a + r8, a + rS) = m since
(a, 8) =0 and (3, 3) =0. By Proposition 17.11 this implies a + r8 G ФКе s.
Conversely suppose /3 = J2;=o kiai G ФКе s. We have a0 = 1, thus <5 = a0 +
J2.=ia;a;. Hence /3 — ko8 = 'ffli=1(ki—koaj)ai. Thus (/3 — k08,/3 — k08) =
(/3, /3) =m. Again by Proposition 17.11 we deduce /3 — £0<5 G Ф°. Let
a = /3 — k08. Then /3 = a + k08 for a g Ф°, k0 g Z.
Thus the short roots in Ф have the required form. We now consider the
long roots. We have Ф;0 с ФКе j.
Let аеФр Then (a, a)=M and so (a + s8, a + s8} =M for all sgZ.
Let a = J2;=1 £;a;. By Proposition 17.13 we have kt 6 Z for i = 1,... , /.
The same proposition shows that a + s8 E ФКе j if and only if sa G Z
for z = 0,1,... ,/. Now (аи af = —, thus the condition is —^stZ for
i = 0, 1,... , /. We note that {a0, a0) = 2 since a0 = 1.
First suppose that a0 is a long root, that is that we are in case (a). Then
(a, a) =2 and so = C:S g Z. Hence tt + .s?> e Ф„,. for all seZ.
Conversely suppose /3 = 22;=o G ФКе P Then
i
/3 — k08 = (kt — koaf at
1=1
and we have (/3 — k08, 0 — ко8} = (/3, f3} =M. Since /3 G ФКс we have
к, G Z for i = 0, 1,... , /. We have кпа; 21’„Р G Z also since (a;,	= —
and (f3,[3}=2. Hence /3 — £0<5 G Ф^1 by Proposition 17.13. Thus /3 = a + k08
for some a e Ф° and k0 G Z.
Next suppose that a0 is a short root, i.e. that we are in case (b) or (c). Let
fa’a'> = n. Then p = 2 in case (b) and p= 3 in case (c). Thus
^C-	s
-—s=ct— since (a, a) =2p.
{a, a)	p
402
Kac-Moody algebras of affine type
Since c0 = 1 this lies in Z for all i = 0, 1,... , / if and only if s is divisible
by p. Thus by Proposition 17.13 a + pr8 e ФКе j for all r e Z.
Conversely suppose /3 = 22-=0 kiai e ФКе Р Then
i
/3 — k08 = 22 (kt — к0а2) at.
1=1
We have {(3 — k08, (3 — k08) = {(3, (3}=M. Since /3 e ФКед we have kt
for z = 0,1,... ,In particular kQ 2c22 = — € Z. We show Aya, 2(2^ e
r	u AA> p	u 1 AA>
for i = 1,... , For koat = 7'ci e since {at, a;) = and {/3, {3} = 2p.
Thus by Proposition 17.13 /3 — Ao<5 e Ф^1. Let a = /3 — k08. Then (3 = a + pr8
for some a e Ф^1, r e Z.
We have thus proved the required result in cases (a), (b) and (c).
(ii) We now suppose that A has type C;'. Then we have Ф° с ФКе; and Ф° c
ФКе j. First suppose a e Ф°. Then (a, a) = m' and so {a + r8, a + r8) = m'.
By Proposition 17.16 a + r8 e ФКе; for all a e Ф°, r e Z.
Conversely suppose /3 = '^!i=okiai e ФКе;. Then k. e Z for
i = 0, 1,... , / by Lemma 17.15, in particular k0 2c’c2 = 2 e Z. Now
^a\a
P 2 (2Д 2 7
We have (/3 — ^8, /3 — ^-8^ = m' and so by Proposition 17.11 /3 — ^8 e
Ф°. Let a = /3 — у <5. Then (3 = a+^-8 = a + r8 for some a e Ф°, r e Z.
We now turn from the intermediate roots to the long roots. Suppose
a e Ф°. Then (a, a) = M and (а + л-З, а + ^З) =M for seZ. Let a =
J2 i к<а>. Then k< 2‘’“‘? e Z for i = 1,... , Now
^^i=L i i	i {a,a)	’	’
I	I
a + s8 = 22	+ Z2 saiai-
i=l	i=0
We wish to know for which seZ we have
(tt;, aA
sat —----2 e Z for all i = 0, 1,... , I.
(a, a}
Now (a(i, aA = — = 1, thus (a, a) =4. Hence sa 2,,a2 = Since
c0 = 1 this lies in Z for all i = 0, 1,... , I if and only if s is even. Thus
by Proposition 17.13 a + 2r8 e ФКе j for all a e Ф°, r e Z.
П.2 The roots of an affine Kac-Moody algebra
403
Conversely suppose /3 = J2-=o	£ ФКе1. Then к; € Z for i =
0, 1,in particular k0	e Z. Now
^a\a
! 2 “Д 2 7
satisfies {/3-^3,/3-^3) = M. Also ^a;^^ = ^c;eZ. Thus by
Proposition 17.13 we have /3— уЗеФ°. Let a = /3 — у<5. Then /3 =
a + 2r8 for some a e Ф°, r e Z.
We now consider the short roots of Ф. There is no root of Ф0 of the
same length as the short roots of ФКе. The squared length of the short
roots of ФКе is one half that of the long roots of Ф0. So suppose a e Ф°.
We consider elements of form | (a + \3) where s eZ. We consider which
of these elements lie in Q. Since the long roots of Ф0 have form
±{a;,2a;_i + a;,... ,2^4---h2a;_j + a;}
and 8 = 2a0 + 2a1-\--h2a;_1 + a; we see that ^(a + sS)eQ if and
only if is odd. Thus we consider elements of Q of form | (a + (2r — 1)3)
with r e Z. We have
(|(а + (2г —1)3), |(a+(2r-l)3)) = i(a, a) = m.
By Proposition 17.11 this implies that |(a+(2r— 1)3) еФКе8.
Conversely suppose /3 = 22;=O ktat e ФКе s. Then k. £ Z for i =
0, 1,, I. Then 2/3 — k08 = J2;=1 (2kt — коа/) at. We have (2/3 — k08,
2(3— k08) =4((3,/3) =4.
This is the squared length of the elements of Ф°. We also have
koat (at, at) k0
(2/3,213)
since k0 e Z and c; = 2 for i = 1,... , By Proposition 17.13 we have
2/3 —£0ЗеФр. Let a = 2/3 — k08. Then /3= | (a + k08~). Since /3 e Q,
k0 is odd. Thus
/3= |(a+(2r—1)3) for some a e Ф°, r e Z.
(iii) Finally we suppose that A has type A). The diagram of A is
for i= 1,... , /
0
1
with a0 = 2,	= 1, c0 = 1, Cj = 2. We also have
(ao,“o) = t («i, ai)=4.
404
Kac-Moody algebras of affine type
Now Ф° С Фкед- Let a e Ф0. Then (a + s8, a + s8) = M. We can write
a = k1a1. Then a+ ,«5 = 2га0 +	+ «) ar. We have k1 € Z and
we consider which \ e Z have the property that sat	e Z for i = 0, 1.
Since sa,	and cn = 1 this lies in Z for i = 0, 1 if and only if
s is even. By Proposition 17.13 we deduce that а + 2г<5еФКе1 for all
аеФ0, reZ.
Conversely suppose [3 = koao + k1a1 lies in ФКе1. Then (3-^-8 =
(k.-^a,. We have {/3 - ^8,/3 - ^8} = M. Also k0^ = ^Z
By Proposition 17.11 we have /3 — y<5 e Ф0. Let a = /3 — y<5. Then /3 =
a+ у <5 = a + 2r8 for some a e Ф0, r e Z.
We now consider the short roots. Suppose a e Ф0 and consider the ele-
ment |(а + ,я5) for seZ. Since a = ±a1 and 3 = 2a0 + a1 this element
lies in Q if and only if s is odd. We have
(l(«+(2r-Wl(a+(2r-l)g))=l(a,a) = l.
This is the squared length of the short roots of ФКе s. By Proposition 17.11
i (a + (2r - 1)<5) e ФКе s for all r e Z.
Conversely suppose /3 = коао + к1а1 6®Res. Then
о ко s / ,	коаг \
P----o= k,--------a,.
*	2 V 2 ) 1
We have (2/3 — k08, 2j3 — k08) =4-(/3, /3) =4. This is the squared length
of the roots in Ф0. So by Proposition 17.11 we have 2/3 — к08еФ°.
Let a = 2/3 — k08. Then /3= | (a + k08). Since /3eQ k0 must be odd.
Hence /3 = |(a+ (2r — 1)3) for some a e Ф0, r e Z. This completes the
proof.
□
17.3 The Weyl group of an affine Kac-Moody algebra
Let A be an affine Cartan matrix and W the Weyl group of L(A). Then
W = (sq,, s^. The subgroup IT" = (sy ...,$)) is the Weyl group of
the finite dimensional simple Lie algebra L (A0). In order to investigate
the structure of W we introduce the element 0 = 8 — aoao = J2;=1 а^. This
element в lies in Q° = Q (A0).
Proposition 17.18 (i) If the affine Cartan matrix A is not of type
B\, Cj, F4, C?2 then в is the highest root of Ф0.
(ii) If A is of type B\, CJ, G) then в is the highest short root of Ф0.
17.3 The Weyl group of an affine Kac-Moody algebra
405
Proof We first show that в e Ф0. We have {в, в) = {8 — aoao, 8 — aoao) =
af (a0, a0) = 2a0. First suppose a0 = 1 and a0 is a long root. Then (в, в) =
(a0, a0) = 2. Also
Thus в e ФР by Proposition 17.13
Next suppose a0 = 2. Then (в, в) =4(a0, a0) =4. Thus в has the same
squared length as a long root. Also at	This lies in Z for i = 1,... , /
since c; = 2 for such values of i. Hence в e Ф/ by Proposition 17.13.
Finally suppose a0 = 1 and a0 is a short root. This occurs for the cases
in (ii). Then (в, в} = (a0, a0). Hence в e Ф) by Proposition 17.11.
Thus we have shown в e Ф0 in all cases. We also have
(0, a;) = (8 - aoao, a;) = -a0 (a0, a;)
_	a0^0i (a0’ ao) _ Л — Л
—	2	~ C0-A-0i ~ ^-Oi-
Thus (в, af > 0 for i = 1,... , Hence в e Co, the closure of the fundamental
chamber for Ф0. This implies that 0 is the highest root of Ф0 in the cases in
(i) and the highest short root of Ф0 in the cases in (ii), by Proposition 12.9.
□
Now let se be the reflection corresponding to the root 0. Then se : H° H°
is given by se(h) = h — 0(Ji)he.
Lemma 17.19 The coroot he is given by he = 4- (c — hf).
Proof. Since в = J). , a;ai; we have -44- = £ , at fff) . 2ai,, hence
J	^i=L i i	(0,0) ^i=l i (0,0)
а0 .=1 a0
Now the affine Weyl group W is generated by И'" and s0, so is also
generated by И'" and sose. We consider the action of sose on H.
Proposition 17.20 Let h-.il Then
sose(h) = h + 8(h)he -	h) +1 (he, hf 8(h)) c.
406
Kac-Moody algebras of affine type
Proof.
sose(h) = s0(h- 0(/i)/i0) = h - a0(h)h0 - 0(/i) (he - a0 (he) h0)
= h — a0(/i) (c - aohe) - 0(h)he + 0(h)ao (hg) (c - aohe)
= h+(aoao(h) - 0(/i) - aod(h)ao (hg)) hg + (0(/i)ao (M - “oW) c-
Now (/i0) = ao(i(c-/io)) = -^. Thus
/2	\
sose(h) = h + (aoao(h) + 0(h))hg — —0(/i) + a0(/i) c
\ao	/
= h + 8(h)he-----(0(/i) + 3(/i)) c
ao
= h + 8(h)hg - ((/i0, h) + j (he, he) 8(h)) c
since (he,h) = ^- = -fd(h) and (he, he) =	= т/тр = у-. We
define th : H -+ H by
4 W = h + 8(h)hg - ((hg, h) + i (hg, hg) 8(h)) c.
Thus we have sose = thf). Hence W is generated by И'" and thf).
More generally, for any .re//1 we define tx : HH by
tx(h) = h + 8(h)x— ((x, h) + | (x, x)8(h)) c.	□
Proposition 17.21 (i) txty = tx+y for all x, у e H°.
(ii) uffur1 = tw(x) for all weW°,xe H°.
Proof The linear map tx : PI H is uniquely determined by the properties
tx(h) = h — (x, h)c when<5(/i) = 0
tx(d) = d + aox — ^a0(x, x)c
since 3 (/i;) = 0 and 8(d) = a0. If 8(h) = 0 then
txty(h) = tx(h- (y, h)c) = h- (x, h)c- (y, h)(c- (x, c)c)
= h—(x + y,h)c since (x, c) = 0
= tz+yth')-
17.3 The Weyl group of an affine Kac-Moody algebra
407
Also
V/O = (d + aoy- \a0(y, y)c)
= d + aox-\ao(x,x)c + ao(y- (x,y}c)-\a0(y,y)(c- (x,c}c)
= d + a0(x + y)-a0 (|(х, x) + (x, y) + ^(y, у)) c
= d + a0(x + y)-a0-^(x + y,x + y)c
= tx+y(d).
Thus txty = tx+y for all x,yeH°.
Now let w e W°, and h e H satisfy 8(h) = 0. Then
wtrir' (h) = w (ir 1 (h.) — (x, w 1 (hf) c)
since 8 (»’ 1 (/?)) = (w8)(h) = 8(h) = Q. Thus
wt^uT1 (h) = h - (w(x), h)c = tw(x} (h)
since w(c) = c. Also w(d) = d for all w e IT" and so
wlpr'id) = wtx(d) = w (d + aox — |(x, x)aoc)
= d + aow(x) — |(w(x), w(x))aoc
h(x) (^)-
Hence wtxw ! = tw(xy	□
Let M be the additive subgroup (i.e. lattice) of H° generated by the elements
w (hf) for all w e W°. Let t(M) = {tm ; me M}.
Proposition 17.22 W = t(M)W° where t(M) is normal in W and t(M) П W° =
1. Thus W is a semidirect product of t(M) and W°.
Proof We know that the e W, hence wthe 1 = tw(hy e W for all w e W°. Thus
t(M) is a subgroup of W. Since W is generated by IT" and thf), W is generated
by t(M) and W°. But IT" lies in the normaliser of t(M) by Proposition 17.21
(ii). Thus W = t(M)W°. Finally t(M)nW° = l since t(M) is a free abelian
group whereas W° is finite.	□
The lattice Me. !/ will be important in understanding the affine Weyl
group W. We shall now identify it in each case.
408
Kac-Moody algebras of affine type
Proposition 17.23 (i) If A is an affine Cartan matrix not of types B\, C\,
Fj, C?2 then M = J2;=1 Z/i;.
(ii) If A has type B\, C\, F^, G\ then
m= £ Z/l;+ У) p’Lhi
aj short	a, long
where p is the ratio of the squared lengths of the long and short roots
(p = 3 for G2 and p = 2 in the other cases).
Proof By Proposition 17.18 в is a long root in the cases in (i) and a short
root in the cases in (ii). Thus he is a short coroot in (i) and a long coroot in
(ii). Thus M is generated by all short coroots in (i) and by all long coroots
in (ii). Now it follows from Proposition 8.18 that the set of all short coroots
generates the coroot lattice J).=1 Z/i;. But the set of all long coroots generates
the sublattice with basis ht for ht long (i.e. at short) and pht for ht short (i.e.
a; long). The result follows.	□
We have been considering an action of the affine Weyl group W by linear
transformations of the vector space H of dimension / + 2. However, we now
show that there is a simpler action of W by affine transformations on the
real vector space II) of dimension /. We recall that the group of affine
transformations of a vector space is generated by the group of non-singular
linear transformations and the group of translations.
We first define HK x = {h e HK ; 8(h) = 1}. The space HK p although not
a subspace of /F, is invariant under W. For
5(w(/i)) = (ur15) (h) = 8(h)
since w(8) = 8. Now we have a decomposition
/А = // ф (IRc + IRc/)
into subspaces of dimension / and 2 which are mutually orthogonal.
For (/i;,c)=0 and (/i;,c/)=0 for z=l,...,/. Since <5(/i;) = 0 for i =
1,... , I, 5(c) = 0 and 8(d) = a0 the elements of Нж which lie in HK1 are
those of form
1	1
J2A;/i; + Ac4----d A;eIR,AeIR.
i=l	a0
Now heHK1 implies Н+цсеНХ1 for /zelR. Since w(c) = c for all we
W, W acts on the quotient space IP j/IRc. Also we have a bijective map
Hv , /IRc	7/2
IK, 1/	IK
17.3 The Weyl group of an affine Kac-Moody algebra
409
given by
i	।	i
IRc + Xff + — d A;/i;
i=l	a0	i=l
and this bijection may be used to define an action of W on H^.
Proposition 17.24 The action of W = t(MfW° on is as follows. The
W°-action on H^_ is that previously considered. For meM, he H^_ we have
tm(h) = h + m. Thus tm acts on H^_ as translation by m. Hence W acts on H^_
as a group of affine transformations.
Proof. If w e IT" then w(c) = c and w(d) = d. This implies that the w-action
on Z7g defined above is the usual w-action. If meM, heHK x then tm(/i) =
h + m + p.c for some /zelR. This induces an action of tm on //j given by
tm(h) = h + m. Thus tm acts on Z7g as translation by m.	□
Corollary 17.25 The action of W on Z7g is faithful.
Proof. Suppose tmw, w eW°, acts trivially on Z7g. Then tmw(0) = 0. This
implies m = 0, that is tm=I. Hence weW° acts trivially on Since IT"
acts faithfully on Z7j| this implies w = 1.	□
Corollary 17.26 Sq acts on Z7g as the reflection in the affine hyperplane
L '«<// : 0(h) = l}.
Proof. For h e IF we have
= hffeQ1) = h~ Kh)he + he = h+ (1 - 0(/i)) he.
This is the reflection in Le P	□
For each a e Ф0 and k e Z let La k be the affine hyperplane given by
7 ... P': I a(Ji) = k}.
Thus the generators s0, sr,... , sl of the affine Weyl group W act on //j as
the reflections in the hyperplanes L01,	0,... , L 0 respectively.
We now introduce a collection of affine hyperplanes whose corresponding
affine reflections will lie in W.
Let 53 = {Lk ; ae<t>°,keZ, p divides к if a is a long root and A is
one of B\, C\, F\,(j2}. Here as usual /7 = 2 in the first three cases and p = 3
for &2.
410
Kac-Moody algebras of affine type
Let sak be the reflection in Lak. Then sa k(h) = h + (k — a(hf)ha. For
2	a’k
and h+ (k — a(hf)ha differs from h by a multiple of ha. Thus sa k = tkhasa.
Proposition 17.27 The reflection sak e W for all Lak e £. In fact sak = sa_k3.
Proof The reflection sa_k3 : IP HK is given by
sa-t8 (h)=h-(a- k8) (h)ha_k3.
Thus the restriction of sa_k3 to IP x is given by
= h- (a(lT) -k)ha_k3.
Since 8 eH* corresponds to с e H under our bijection between H and IP we
have ha_kS = ha--^c. Thus the action of sa_kS on H^ffRc is sa_kS(h) =
h — (a(h) — k)ha and the action on III is given by the same formula. Thus
sa_k3 = sak on H°,. Moreover we know from Theorem 17.17 that sa_k3 e W
whenever Lak e £.	□
We note that Lffl, L 0,... , L 0 all lie in £. For by Proposition 17.18 в
is a short root when A has one of the types B\, C\, F\, G\.
Definition The connected components of the set H^_ — Ulo La,k are called
alcoves.
Proposition 17.28 The set
Д {//:// : a;(/i)>0 for i= 1,... , I, 0(/i)cl}
is an alcove.
Proof. We show AC\La k = ф for all LffJfce£. Let	We may
assume a e (Ф°)+- Suppose в is a long root. Then 0 < a(h) < 0(/i) < 1 by
Proposition 12.9 and so h cannot lie in Pnt for к e Z. So suppose в is a short
root. If a is a short root we again have 0 < a(h) < 0(/i) < 1, so h cannot lie
in La k with к e Z. Thus suppose a is a long root. Then a(h) < 0;(/i) where
0l is the highest root of Ф0. Let 0l = J2;=1 Then we have
i
Ql = '^2, bjCti is the highest root of Ф0
i=l
I
0 = '^2,aiai is the highest short root of Ф0.
17.3 The Weyl group of an affine Kac-Moody algebra
411
By considering the coroot of the highest root, or by a case-by-case check,
one may show
ai
pai
if at is long
if a; is short.
In particular bt < pat for all i. Hence
0 < a(h) < djh) < pO(h) < p.
Thus h cannot lie in La k with k e Z divisible by p.
Thus A lies in an alcove. But
q UL01.
This shows that A cannot be properly contained in an alcove, since
L q, ... , Lai 0, Le j lie in £. Thus A is an alcove.	□
Let ?I be the set of alcoves. We show that W acts on ?I. Since W is
generated by ... , sh se it is sufficient to prove the following lemma.
Lemma 17.29 (i) st (Lak) = LSi(a),k for / = 1,... , I.
(ii) se = ^s0(a),k+a(he)- Also if Lak g£ then LSo^a^k+a^h^ G £.
Proof (i) Let // ://. Then hELak if and only if a(li) = k, and this is
equivalent to (^(a)) (st(h)) = k, that is st(h) G LSi(a),k. Thus st (La k) =LSi(a),k.
(ii) se(h) = soth (/i) = s0 (h+hf) = s0(h) + s0 (/i0). Thus a(h) = k if and only
if ЫО CsbW) = *> that is ЫО C?oW + >?o (M) = ^ +“ (M- 11 fol-
lows that Sg (La k) = Ls^a-f^+afhg)-
Now suppose La k e £. Then к is divisible by p if a is a long
root and A e {Bj, CJ, Ff G)}. If we are not in this special case then
^so(a) k+a(he) e & since a (/i0) G Z. So suppose A is one of the above
four possibilities and a is a long root. We know p divides к and must
show p divides a (Jig). Now he = (c — h0). We have a0=l in the
given cases and a(c) = 0, thus a (/i0) = — a (/i0). Let a = J).=1 kiai. Then
a (/i0) = J2;=1 ktat (/i0) = J2;=1 Aoikt. There is precisely one zg{1,...,/}
with Aoi For this i, Aoi = —2 in type C) and Aoi = — 1 in the other cases.
In the latter cases at is a short root. Thus
(at, a J	kt
k> 1	=-^.
(a, a) p
412
Kac-Moody algebras of affine type
This shows that p divides J2;=1 Aoikj, and so p divides a (/i0) in all cases.
Thus LSo(a^k+a(he) e £.	□
Corollary 17.30 If w e W, A' e ?I then w (A') e ?I.
Proof This follows from the definition of alcoves, together with the fact that
the elements of W permute the affine hyperplanes in £.	□
We define L; = La. 0 for i = 1,... , / and L0 = Lei. Thus L0,L1,... ,Lt
are the walls bounding the alcove A and s0, sr,... , s, are the reflections in
Lo, Lx,... , L[ respectively.
Given weW we say that L; separates the alcoves A and w(A) if these
alcoves lie on opposite sides of L;.
Lemma 17.31 L; separates A and w(A) if and only if l(w) = / (s;w)+ 1.
Proof. First suppose w' e W has the property that w'(A) lies on the same side
of L; as A but w'Sj(A) lies on the opposite side of L; to A. Then w'(A), w'Sj(A)
lie on opposite sides of L; so A, Sj(A) lie on opposite sides of w'1 (Lj). This
implies w'1 (Lj) = Lj so Lj = w' (Lj. Hence si = w,SjW,~l and w,Sj = siw'.
Now suppose w e W is such that w(A) is on the opposite side of L; to A.
Let w = st ... st be a reduced expression for w. Then there exists q > 1 such
that Sj ... Sj (A) lies on the same side of L; as A but st ... st (A) lies on the
opposite side of L;. Then we have
as above. Hence
SjW = SjSj ...Sj =Sj ...Sj Sj ...Sj
1	1 li	lr 4	lq-i lq+i	lr
and so / (SjW) < l(w).
If w(A) is on the same side of L; as A then SjW(A) is on the opposite side.
Hence / (Sj  stw) < I (stw), that is / (stw) > l(w).	□
Theorem 17.32 The map w(A) is a bijection between the elements of
the affine Weyl group W and the set A of alcoves.
Proof. Given any alcove A' e Я we can find a sequence of alcoves
A = A1,A2,... , Ar = A'
17.3 The Weyl group of an affine Kac-Moody algebra
413
such that Ai is obtained from A;-1 by reflection in a common wall. Such
reflections lie in W by Proposition 17.27. Hence A' = w(A) for some w e W.
Thus the map w(A) is surjective.
Next suppose w(A) = w'(A). Then w'-1w(A) = A. We show w'-1w= 1. If
this is not so then
w'^1w = siw" with / (»’' 1»’) = / (у,»’' 1 n j + I
for some i. By Lemma 17.31 L; separates A and w'~1 w(A). This is a contra-
diction so w'^w= 1 and w = w'.	□
Theorem 17.33 The closure A of A is a fundamental region for the action
of the affine Weyl group W on H1^, i.e. each W-orbit on intersects A in
exactly one point.
Proof. Each point in lies in the closure A' of some alcove A'. By Theo-
rem 17.32 A' = w(A) for some weW. Thus the W-orbit of the given point
intersects A.
Now suppose x,yeA satisfy y = w(x) for weW. We shall show x = y
by induction on /(w). If /(w) = 0 then w=l so x = y. So suppose /(w)>0.
Then w = S/W' with / (stw) < l(w). By Lemma 17.31 L; separates A and w(A).
Thus АП w(A) c L;. Now у e АП w(A) hence у e L;. Thus sfy) = y. But then
st (y) = w' (x) so y = w'(x). Since / (w') < /(w) we deduce x = y by induction.
□
Remark 17.34 We may also define an action of the affine Weyl group W
on II in a way which is compatible with the bijection II II determined
by the standard invariant form (,) on H. Under this bijection the element
h№ e H corresponds to
For each a e (/7°)* we may define ta : H* H* by
tff(A) = A-|-A(c)a-((A,	+	a)A(c))8.
Then we have sose = l(i/ao)0 on H*. Moreover we have tatp = ta+p and
wt/fw ' = tw^ for w e W°. It follows that we have a semidirect decomposi-
tion W = t (M*) W° where t (M*) is the set of ta for a e M* and M* is the
sublattice of spanned by w ( — в j for all w e W°.
414
Kac-Moody algebras of affine type
The lattice M* is given explicitly as follows.
i
M* = ^at f°r types Ah D[, Ё6,Ё7, Es
1=1
M* = 22 ^ai+ У? P^ai for types B[, Ct, F\, G2
a.j long	a, short
I
M* = 22 Za; for types В\, Cf Ё[, G\
i=l
M* = 22 2^“;+ 52 ^ai for type c;
a, long	a, short
M* = IZttj for type Af
Now the affine Weyl group W acts on the subset
Я*д = {АеЯ* ; A(c) = l}
and this induces an action on the orbit space //i ^IRS. However, there is a
natural bijection between this orbit space and (Я^)*. This defines a W-action
on (flj)*. The IT,-action on (fljj* is just as before, and the remaining
generator Sq of W acts as the reflection in the affine hyperplane
Ч,1/ао = {Ае(як)*; a(m=i/«o}-
The element ta, aeM*, acts on (Hf)* as translation by a, thus W acts on
as a group of affine transformations.
We may also introduce alcove geometry in (Hf) . We define a set £* of
affine hyperplanes in (Hf) as follows.
£* = {/.j	; a e Ф0, k as below}
where L*h k= {Ae (Hf)* ; A(/iff) = A}. The number k runs through the set
given as follows.
For types At, if, Ё6, Ё7, Ё$ keZ.
For types B;, C;, F4, G2
For types Bf C\, if &2
For type C;'	Ae|Z
AeZ
For type Af	kefZ.
к e Z if a is long
к e />Z if a is short.
AeZ.
if a is long
if a is short.
17.3 The Weyl group of an affine Kac-Moody algebra 415
Then the elements of the affine Weyl group W permute the set £* of affine
hyperplanes. The connected components of
U L*
are called the alcoves of IK. The set A* given by
Л={Ле (//',') ; A (/i;) > 0 for i= 1,... ,A(/i0)cl/ao}
is an alcove called the fundamental alcove. The group W acts on the alcoves
and the map w (A*) is a bijective correspondence between elements of
W and alcoves. Moreover the closure A* is a fundamental region for the
W-action on (flj)*.
We omit the proofs of these facts, which are entirely analogous to the
corresponding results for the W-action on H, or may be deduced from these.
18
Realisations of affine Kac-Moody algebras
18.1 Loop algebras and central extensions
Let A0 be an indecomposable Cartan matrix of finite type. We have A0 = (Ay)
for i, j = 1,... , Let L° = L (A0) be the finite dimensional simple Lie algebra
with Cartan matrix A0. We may construct an (/+1) x (/ +1) affine Cartan
matrix A from A0 by adding an additional row and column, labelled by 0, as
follows. Let в = E'=1 atat be the highest root of L° and he = E/=i c;/i; be the
coroot of 0. We then define A by:
A;> = A° if i,je{l,...,/}
A;o = -£a/°	if ie{l,...,/}
7=1
A0J = -£C;A°	if je{l,...,/}
i=l
^oo = 2-
Proposition 18.1 A is an affine Cartan matrix. The type of A is as follows.
Type of A° : A;, Bh Ch Dh E6, E7, Es, F4, G2
of A :Ah Iff (ff Iff E6, Iff Iff Iff G2
Proof. We have
/°\
0
W
where a0 = 1.
416
18.1 Loop algebras and central extensions
417
For '^!j=0Aijaj = Ai0 + Ej=i Atjaj. If z^O this is 0 by definition. If z = 0 we
have
nAojaj=2+nAojaj=2~m2ciA°jaj-
7=0	7=1	i=l 7=1
However, Eki Ej=i ciA°jaj = (Ej=i (ELi ciht) = 0 (M = 2> thus
Ey=o Aojaj =0.
A similar argument shows that
(c0Cj... c;) A = (00... 0) where c0 = 1.
Now A is determined by A0 and the relations
/°\
0
\o/
(c0Ci...c;)A=(0...0).
But the affine Cartan matrix A of type Lo, where Lo is Ah Bh Ch Dh E6, E-j,
Es, F4, G2 gives A0 when row and column 0 are removed, and satisfies the
above two relations, by Proposition 17.18 and Lemma 17.19. Thus our given
matrix A is the affine Cartan matrix of type Lo.	□
Definition An affine Cartan matrix A is of untwisted type if it is one of
A-i, If, C[, D[, Ё6, Ё2, Eg, if, G2.
Since any affine Cartan matrix A of untwisted type can be constructed as
above from a Cartan matrix A0 of finite type by the addition of an extra row
and column, it seems natural to ask whether the affine Kac-Moody algebra
L(A) can be constructed in some way from the finite dimensional simple Lie
algebra L° = L (A0). We shall now describe a method of doing this.
Let C [t, f-1] be the ring of Laurent polynomials E;ez id' f°r it e
finitely many Let
£(L°)=C[f, r1]®^0.
Then £ (/.") may be made into a Lie algebra in a unique way satisfying
[p®v, q®y] = pq® |vv|
418	Realisations of affine Kac-Moody algebras
for p, q e C [t, i-1], x, yeL°. This Lie algebra £ (L'j is called the loop
algebra of L°.
We now wish to construct a 1-dimensional central extension of £ (lP)-
Lemma 18.2 Let L be a Lie algebra over C and L be the set of elements
x + Ac withxeL and A e C. Let к : LxL^> C be a bilinear map satisfying
к(у,х) = -к(х,у) for X, yeL
*([xy], z) + *([yz], x) + k([zx], у) = 0 for x,y,ze L.
(к is called a 2-cocycle on L.) Then the Lie multiplication
[xT Ac, y + /zc] = [xy] + k(x, y)c
makes L into a Lie algebra.
Proof This is elementary. The two relations satisfied by к give anticommu-
tativity and the Jacobi identity on L.	□
We note that L is a 1-dimensional central extension of L, i.e. there is a
surjective homomorphism
в : L^L
given by 0 (x + A c) = x, such that dim (ker ff) = 1 and ker 0 lies in the centre of L.
We apply this idea to construct a 1-dimensional central extension of £ (L'j
by taking a 2-cocycle on £ (L°). Let (,) be the invariant bilinear form on L°
satisfying (he, hf) = 2. Since an invariant bilinear form is determined up to a
scalar multiple on L°, this condition determines it uniquely. In fact this form
on L° is the restriction to L° of the standard invariant form on L = L(A),
since for the standard form we have (в, в) = 2 as in Proposition 17.18, hence
I 2в 2в \_ 4
(ff^hf)-^ Off (0, (y/~ (y ~2'
We next define a bilinear form
by {p®x, q® y)t = pq{x, y). We define the residue function
Res : C [t, / '] C
by Res (E^t;)=^p
18.1 Loop algebras and central extensions
419
Lemma 18.3 The function к: £ (L°) x £ (L°) C defined by
к(а, b) = Res
da \
( —, b)
\ dt /
is a 2-cocycle on £ (/.").
Proof. To show that к is anticommutative it is sufficient to verify that
к (t'®x, P ®y) = —к (P ®y, t'®x).
Now
к (t'®x, f ®y) = Res (it! ' ® x, P ®y}t
= Res (z7!+-'-1(x, y))
_ i(x, y) if i + j = 0
0 if i+j^O.
The anticommutativity follows.
We also need
к ([t!®x, t^®y], tk®z) + к (1У ®y, Л®г], t'®x)
+ к ([?®z, /'0x], P ®y) =0.
Now we have
к ([t!® x, P ®y], P ®z) = к (t'+j® [xy], tk®z)
= Res((z + j')tl+j-1® [xy], tk®z\
= Res ((/ +	([xy], z})
_ G +	ifz + j + £ = O
0	ifz + j + £^O.
If i + j + k^£O the required property is clear. If i + j + k = O the required
sum is
-£([xy], z) - z([yz], x) -j([zx], y)
=	z) - z([xy], z) - j([vy], z)
= 0
since the form is symmetric and invariant.	□
420
Realisations of affine Kac-Moody algebras
We may therefore construct the 1-dimensional central extension £ (L°) of
£ (L°) given by
£(L°) = £(L°)®Cc
whose Lie multiplication is given by
[a + Xc, b + p^c\ = [а, Ь]0 + к(а, K)c
where a,b eH (L°) and [a, b]0 is the Lie product of a, b in £ (L°).
We next wish to adjoin to £ (L°) an element d which acts on £ (A") as a
derivation.
Lemma 18.4 The map Д: £ (L°)	£ (L°) given by Д (a + Ac) = t^ for a e
£ (L°), A e C, is a derivation.
Proof Since [a + Ac, b + p.c\ = [a, b]0 + к(а, b)c we must show that
t+;[a,b]0 = t—,b + fic
dt	at
v db~
a + Xc, t—
dt
that is
/ dZA
4-K a, t— c
\ dt /
that is к (t^, Ь) + к (a, t^y) =0. It is sufficient to prove this when a = p® x,
b = q®y with p, q e C [t, / 1 ], x, у e L°. Then
(da	\	/	dZA	(dp	\	(	dq
t—, b +k a, t— = к t— ®x,q®y +k p®x, t— ®y
dt	)	\	dt)	\ dt	)	\	dt
(dq	\	(	dp
p®x, t— ®у I — к a®y, t— ®x
dt	)	\	dt
I dp	da	\	Idq	dp	\
= Res( — t—®y) —Rest — ®y, t—®x)
\ dt	dt	/t	\dt	dt	/
/d/7da \ (dpdq A
= Res t----(x,y) I — Res t-(x, у)
\ dt dt '	\ dt dt '
We now define £ (L°) by
£(L0) = £(L°)®Crf
18.2 Realisations of untwisted affine Kac-Moody algebras
421
and make £ (L°) into a Lie algebra by defining the Lie product as
[a + Xd, b + p.d\ = [a, b] + АД(Ь) —/хД(а).
This is clearly skew-symmetric, and the Jacobi identity follows from the fact
that Д is a derivation. In particular we have
[(/'Or) + Xc + fid, (ff ®y) + X'c + ffd]
= (tt+i® |vv|) + p.j (ff ®y}-p,'i ^tt®x)+8i_ji(x, y)c
for x, yeL0, A, gL, A', ff e C.
18.2 Realisations of untwisted affine Kac-Moody algebras
We aim to show that £ (L°) is isomorphic to the affine Kac-Moody algebra
L(A). Thus L(A) can be constructed from L" = L(A") by the following
procedure. First form the loop algebra £ (L°). Then form the 1-dimensional
central extension £ (L°). Finally extend this Lie algebra by a derivation to
give £ (L°).
Theorem 18.5 Let L° = L (A0) be a finite dimensional simple Lie algebra
and let A be the untwisted affine Cartan matrix obtained from A0 as in
Section 18.1. Then L(A) is isomorphic to £ (L°).
Proof. We shall define elements e0, ex,... , e;; f0, ff,... ,	; h0, ff,... , /i;
in £ (L°) with the aim of using Proposition 14.15 to show that our Lie algebra
is isomorphic to L(A).
Let Eb ... , E; ; Eb ... , E; ; Iff,... , Hl be corresponding generators of
L°. We define
e; = l®E;, /; = 1®Е;, f=\®If
for i = 1,... , /. Then [eff] = ht for each i. We must also define e0, f0, h0 e
£(E°).
We consider the root spaces L0, L°_0 where 0 is the highest root of L°.
We have dim L0 = dimL°0 = 1, and the map L0xL°0^C given by the
invariant bilinear form {,) on L° defined in Section 18.1 is non-degenerate.
Let <y° be the automorphism of L° satisfying <y° (E;) = — E;, <y° (E;) = — E;.
Then <y°(L0)=L°0. We claim it is possible to choose elements EoeL0,
Eo e L°_0 such that <y° (Eo) = —Eo and (Eo, Eo) = 1. First choose any non-zero
element /(' e L0 and let E'o = —<y° (Ff). Let (Eq, E'o) = Then we have d 0.
422
Realisations of affine Kac-Moody algebras
Now let F0 = AFq and F0 = AFq for AeC with A^O. Then we have Fo =
—<y° (Fo) and (Fo, Fo) = A2£. By a suitable choice of AeC we can ensure
that A2£= 1.
We now define e0 = t® Eo and f0 = F1 ®F0. Let H° be the subspace of L°
spanned by hx,... , ht and
H={\®H°)®Cc®Cd.
We define h0 e H by
hQ = (1 ® (—Hff) + c.
Then we have
[eo/o]= P® Eq, t ® Fo] = (1 ® [F0F0]) + {Eo, Ff)c.
But
[F0F0] = {Ео, Fq}H'_№ = H'_e = H_e = -He
by Corollary 16.5, since {в, 0} = 2. Thus
Wo] = (l®(-0 + c = /i0-
We also define elements a0, ax,... , al e El*. We have elements
a1,... , al e (// ") * and we extend these to El* by saying that at (c) = at {d) = 0
for i = 1,... , We also define OeEI* similarly, saying that 0(c) = 0{d) = 0.
Let 8 eEI* be the element defined by
<5(x) = 0 forxe№, 8(c) = 0, 8(rf) = l.
We then define a0 eEl* by a0= —0 + 8.
We now show that {El, П, IP) is a realisation of A where П =
{a0, cij,... , aj and ГГ = {h0,	... , /ij. П is linearly independent since
cij,... , al are linearly independent and a0{d) 0, at{d) = 0 for i = 1,... ,
IP is linearly independent since hx, .. . ,ht are linearly independent and h0
involves c whereas ht does not for i = 1,... ,
We show that (/i;) = Atj for i, j e {0, 1,... , /}. This is clear if i 0.
Also for i 0 we have
i	i
“o (^i) = -0 (hf) + 8 {h^ = -0{h^ = -^ajaj (M = -EAijaj = Aio-
j=i	j=i
Similarly for j 0 we have
aj (M = “y {~he + c) = -Uj {hf) = - CiUj = - 52	= A0j.
18.2 Realisations of untwisted affine Kac-Moody algebras
423
Finally a0 (/i0) = (—0 + 3) (—he + с) = в (/i0) = 2. Thus (H, П, IIV) is a real-
isation of A.
We next verify the relations
[eifi\ = hi
| де J = afxfe	for xeH
|x/j | = —a; (x)f	for x E H
where i = 0, 1,... , I. These relations certainly hold when i 0 and j 0. We
have shown above that [eo/o] = ^o- For i/0 we have
[cJo] = [1 ®	/ 1 0 / <>] = r1 ® [F;F0] = 0
since Fo e L0 and 0 is the highest root of L°. Similarly for j 0 we have
[eofj] =	1 ®Fj] = t® [E0Fj] =0.
Now let x = x0 + Ac + pud E H where x0 e H° and A,/t eC. Then
a0 (x) = - 0(x) + 3 (x) = - в (x0) + /i
since 0(c) = 0(d) = 0, 3 (x0) = 3(c) = 0, 3(d) = 1. Also
[xe0] = [x0 + Ac + fid, t ® Fo] = ft ® [x0F0]) + fi ft ® Fo)
= —0 (x0) (t® Fo)+ /z (t® Fo)
= М-Фо-
Similarly one shows [x/0] = — a0(x)/0. Thus the required relations are all
satisfied.
We show next that e0, eb ... , e;, f0, fx,... , and H generate £ (Lo).
Let M be the subalgebra of £ (/.") generated by this subset. Since
Fx,... , F;, Fx,... , F; generate L°, ex,..., e;,generate 1®L°.
Thus Ig/.'c.W.
Let 1° = {x eL° ; t®x e.Wj. Since e0 = t®E0 we have Fo ef° so /" ^0.
Also if x e 1°, у e L° then [xy] e 1° since
t®[xy] = [t®x, l®y] eM.
Thus 1° is a non-zero ideal of L°. Since L° is simple we have I° = L°. Thus
t ®x E M for all x E L°. We may now use the relation
[t®x, /* ' 0y] = //0|.vy|
424
Realisations of affine Kac-Moody algebras
to deduce by induction on k that tk ® x e M for all x e L° and all k > 0. In an
analogous way, starting with ./[, = / '0/\, we can show t~k®xEM for all
x e L° and all к > 0. Now
£ (L°) = H + (1 ® L°) + £ (tk ® L°) + £ (tk ® L°)
k>0	k<0
hence M = £ (L°).
It remains to show that £ (L°) has no non-zero ideal J with Jr\H = 0. Let
/. = £(/.'') = //© £
МЖ)
summed over i e Z, a e (#0)* with (z, а) ф (0, 0). We claim that this is the
weight space decomposition of L with respect to ll. For let h e H, x e (L°) .
Then h = h0 + Xc + fid with h0 e H°, A, /х e C. Thus
[/z, /' 0x] = [h0 + Xc + /id, /' 0x] = (t! ® |/z,rv|) + fii (t! ®x)
= (a (/z0) + /ii)
= (a(h) + i8(h)) (tl®x}
= (a + i&)(h) (t!®x)
since a(h) = a (/z0) , 8(h) = fi. Thus t'®x is a weight vector with weight
a + i8. Thus we have
L = Lq® La+iS
WW,0)
where L0 = H and La+i3 = f® (L°)a.
Let J be a non-zero ideal of L with Jh\H=O. By Lemma 14.12 we have
J = (Lonj)© £	П./).
МЖ»)
Since L0C\J=O we have 1.,,-Г'\.1 f() for some (a, i). Let t'®xEJ for
xe (L°)a with x^O. Then there exists у e (A1) with (x, y) ^0. Thus
[/' 0 x, / ' ® y] = [xy] + i (x, у) c
lies in J Г\ H, and hence
[xy] + z(x, y)c = 0.
Since [xy] eH° and (x, y) #0 we must have i = 0. But this implies [xy] = 0,
whereas we have
[xy]=(x, y)/i'ff#0
18.2 Realisations of untwisted affine Kac-Moody algebras
425
by Corollary 16.5. This gives us the required contradiction. Thus jr\II = O
implies J = 0.
We have now verified all the conditions of Proposition 14.15. We may
therefore deduce that £ (L°) is isomorphic to If A).	□
We can deduce from this theorem the multiplicities of the imaginary roots
of I .(A). These multiplicities were not obtained in Chapter 17. We recall from
Theorem 16.27 (ii) that the imaginary roots of IfA) have form k8 where
k e Z and 0.
Corollary 18.6 Let A be an indecomposable affine GCM of untwisted type.
Then the multiplicity of each imaginary root kS, kf(), is / = rank A.
Proof. We use the realisation IfA) = £ (/."). The weight space decomposition
of £ (/.") shows that the root space for the root k8 is tk ®H°. The multiplicity
of k8 is the dimension of this root space, which is dim 77° = 1.	□
We now make some comments on the isomorphism between IfA) and
£ (L°) which we have obtained. Firstly the standard invariant form on IfA)
maps under this isomorphism to the form on £ (L°) given as follows:
f'0vj = O	ifjjL— f for x, yeL0
(tl®x, rl®y) = (x, y)
(tl®x, c) = 0
(t'®x, d] = Q
{c, c) =0
(d, d}=0
{c, d) = 1.
For it is readily checked that the form defined in this way on £ (L°) is invari-
ant. Moreover we also see that the above form on the subspace (I
СсфСй? of £ (L°) agrees with the standard invariant form on the subspace II
of L(A) under our isomorphism between these subspaces. However, the proof
of Theorem 16.2 shows that a symmetric invariant bilinear form on L(A) is
uniquely determined by its restriction to H. Thus the above form on £ (L°)
corresponds to the standard invariant form on L(A).
We also observe that the element c e £ (L°) corresponds to the canonical
central element in L(A) under the isomorphism. For we have
h^ = (f®+c in£(L°)
426
Realisations of affine Kac-Moody algebras
and hence cihi = c in £ (L'f It follows that the image of c under the
isomorphism is the canonical central element of L(A).
Also, since (w(d) = 1, afdf = 0 for i=l,...,l the element </e£(L°)
corresponds under the isomorphism to an analogous scaling element d
for L(A).
18.3 Some graph automorphisms of affine algebras
We now wish to find realisations of the remaining affine Kac-Moody alge-
bras L(A) where A has type B\, C\,F4, G^A^ or C;'. These are called the
twisted affine Kac-Moody algebras. We shall obtain realisations for them as
fixed point subalgebras of certain automorphisms of untwisted Kac-Moody
algebras. Before doing so, however, we consider the graph automorphisms of
the untwisted algebras which fix the vertex 0 and therefore arise from graph
automorphisms of the corresponding finite dimensional simple Lie algebras.
The graph automorphisms of the finite dimensional simple Lie algebras were
considered in Section 9.5. We recall from Theorem 9.19 that if cr is a graph
automorphism of the finite dimensional simple Lie algebra IfA) then L(A)a
is isomorphic to the simple Lie algebra L (A1) where A1 is obtained from A
as follows.
A	•	A2k	Л-2к-1	^/1	^4	^6
Order of	a	:	2	2	2	3	2
A1	:	Bk	Ck	Bk	G2	F4
We shall now prove an analogous result to Theorem 9.19 for affine algebras.
Theorem 18.7 Let A be an affine Cartan matrix of type A2k_r, Dk+1, If or E6
and let <j be a graph automorphism of the Kac-Moody algebra L(A) which
fixes vertex 0 and has order 2, 2, 3, 2 respectively. Let A0 be the corresponding
finite Cartan matrix and A1 be the finite Cartan matrix associated with A0 as
above. Let A1 be the untwisted affine Cartan matrix obtained from A1. Then
L(A)<7 is isomorphic to L (A1).
Specifically we have
В (^2£-l) — В (C.)
L(Dk+iy = L(Bk)
L(D4y = L(G2)
L(E6)a = L(F4)
18.3 Some graph automorphisms of affine algebras
427
Proof. The algebra L (A0) has a Cartan decomposition
/.(Л11) = //''© £ L°a.
аеФ0
Thus L(A) has a corresponding decomposition
L(A) = H° ф Cc ф Cd ф	(tk ® H°) + (tk ® L°a).
fc/0	k,a
Similarly we have decompositions
аеФ1
/. (л1) =7/1 ®Cc®c^®52(/'0//') ®	)•
fc/0	к,a
Consider the graph automorphism cr : L(A)	L(A). We have
a(H°)=H°, a(tk®H°) = tk®H°,	a (tk ®L°) = tk ®L°a(a),
a(c) = c, cr(d) = d.
Hence
ffiAY = (Н°У ф Cc ф Cd ф £ (tk ® (H°) 3 ф £ (? ® (L°s) 3
k^O	k,S
where 5 is an equivalence class of roots in Ф0 and L°s = L°a (cf. Propo-
sition 9.18). Now the isomorphism L (Л"),г^ L (A1) of Theorem 9.19 gives
rise to bijective maps
(L°sy I. ', where a e Ф1 corresponds to 5
tk ® (н°У tk ® h1
tk®{L°sy ^tk®L\.
These maps, together with c c.d d. determine a bijective map
ф : L(A)<7 L (A1). We wish to show this map is an isomorphism.
Under this bijection ф the subalgebra (Н°У ф Cc ф Cd maps to H1 ф Cc ф
Cd and both are abelian. The action of (Н°У on tk® (Н°У and tk® (1УУ
agrees with the action of H1 on tk ®HX and tk ®L1a respectively. The element
c lies in the centre on both sides. The action of d on tk ® (Н°У and tk ® УУУ
(i.e. multiplication by k) agrees with the action of d on tk ® H1 and tk ® I.\t
respectively. Thus it is sufficient to compare the multiplication of the root
spaces on both sides. These multiplications are trivially preserved by ф unless
428	Realisations of affine Kac-Moody algebras
we take two roots whose sum is 0. So suppose x, у e (H°Y and xx, yx are the
corresponding elements of H1. We have
[t* ®x, t~k®y\=k{x, y)oc
where {, )0 is the standard invariant form on L (A0) and {, the standard
invariant form on L (A1).
Also if x e у G (L°_SY and xk, yr are the corresponding elements of
L*, L1_a then we have
[?®x,	= [xy] +k(x, y)oc
|7 ® xx, Гк ® У1 ] = [xx yx ] + к (xj, yx) x c.
Thus to show that ф is an isomorphism it is sufficient to show that the
isomorphism L (A0)"7 L (A1) preserves the standard invariant form, that is
if x —> xb у yx then (x, y)0 = (xj, У1)р Since any two symmetric invariant
bilinear forms on a finite dimensional simple Lie algebra are proportional it
is sufficient to check this for just one non-zero value. To do this we choose
a 1-element orbit (z) of <j on {1,... , /}. Such a 1-element orbit exists in all
the cases being considered. Then we have an element /z;eL (A")7 mapping
to an element ht e L (A1). We have
(/i;,/i;)0 = 2 and (hi,hl\=2di = 2ai/ci.
A glance at the values of a;, c; for L (A1) for i coming from 1-element orbits
of <j shows that a; = c; in these cases, so dt = 1. Hence (/z;, /z;)0 = (/z;,
and it follows that the isomorphism L (A0)"7 L (A1) preserves the standard
invariant forms. This completes the proof.	□
Note The reader will have noticed that the case L (Л2к) has not been included
in this theorem. The above proof breaks down in this case because cr has no
1-element orbit on {1,... , /}. The diagrams of A0, A1 are as shown.
1	2	к
О-----О-----О------О-----О
1	2	к
О-----О-------О----о > о
о-----о-----о------о-----о
2к 2к-\	£+1
А0	А1
In fact, if we take the сг-orbit (к, k + 1) on {1,... , 2k}, then the isomorphism
L (Af L (A1) of Theorem 9.19 maps
2(hk + hk+1)eL(A°y to /zteL(A1).
18.4 Realisations of twisted affine algebras
429
We have
(fyt’ =2’	^lt+l)o = 2’ (fyt’ ^k+l)o = — 1-
Thus
(2 (Jh + hk+i) ’2(hk + fyt+i))o= 8-
On the other hand
(hk, hk)1 = 2— = 2dk=4.
ck
Thus
(2 (fyt + fyt+i) ’ 2(^ + ^+i))o^ (hk, hk}v
and so the isomorphism between L (Alk)a and L (Bk) does not preserve the
standard invariant form. It does not therefore lead to an isomorphism between
L(A1кУ and L (lif in the manner described in Theorem 18.7.
18.4 Realisations of twisted affine algebras
In order to obtain realisations of the twisted Kac-Moody algebras IfA)
where A has types B\, C\, Ff Af (f we must consider the fixed point
subalgebras of so-called twisted graph automorphisms.
Let L° = L (A0) be a finite dimensional simple Lie algebra and cr : L° L°
be a graph automorphism of L°. Then cr extends to a graph automorphism of
£ (L°) = £ (L°) ф Cc ф Cd given by
a (tl ® x) = tl ® cr(x)	for x e L°
a(c) = c, cr(d) = d.
Suppose <j has order m and let b = e2"l/iH. Then we may define an automor-
phism г of £ (L°) by
г (tl ® x) = 8~ltl ® <t(x)	for xeL0
7(c) = c, т(сГ) = d.
7 is called a twisted graph automorphism of £ (/. "), and also has order m.
In fact m = 2 or 3 in the cases which can arise. We shall consider the fixed
point subalgebras £ (L°)T. In order to do so we first obtain more information
about the action of <j on L°.
430
Realisations of affine Kac-Moody algebras
Proposition 18.8 (i) Let L° be a simple Lie algebra of type A2l_3,Dl+1 or
E6 and cr be a graph automorphism of L° of order 2. Let (L°)_1 be the
eigenspace of a on L° with eigenvalue —1. Then
L°=(L°)<7©(L°)_1
and (L°Yi is an irreducible (L°Y-module.
(ii) Let L° have type D4 and cr be a graph automorphism of L° of order 3.
Let (L0)^, (^°)ю2 be ,l]e eigenspaces of a with eigenvalues a>, a>2 where
a> = e2m/3. Then

and	(^°)c o2 are both irreducible (L°Y-modules.
Proof. Let x e (E'Y- у e (L°)e where e is an eigenvalue of a. Then
cr[xy] = [cr(x), cr(y)] = £[xy].
Thus |a v] e (L°Y and so (L°)e is an (L°)<7-module.
Suppose first that cr has order 2. Let (a, /3) be a 2-element orbit of cr on
Ф0 and Ea,EpEL° be root vectors such that cr (Ea) = Ep. Then Ea—EpE
(L°Yi and the weight spaces of (L°)_1 are spanned by such elements for all
2-element orbits (a, /3). The roots а, /3 с (№)* have the same restriction to
((H°Y)* and	is the weight of Ea — Ep. The highest weight of the
(L°) ^-module (L°) x is obtained from the highest 2-element orbit (a, /3). Let
us choose the labellings
for the Dynkin diagrams of A2l_r, Dl+l, E6. Then the highest 2-element
orbits are
(«! + a24-----ha2l_2,	a23-----ha2l_2 + a2l_f) for A2;_j
+	----ha;_i + a;, ai + a2H-----H-i + “i+i) for Dl+1
(a1+2a2 + 2a3 + a4 + a5 + a6, a3 + 2a2 + a3 + a4 + a5 + 2a6) for E6.
18.4 Realisations of twisted affine algebras	431
In these three cases the subalgebra (L°Y has type ChBl or F4 respectively
by Theorem 9.19. We choose the labellings
12	Z-l I 1	2	l-l I 1	2	3	4
o----О---О---О----<1 < <1 О----О----О----О---1 > > О О---О > < 1---о
for these Dynkin diagrams. Thus the highest weights for the -modules
(/-") ! are
aj + 2a2 4---h 2a;_1 + al for C;
“1 + а2“1----'rai-i + ai f°r Bl
a1+2a2 + 3a3 + 2a4 for F4.
Using the equation a; = 'ffj , we see that these highest weights are <y2
for C;, и, for Bl and a>4 for F4.
Now dim ( =dim Л" — dim (L°Y and this is 2/2 — I — 1 = Qz) — 1 for
C;, 2/ + 1 for B[, and 26 for F4. However, we also have
z ч /2/\
dim L (w2) =11 — 1 in C;
dim Л (оз, ) = 2/+I in Л’,.
dim L (w4) = 26 in F4
by Weyl’s dimension formula. Thus in each case dim (L ) ( is the dimension
of the irreducible module with the appropriate highest weight. Thus (L) (
is isomorphic to this irreducible (Lo) -module.
Now suppose cr has order 3. Then L° has type D4. Let (a, /3, y) be a
3-element orbit of <j on Ф0 and Ea, Ep, Ey be root vectors such that cr (Ea) =
Ep, <j (Ep) = E . Then we have
+ ы2Ер + a>Ey e (L°) м
Ea + a>Ep + ы2Еу e (L°)
where a> = e2"l/3, and the weight spaces of the (L°) ^-modules (L°) and (L°)ffl2
are spanned by such vectors for all 3-element orbits. We choose the labelling
<O 2
О 3
О 4
for the Dynkin diagram of D4. The highest 3-element orbit of cr on Ф0 is then
(a1 + a2 + a3, a3 + a2 + a4, a1 + a3 + a4).
432
Realisations of affine Kac-Moody algebras
The subalgebra (L°Y has type G2, for which we take the labelling
в
1	2
Thus the highest weights of the G2-modules (L°) and (L°)ffl2 are both
a1+2a2. Now in G2 we have a1+2a2 = <y2 and dim A (w;) = 7. However,
we also have
dim (L°) ш = dim (L°) щ2 = - (dim L° — dim (L°)<7) = 7.
Thus the G2-modules (L°) and (L°)ffl2 are both irreducible and isomorphic
to L (<u2).	□
Theorem 18.9 Let L° be a simple Lie algebra of type Dl+1, E6 or i)4
and let cr be a graph automorphism of L° of order 2, 2, 2, 3 respectively. Let
t be the corresponding twisted graph automorphism of £ (/."). Then the fixed
point subalgebra £ (L°) t is isomorphic to a twisted affine Kac-Moody algebra.
Explicitly we have
й{А21_У) =L{B^
й(рмУ = L^
& (E6)T = L(F‘)
£(Z>4)t = L(G‘).
Proof. The method of proof is broadly similar to that of Theorem 18.5 giving
the realisations of the untwisted affine Kac-Moody algebras. The basic idea
is to show that the given subalgebra of т-invariant elements satisfies the
conditions of Proposition 14.15, and is therefore isomorphic to the appropriate
twisted affine Kac-Moody algebra.
We have
£ (L°) = (Л ® L°) Ф Cc ф Cd.
If cr has order 2 we have
£ (L°)T = £ (r“ ® (L°f) ф £ (t2k+1 ® (L°)J ф Cc ф Cd
whereas if cr has order 3
£ (L°y = E (t3k ® (r°D ф E y3k+1 ® (П) ® E У3к+2 ® (L°yy
ke%	ke%	ke%
ф Cc Ф Cc?.
18.4 Realisations of twisted affine algebras
433
Let Ex,... , Ek, Fr,... , Fk, Ff,... , Hk be standard generators of L°. We wish
to define analogous elements
e0, еъ ... , eh f0,	, fh h0,h1,...,hl	in£(L°)T.
We pick a representative d" e Ф0 of the highest 2- or 3-element сг-orbit on Ф0.
Specifically we have
+	----ha2;_2 inA^j
0o = a1 + a24-----ha; inF>;+1
0° = a1+2a2 + 2a3 + a4 + a5 + a6 inF6.
The elements e;, f, ht are then chosen as follows.
Type Л2/ ,
12	l-l
О-----О-----О-----о-------о------о-.
1
о-----о-----о-----о-------о------сг
21-1	21-2	/+1
<h = 1 ® (Ei +E2;_i) ,... , e;_i = 1 ® (£ц+М , е; = 1 ®F;
/1 = 1 ® (Fi +F2/-i) , • • • , fi-i = 1 ® (^-1+^+1) , fi = 1 ®Ft
hr = 1 ® (If +H2i_f) , • • • , /i/_i = 1 ® (Hi-i +Hl+1) , hl=\®Hl
/i0
f0 = t ^^Ego-E^go^
1	2
О--------О--------О-
Type Dl+1
о
Si — 1 ®ЕХ,... , е[_х — 1 ®E[_r, е; — 1 ® (F; + F;+1)
f1 = l®F1,..., ft_, = \®Fl_l, fl = \® (F; + F;+1)
/ii = 1 ® Hk,... , /i;_! = 1 ® Ht_k, /i; = 1 ® (77; + 77;+1)
e0 = t®(Ffp-Fa^ ,
f0 = t ^(^Ego-E^go^
/i0
-Ego -H^^go^ +2c.
= 1® (
434
Realisations of affine Kac-Moody algebras
Type E6
3	4
6	5
ех = \®Еъ e2 = l®E2, e3 = 1 ® (F3 + E6) , e4=l®(E4 + E5)
=	f2=l®F2, /3 = 1®(F3+F6), /4=1®(F4 + F5)
/^ = 1®/^, h2 = l®H2, h3 = l®(H3 + H6), h4 = 1 ® (7/4 + Hf)
e0 = t® (f0O — F^go)) , fo = t 1 ® ^F0O — ^(go))
h0 = 1 ®	—Ha(0>>^ + 2c.
Type D4
e1 = l®E1, e2 = 1 ® (F2 ф F3 ф F4)
/1 = 1®^, /2= 1 ® (F2 + F3 + F4)
hx = 1 ® //j,	/z 2 — 1 ® (^2 -I- нз ^4)
Ф = / ® (f0O + w:Fr(goj + a>Fa2(0O))
fo = t 1 ® (f0O + ыЕа^ + ti>2Ea2(ffl)^
h0= 1® (^—EIff> —Ha^ —	+ 3c.
Let Яс£(Ь°) be given by /7 = (I 0/7")фСсфСФ We define maps
a0, ax,... , U[ : EK -+ C. We have roots at e El* and such roots in the same
сг-orbit have the same restriction to EK. We define ax,... , al e (EK)* to be
the restrictions of the corresponding roots in El*. We also define a0 e (EK)*
by a0= -в° + 8.
Let £ (L°) =L(A) as in Theorem 18.5. We wish to show that £ (L°)T =
L (A') where A' is the affine Cartan matrix of type given below:
A :	A2l_ 1	Dl+1	F6	D4
A':	B]	q	K4	G\
18.4 Realisations of twisted affine algebras
435
We shall first show that (1ГГ. П, IIV) is a realisation of A' where
IT = {h0,	П = {«,,, aj C (H<y .
We know from Theorem 9.19 that = for i, je ,/}. This
/ x / matrix is non-singular and so hx,... , /i; and a1,... , a; are linearly
independent. The element h0 involves c whereas hx,... , /i; do not, thus
h0, hx,... , hl are linearly independent. We have afd)= 0 f°r * = 1, •• • , I
but a0(</)^0, thus a0, ar,... , al are linearly independent. We must show
that
«o(^) = ^o z=l,...J
ao (M = 2.
We recall the integers a;, c; associated with the affine Cartan matrix A', which
are as follows.
0’
1, • • • у Cli
(5 / t)—n-О--О---О > n
1	2	2	2	2	2	1
(>	< О-----------О---------О----------О---------(>	> 1)
1	2	3	2	1
О---------<> > <>---------------о----------о
2	4	3	2	1
О-----О ) О-------о-----о
1	2	1
ч—о---о
3	2	1
О > О--------О
We then note that the following significant equations hold in each of the cases
being considered:
i
E aiai = 8
1=0
I
'^2,cihi = me where m is the order of <j.
i=0
We then have
i	i
“o (^) = - E ajaj (M = - E A'ijaj = А'юао = A'io
7=1	7=1
436
Realisations of affine Kac-Moody algebras
for i = 1,... , / since a0 = 1 in the cases being considered. Similarly
ai (.ho) = - E ciaj (hi) = - E ciA'ij = coA'oj = A'Oj
i=l	i=l
for j = 1,... , I. Also
(hf = (-0° + <5) (1 ® ( 7/0o - Ha^ - • • •) + me) = 0° (Я0о) = 2.
We also note that A' is an (/+1) x (/ + 1) matrix of rank / and that dim//'7 =
1 + 2. Thus we have shown that (Ha, П, IIV) is a realisation of A'.
We next verify the relations necessary for applying Proposition 14.15. We
first show that
\h^=A^ Ы] = -А'^
for i, j e {0, 1,... , /}. These are known for i, j e {1,... , /} by Theorem 9.19.
So we must check
[h0f}] = -A'0jf}
[hieo] = A'ioeo, [Vo] = -^o/o	i=l,...,l
[Vo] = 2e0, [Wo] = —2/0.
For j = 1,... , / we have
[ M] = - E ci [hiej] = (- E ciA'ij) ej=A'Ojej
and similarly for i = 1,... , I we have	= ~Aojfj- Also
[Vo] = t® (F0o +£ 1/?сг(-0о-) 4----)]
= t® — 0° (h) (F0o + £ 1^7Cr(0O)H-)
= -0° (V eo = (-Eaj“j<A)) eo= (-EAbay) eo
\ 7=1	/	\ 7=1	/
= Ао^о-
Similarly we have	= — A'iofo. We also have
[Vo] = [1 ® (~V ~--------------), t®(FeP + s 1Fo.(e0)-\----)]
= 2t®(F0o+£ 1F(7(-0o^ 4---)=2e0.
Similarly we have [/i0/0] = — 2f0.
18.4 Realisations of twisted affine algebras
4Ъ1
Finally we have relations
[ce;] = 0 = a;(c)e;	i=l,...,l
[c/;] =0 = — a;(c)/;	i=l,...,l
[det] =0 = ai(d)ei	i=l,...,l
[<]=0 = -a;(7)/;	i=l,...,l
[de0] = e0 = a0(d)e0
[^/o] = -/o = -“o(^)/o
Since FT = (1 ® ф Cc ф Cd we have now verified all relations neces-
sary for applying Proposition 14.15.
We next show that the elements e0, eb ... , e;, f0, fx,... , together with
FT generate £ (L°Y  know that eb ... , eh fv, ... , fl generate {L°Y by
Theorem 9.19. Since
£ (b°)7=E E® E'E E (/2/ l ® (L°) J ф Cc ф Cd
when a has order 2 and
£ (L°)T = E Yk ® (L°Y) ф £ (/3il ® (F°) J ф £ (t“+2 ® (F°) J фСсфЫ
keZ	keZ	keZ
when a has order 3, it is sufficient to show that the subspaces (t2k® (F0)"7)
for kf() and t2k+1 ® (F°) lie in the subalgebra generated by the above
elements when cr has order 2, and the subspaces	) for k^£Q,
(t3k+10 (F°) ) , (t3k+2 ® (C'E) lie in this subalgebra when a has order 3.
Let M be the subalgebra of £'(/.")7 generated by e0, eb ... , eh
f0,fx,... , fh FT. Suppose first that <j has order 2. We have
e0 = t®	eM and F0O e (F°)_1 •
Now if x e (L°Y, у e (C). then
[1 ® x, t®y] = t® [xy] e t® (F°)_1 .
Thus the elements у e (C) , for which t® у e M form an (Lo)^-submodule of
(L'Ep However, (L°)_ x is an irreducible (L°)<7-module by Proposition 18.8.
Thus t® (C0) ^ lies in M. Now we can find elements x, у e (F0)^ such that
|л у | 0. Then [t ® x, t ®y\ = t2 ® [xy] is a non-zero element of M. However,
438
Realisations of affine Kac-Moody algebras
the set of ze (L0)17 such that t2®zeM is an ideal of (Е°У and (L°)a is a
simple Lie algebra. Thus t2® (Lo) lies in M. The relations
[t2 ® x, t2k ® у] = t2k2 ® [xy]	x, у e (L°)a
\t2® x, t2k+1 ®y] = /2/30|xy| xe (L°y, у e (A") !
can then be used to show by induction on к that t2k ® (Е°У С M and
t2k} 0 (L°) c.W when k>Q. Starting with /0 instead of e0 will similarly
show this when к < 0. Thus M = £(L°)T .
Now suppose that <j has order 3. We have
eo = I® (Fgo + w2F00o) +
F0O Tu/F^go) + 60^2(00) e (A°)M •
An argument similar to the above shows that t® (L°) с M. Now there exist
elements x, у e (L°) with [xy] ^0. Then
[t®x, t ® y] = t2 ® [xy] et2® (F°)m2 .
We can then show as above that t2®(L°) 2C.W. There exist elements
x e (L°)m, у e (L°)m2 with [xy] #0. Then
[t®x, t2®y] = t3 ® [xy] e t3® (L°y.
We then see as above that t3 ® (Ь°У С M. Induction on к can then be used
to see that the subspaces
t3k ®(L°y, t3k+1 ® (L°) a , t3k+2 ® (L°) m2
all lie in M when к > 0. A similar result is obtained when к < 0 starting
with /о instead of e0. Thus М = й(к°у. Hence in all cases the elements
e0,	• • • , eh f0,fi,--- ,fi and H7 generate £(L°)T.
Finally we must show that £ (L°) has no non-zero ideal J with JC\H7 = 0.
To see this we decompose £ (L°)T into root spaces with respect to H7. We
first suppose cr has order 2. For each 1-element orbit (a) of cr on Ф0 we
choose Ea&L°a. We showed in the proof of Theorem 9.19 that <j (Ea) = Ea.
For each 2-element orbit (a, /3) of cr on Ф0 we choose Ea e L°a, E^e If such
that a(Ea) = Ep. Then £(L°)T is the direct sum of H7 and the following
weight spaces.
18.4 Realisations of twisted affine algebras
439
t2k ® (Н°У with weight 2k8
t2k+1 ® (H0)^	with weight (2k +1)5
t2k®CEa with weight a + 2k8 where (a) is a 1-element orbit
t2k ® C (Ea + Ep)	with weight a + 2k8 where (a, (3) is a 2-element orbit
tlk' ® C (Ea — Ep) with weight a+ (2^4-1)5 where (a, /3) is a
2-element orbit.
By Lemma 14.12 the ideal J is the direct sum of its intersections with these
weight spaces. Thus J has non-zero intersection with one of these weight
spaces. Taking a non-zero element x in this weight space and in J we can
find an element у in the negative weight space such that [xy] is a non-zero
element of HT, and this contradicts J Г\1Г = 0.
When <j has order 3 a similar argument can be applied. This time the
weight spaces are
t3k ® (H°)a with weight 3k8
t3k+1 ® (Н°)ш with weight (3k +1)5
t3k+2 ® (H°)щ2	with weight (3k + 2)5
t3k®CEa with weight a + 3£5 where (a) is a 1-element orbit
t3k®C (Ea + Ep+ Ey) with weight a + 3k8 where(a,/3,y) is a
3-element orbit
t3k+1 ® C (Ea + ofEp + a>Ey) with weight a+(3£+l)5 where (a, /3, y)
is a 3-element orbit
t3k+2 ® C (Ea + o)Ep + ofEy)	with weight a+ (3£ + 2)5 where (a, /3, y)
is a 3-element orbit.
Any non-zero ideal J with J Г\1Г = () must have non- zero intersection with
one of these weight spaces. We can then multiply it by an element of the
negative weight space to give a non-zero element of HT, and this contradicts
JC\ET = 0. Thus we deduce that J =0.
We have now verified all the conditions of Proposition 14.15 and can
conclude that £ (E°)T is isomorphic to L (A')-	О
As a corollary we obtain the multiplicities of the imaginary roots of the
affine Kac-Moody algebras of types B\, C\, Ff G^.
440
Realisations of affine Kac-Moody algebras
Corollary 18.10 The multiplicities of the imaginary roots are as follows.
Type B\. The roots 2k8 (k^O) have multiplicity I and the roots (2£+l)<5
have multiplicity I — 1.
Type CJ The roots 2k8 (k + 0) have multiplicity I and the roots (2A'+I)3
have multiplicity 1.
Type F4 The roots 2k8 (k^O) have multiplicity 4 and the roots (2£+l)<5
have multiplicity 2.
Type G[ The roots 3k8 (kfO) have multiplicity 2 and the roots (3£+l)3
and (3k + 2)8 have multiplicity 1.
Proof For types	Theorem 18.9 shows that the multiplicity of
2k8 (kfb)) is dim ( ll ' f. which is equal to /. The multiplicity of (2k +1)3
is dim (№) which is / — 1, 1,2 in the three cases respectively.
For type Gz the multiplicity of 3k8 (k + +) is dim(№)<7 = 2, and the
multiplicities of (3k + 1)3 and (3k+ 2)8 are dim (/7")^ = dim (//")^; = 1- О
We now make some comments on the isomorphism between L (A') and
£ (L°y which we have obtained. The standard invariant form on L (A') does
not map under this isomorphism to the restriction of the standard invariant
form on £ (L°). For let (z) be a 1-element сг-orbit on {1,... , /}. Such an orbit
exists in each of the cases. Then hteL (A') corresponds to 1 G £ (L°)T.
We have
(1®Я;, \®Hi}=2
(ht, hj)' = 2af ct.
However, we may check that c; = ma; for all 1-element сг-orbits, hence
(ht, ht)' = 2/m
where m is the order of cr. Thus
(1 ® //,, l®Hj) = m (ht, hi)'.
Also the isomorphism does not map the element c G £ (L°)T to the canonical
central element c+L (A')- We noted in the proof of Theorem 18.9 that
1
c,7z; = me in£(L°)T
i=0
and hence our isomorphism maps me to c'.
However, the scaling element d^fl (L°)T maps to a scaling element d' G
L(A'). For we have a0 = — в° + 8 and so a0(d) = 3(d) = 1. Also a;(d) = 0
for i = 1,... , /.
18.4 Realisations of twisted affine algebras
441
We also see that, if x, у e £ (L°)T map to x',y'eL (A'), then
(x, у) = т(х',уУ
where (,) and {,)' are the standard invariant forms. This is true for x', y' e
L (A'f since any two symmetric invariant forms are proportional on L (A'f.
However, it is also true when y = d,y' = d' since
(/i;,d) = 0 for/ = 1,... ,/
(c, d) = 1
(d, d}=0
Thus it is true for all x', y' eL (A')-	О
We now wish to obtain realisations of the remaining twisted Kac-Moody
algebras of type C;' for l>2 and A[. These will be obtained as fixed point
subalgebras of the untwisted Kac-Moody algebra of type A2l under the twisted
graph automorphism r. We begin by recalling from Theorem 9.19 that the
fixed point subalgebra of the finite dimensional Lie algebra L (A2;) under its
graph automorphism <j is given by
L(A2;f = L(B;).
12	Z-l I
О----------О----------О---------О----------<1 > <>
В/
In order to show that L (A2;)t = L (C,) if I > 2 and L (A2)t = L (A'J we shall
compare the diagrams
o------о------о-
442
Realisations of affine Kac-Moody algebras
We note that the numbering of the vertices of the graph for C;' is not the
same as the numbering previously used when C;' was constructed from C;
by adding an extra vertex labelled 0. Here we are starting from the finite
dimensional simple Lie algebra rather than C;. In Theorem 17.17 (d) and
(e) we obtained the roots of C;' and A\ in terms of those of C;. For our present
purpose we require these roots in terms of those of If.
We consider the diagram of C;' labelled as follows.
<> > <>-----О------О------О-----О-----О------<j > о
0	12	1-2	1-1 I
and let Л’,. be the subdiagram obtained by omitting vertex 0 and Cl be the
subdiagram obtained by omitting vertex /.We recall that
<5 = a0 + 2a1 4--h 2a;_1 + 2a;.
The following lemma will be useful by relating the roots of If and C;.
Lemma 18.11 (i) Each long positive root of Cl involves a0. Each short
positive root of If involves ctj. There is a bijective correspondence a++ /3
between long positive roots of Cf and short positive roots of If satisfying
a + 2(3 = 8.
(ii)	There is a bijective correspondence a++ /3 between short positive roots of
Cj and long positive roots of If. If a, (3 do not involve a0, al respectively
this correspondence is the identity map. If a involves a0 and /3 involves
al the correspondence is given by a + (3 = 8.
Proof. This follows immediately from expressing the roots of and C; in
terms of the fundamental roots ax,... , al and a0, aT,... , respectively.
□
Example The above bijection between Ф+ (C3) and Ф+ (B3) is as given
below.
0	12	3
<1 ) O------<> ) t)
Ф+ (C3)	Ф+ (Дз)
“О “1 + “2 + “3
а0 + 2а1	«г + ^з
18.4 Realisations of twisted affine algebras
443
aQ + 2.ax+2.a2
“i
“2
«, +«2
аз
“i
“2
“o + “i
“o + “i + “2
a0 + 2a1 + a2
«, +«2
«j + 2a,2 + 2a'3
а,1 + а,2 + 2а,з
a,2 + 2a,3
By using Lemma 18.11 together with Theorem 17.17 we may express the
real roots of C;' in terms of the roots of B;.
Proposition 18.12 (i) The real roots of L (C;'), / >2, are
ФКе8 = {o' + rS ; аеФ°, reZ}
ФКе1 = {a + rd ; аеФ°,ге%]
ФКе1 = {2a+(2r+l)<5 ; аеФ>е2}
where ФКе s, ФКеФКс2 are the short, intermediate and long roots respec-
tively, and Ф°, Ф^1 are the short and long roots of If.
(ii) The real roots of L (A'J are
ФКеs = {« + rS ; а,еФ°,ге2)
ФКе1 = {2a+(2r+l)<5 ; аеФ°,ге2}
where Ф° is the root system of type Ax obtained from the short funda-
mental root of A).
Proof, (i) We know from Theorem 17.17 that
фке.з = {|(а + (2г-1)3) ; « e Ф/(Q), r e Z
фке,1 = {« + r8 ; а е Ф8° (С;), г е Z}
фке,1 = {« + 2г<5 ; а е Фр (С;), г е Z}.
We make use of the bijection a off — a о — /3 of Lemma 18.11 where
a e Ф+ (C;), /3 e Ф+ (B;). For each a e Ф0 (C;) we choose the corresponding
444
Realisations of affine Kac-Moody algebras
/3 e Ф0 (/>’,). First suppose a e Ф° (C;). The corresponding /3 e Ф° (/>’,) is given
by a+ 2/3 = 3 if a is positive and a+ 2/3 = —8 if a is negative. Thus
|(a+(2r-l)3) =
—/3 + rS if a is positive
—/3 + (r — 1)3 if a is negative
—2/3+(2r+l)3
—2/3+ (2r — 1)3
if a is positive
if a is negative
This gives the required formulae for ФКе s and ФКе j as r runs through Z. Next
consider ФКе P If a e Ф° (C;) does not involve a0 we have /3 = a e Ф° (B;). If
a e Ф° (C;) does involve a0 the corresponding /3 e Ф)1 (/>’,) satisfies a + /3 = 3
if a is positive and a + /3 = —8 if a is negative. Then
a + r8 = /3 + r8 in the first case
a + r8 =
—/3+ (r+1)3
—/3+ (r — 1)3
if a is positive
if a is negative
in the second case. This gives the required formula for ФКе; as r runs
through Z.
(ii) In type A( the argument is exactly the same except that Ф° = Ф° has type
Aj and Ф° is empty. Thus ФКе; is empty in this case.	□
We shall next prove the analogue of Proposition 18.8 in our present case.
Proposition 18.13 Let L° be the simple Lie algebra of type A2l and cr be
its graph automorphism of order 2. Let (L°}_x be the eigenspace of cr on
L° with eigenvalue —1. Then L° = (L°} ®(L°) r The eigenspace (L0)^ is
an irreducible (/.") -module. The algebra (/.") is isomorphic to L (B;) and
(L0)^ is isomorphic to its irreducible module L (T.af).
Proof. It is clear that (L°)_1 is an (L°) ^-module and that L° = (/."p® (Tci)_1.
We have
dim L° = dim L (A2;) = 2/ (2/ + 2)
dim (L°y = dim L (B;) = /(2/ + 1).
Thus dim (/.") ( =dim Л" — dim (Л") г =/(2/ + 3). Let в = а14---ha2; be
the highest root of A2;. Then a corresponding root vector Ee lies in (C) (
and gives the highest weight of (L0)^. It gives rise to the weight 2a1d-
+2a; of L (B;). By considering the Cartan matrix of we see that a1 -I---
+al = a>1 and so the highest weight of the L(B;)-module (T") ( is 2w,.
18.4 Realisations of twisted affine algebras
445
Weyl’s dimension formula shows that dim/. (2W| ) =/(2/ +3). Since this is
also the dimension of (/-"), we see that (/-"), is an irreducible L(F?Z)-
module isomorphic to L (2<m>,).	□
By using this result we can prove the analogue of Theorem 18.9 in the
A2l case.
Theorem 18.14 Let L° be a simple Lie algebra of type A2l with I > 2 and
cr be its graph automorphism of order 2. Let т be the corresponding twisted
graph automorphism о/£ (E°). Then the fixed point subalgebra £(L°)T is
isomorphic to L (Cz).
When I = 1 the fixed point subalgebra is isomorphic to L (A'J.
Proof The general idea of the proof is like that of Theorems 18.5 and 18.9.
We aim to obtain the result by applying Proposition 14.15.
We first suppose / > 2. We have
£ (L°) = (tk ® L°) ф Cc ® Cd
2 (L°)T = £ (t2k ®(L°y) + Z (t2k+1 ® (L°) J ф Cc ф Cd.
Let Eb... , E2l, Fx, ... , F2l, Ef,... , H2l be standard generators of L°. We
wish to define analogous elements e0,	, eh f0,fx,... , fh h0, hr,... ,ht
in £ (L°y. These are chosen as follows.
ez = 1 ® (Ez+E2Z) ,	e;_j = 1 ® (E;_j+E;+2) ,	ez = 1 ® ^/2 (Ez + Ez+1)
/1 = 1®(E1+E2Z),	/H = l®(FH+F1+2), /z=1®V2(Ez + Ez+1)
/,, = 1®^+^),	^ = 1®^+^),	//,„=102
e0 = f®Ee, f0 = t~1®Ee, h0 = — (I0//J + C
where в is the highest root of L°.
Let FFg£(E0) be given by H = (l®H°)®Cc®Ci We define maps
a0, a1,... , al: Ha C. For i = 1,... , / these are the restrictions of the roots
o';: H C. For i = 0 we define a0 = —в + 8.
Let I lv =	G FF'r and ll={a'„ ur,... ,sz}ci7/")'. We
show that (FFC П, IP) is a realisation of the Cartan matrix A' of type Cz.
We know from Theorem 9.19 that a.j (hf) = AC for i, j e {1,... , /}. This
/ x / matrix is the Cartan matrix of type Bz. In particular it is non-singular.
Thus hr,... , hj and ar,... , al are linearly independent. Now h0 involves
c whereas hx,... , /iz do not, thus h0,h2,... , /iz are linearly independent.
446
Realisations of affine Kac-Moody algebras
Also we have a0 (d) 0 and a; (</) = 0 for i = 1,... , / thus a0, ar,... , al are
linearly independent. We must show in addition
«,(//.,) = Л',	j=l,...,l
“o(M = V	i=l,...,l
ao (M = 2.
We recall that the integers a;, c; associated with the affine Cartan matrix
A' are
a0,	c0,cl,...,cl
<> ) <>--o---o-----и > <>	<1 > -о----о---<> у о
122222222221
In particular we have a0 = 1, c0 = 2. (The change of labelling explains the fact
that c0 is not 1, as it usually is.) We note that
У) a;a; = <5
i=0
У) C;/l; = 2c.
i=0
We then have
«о(V =	(hi) = -I2A'ijaj=A'ioao = A'io
7=1	7=1
1 1 1 1 1
“у (M = Eciaj (hi) = ECiA'ij = 2C°A'0j = A'0j
a0 (M = (-0 + g) ((1 ®—Hf) + с) = в (Hf) = 2.
We observe that A' is an (/ + 1) x (/ + 1) matrix of rank / and that dim Ha =
1 + 2. Thus we have shown that (Ha, П, IIV) is a realisation of A'.
We next verify the relations
1^]=А^ Ы] = ~А^
for i, j e {0, 1,... , /}. We know this already for i, j e {1,... , /} by Theo-
rem 9.19. Thus we must verify
[ Vj] = AOjej	[hofj ] = ~AOjfj
[hi eo] = Аюео	[Vo] = Vo/o
[^oeo] = 2£o	[^o/o] = —2/0.
18.4 Realisations of twisted affine algebras
447
Now
1 1	/ 1 1	\
[M]=-zEci[M=	)ej=A'ojej
i=i	\ i=i	/
and similarly = ~Aojfj- Also
[Vo] = [ht, t®Fe] = t®(-0(hi)Fe) = -0(hi) e0
= \~I2ajaj(hi') eo = I -	e0=A'i0e0-
\ 7=1	/	\ 7=1	/
Similarly we have [/i;/0] = ~Аю/о- also have
[Vo] = [- (1 ®He) + c, t®Fe\ = -t®[HeFe\
= 2t®Fe = 2e0
and similarly [/i0/0] = — 2f0. Finally we have
[ce;] = a;(c)e; = 0 i = 0, 1,...,/
[с/;] = -а;(с)Л = 0	/ = 0,1,...,/
[de;] = ai(d')ei = 0	/=1,...,/
[dfj\ =—afd2)fi = Q i=l,...,l
[de0] = a0(dfe0 = e0
[Vo] = -“o(^)/o = -/o-
Since HT = (1 ® (H°Y} фСсфСй? we have verified all relations necessary
for the application of Proposition 14.15.
We next show that the elements e0, ex,... , e;, f0, fx,... , together with
HT generate £ (L°)T  By Theorem 9.19 ex,... , e^f^ ... , fl generate (f.’Y 
Since £(L0)T = £tez(f“®(L0)<7)e£tez(f»+1®(L0)_i)eCc©Crf it is
sufficient to show that the subspaces
(t2k®(L0Y) for к ^0 and (/2/l ® (L°)_1)
lie in the subalgebra M generated by e0, ex,... , e;, /0, /x,... , Нт.
Now e0 = t ®Fe lies in M and Fe e (T0) ^ If x e (L°y, у e (/- "), then
[1 ® x, t®y] = t® |vv] e t® (T0)^ •
Thus the elements ye for which t®y eM form an (L°) ^-submodule
of (L0)^. This submodule contains Fe so is non-zero. Since (T ") , is an
448
Realisations of affine Kac-Moody algebras
irreducible (L°)<7-module by Proposition 18.13 this submodule is the whole
of (L0)-!. Thus t® (L0)^ lies in M.
Now we can find elements x, у e with [xy] ^0. (For example, x =
Fai —F ,y = Eai+ ^) Thus [t®x, /0y| = /20|xy| is a non-zero element
of M. However, the set of ze (T")r for which t2®zEM is an ideal of (C)r
since
[t2®z, l®w] = t2®[zw] for we(L°y.
Since [xy] e (L°y this is a non-zero ideal of (L°)a and since (L°)a is a simple
Lie algebra it is the whole of (L°)a. Thus t2® (F f lies in M.
Now the relations
[t1 ® x, t2k ® y] = t2k+2 ® [xy],	x, у e (L°)a
[t2®x, /2/l ®y] = t2k2® [xy], xe (L°)a, у e (A ") !
can be used to show by induction on к that t2k ® (L°)a с M and
t2M1 ® (L°)_1 CM when Л >0.
Starting with /0 instead of e0 will similarly show this for к < 0. Thus
M = £(L°)T as required.
Finally we must show that £ (L°)T has no non-zero ideal J with J FFT = 0.
To see this we decompose £ (L°)T into weight spaces with respect to H7.
Any non-zero ideal J with JF\H7 = 0 must have non-zero intersection with
one of these weight spaces, by Lemma 14.12. Let x be a non-zero element in
such an intersection. Then there exists у in the weight space corresponding
to the negative of this weight such that [xy] ^0, by Corollary 16.5. But then
[xy] e J П FT and so J C\FT O, a contradiction. Hence J = 0.
We have now verified all the hypotheses of the recognition theorem Propo-
sition 14.15 and so can conclude that £ (T°)T is isomorphic to L (C[).
We now consider the case 1=1. This time the graphs are
and the Cartan matrix A' of A[ is
/ 2 -1\ 0
A'=\-4 2/1.
0	1
18.4 Realisations of twisted affine algebras
449
The elements e0,	/0, /j, h0, h2 of £ (L°)T are
el = l®s/2{El+E2), fl = l®s/2{Fl+Ff), hl=l®2{Hl+H2)
e0 = t®Fe, f0 = r1®Ee, h0=-(l®He) + c
where в = а1 + а2 is the highest root of A2. We have
n = {/i0, M H = {«,,,«l}
where a0 = —в + 8 and ar e (7/'r) is the restriction of u.-JI'. The integers
a0, ax, c0, cx for are
a0=l,	ax=2, c0 = 2, c1 = 1.
We have
aoao + a1a1 = 8
coho + c1h1 = 2c.
We can then check that (1ГГ. П, IIV) is a realisation of A'. We also check the
relations
[/i0e0] = 2e0,	|V| = -T [^e0] = -4e0, [h1e1] = 2e1
[h0f0] = —2f0,	[Wi]=/n M=4/0,	[h1f1] = -2f1.
The facts that FT, e0, ex, f0, fr generate £ (L°)T and that £(L°)T has no
non-zero ideal J with J П1Г = 0 are proved just as before. Thus applying
Proposition 14.15 shows that £ (T°)T is isomorphic to L (Aj).	□
We shall describe explicitly the weight space decomposition of £(L°)T
with respect to FT. We recall from Proposition 9.18 that there is a bijective
correspondence between roots of (E0)"7 = L(Bl~) and equivalence classes of
roots of L° = L (A2;). Each equivalence class has 2 or 3 elements. Equivalence
classes with 2 elements have form (a, /3) where <т(а) = /3, <r([E) = a and a + /3
is not a root. Equivalence classes with 3 elements have form (a,/3, a + /3)
where cr(a) = /3, <r([E) = a and <т(а + /3) = a + /3. Equivalence classes with
2 elements correspond to long roots of and equivalence classes with 3
elements correspond to short roots of B;. For each 2-element equivalence
class we can choose root vectors Ea,Ep with <j (Ef) = Ep. For each 3-element
equivalence class we choose root vectors Ea, Ep, Ea+p with <j (Ea) = Ep and
[EaEp]=Ea+p. Then
o’ (Ea+p) = a [EaEj = [E^eJ = -E„,:
thus Ea+jS e (E°)_1.
450
Realisations of affine Kac-Moody algebras
The Lie algebra £ (L°)T is the direct sum of IГ and the following weight
spaces.
t2k ® (Е[°У with weight 2k8
t2k®C(Ea + Ep)
t2k+1 ®C (Ea—E^
t2k0C(Ea + Ep)
t2k+x®C(Ea-Ep)
with weight (2k + 1)3
with weight a + 2k8 for each 2 element equivalence
class (a, /3)
with weight a + (2k + 1)3 for each 2 element
equivalence class(a, /3)
with weight a + 2k8 for each 3 element equivalence
class (a, /3, a + /3)
with weight a + (2k + 1)3 for each 3 element
equivalence class(a, /3, a + /3)
with weight 2a + (2k + 1)3 for each 3 element equivalence
class(a, /3, a + /3).
The weights listed above correspond to the roots of L (C() as described in
Proposition 18.12.
In the case / = 1 the weight spaces of L (A2) are
with weight 2k8
t2k+1®CEa+/3
t2k®C(Hr+Hf)
t2k+1®C(H1—H1)
t2k®C(E1+E2)
t2k+1®C(E1-E2)
t2k+1®CEai+a2
t2k®C(F1+F2)
t2k+1
®C(Fi-F2)
t2k+1

with weight(2£ + 1)3
with weight a1+'lk8
with weight ar + (2k + 1)3
with weight 2a r + (2k + 1)3
with weight — ar + 2k8
with weight — ar + (2k +1)3
with weight —2tt| + (2k+ 1)3.
Corollary 18.15 (i) The multiplicities of the imaginary roots k8 of are
equal to I.
(ii) The multiplicities of the imaginary roots k8 of A\ are equal to 1.
Proof, (i) The multiplicity of 2k8 is dim (ll ' f. which is equal to /. The
multiplicity of (2k+ 1)3 is dim (№) which is also equal to /.
(ii) The same applies to when I = 1.	□
18.4 Realisations of twisted affine algebras
451
We note that the isomorphism between L (A') and £ (A2;)T does not map
the standard invariant form (,)' on L (A') to the restriction of the standard
invariant form {,) on £ (A2;). For hl e L (A') corresponds to 1 ® 2 (Hl + 77;+1)
in £ (A2;). We have
(h^y = 2—=4
ci
(1 ® 2 (Ht + Я;+1) , 1 ® 2 (Ht + Я;+1)) = 4 (Ht + Hl+1, Ht +	= 8.
Thus the form is not preserved by the isomorphism.
Also the canonical central element c e £ (A2;)T does not map to the canoni-
cal central element o' eL (A')- Since we showed that J2;=o c;/i; = 2c it follows
that 2c corresponds to c' under our isomorphism.
Comments on notation
An alternative notation is sometimes given to the affine Kac-Moody algebras
of twisted type, based on the results of this chapter. The twisted affine
algebra can be specified by the type of the untwisted affine algebra from
which it is obtained, together with the order of the automorphism of which
it is the fixed point subalgebra. This is the notation used by Kac in his book
Infinite Dimensional Lie Algebras. The alternative notation in each case is
shown below.
~В\	2A21-i
c\	2Dl+1
П	2E6
3b4
C'i	2~a21
A'	2A2
/>3
/>2
/>2
19
Some representations of symmetrisable
Kac-Moody algebras
19.1 The category О of L(A)-modules
We now turn to the representation theory of Kac-Moody algebras. We shall
not consider arbitrary representations, but restrict attention to those in the
category О introduced by Bernstein, Gelfand and Gelfand. Let
I.(A) = N ®H®N
be a Kac-Moody algebra and V be an L(A)-module. We say that V is an
object in the category 0 if the following conditions are satisfied:
(i)	У =	where VA = {и e V ; xu = A(x)u for all x e H}
(ii)	dim VA is finite for each Л eH*
(iii)	there exists a finite set Ap ... , XseH* such that each A with VA ф О
satisfies A < A, for some i e {1,... , \}.
The morphisms in category 0 are the homomorphisms of L(A)-modules.
Thus each module in О is a direct sum of its weight spaces and these
weight spaces are finite dimensional. Moreover all the weights are bounded
above by finitely many elements of H*.
We now give some examples of modules in category 0. For each XeH*
we may define the Verma module M(A) with highest weight A. This is
defined in a manner analogous to that in which we defined Verma modules for
finite dimensional Lie algebras in Section 10.1. Let 1I(L(A)) be the universal
enveloping algebra of /.(A) and KA be the left ideal of II(/.(A)J generated
by N and x — А (л) for all xeH. Thus
Ka = U(L(A))V+£1I(L(A))(x-A(x)).
x®H
Then M(A) = 1I(L(A))/Ka is an L(A)-module called the Verma module with
highest weight A. Let mA eM(A) be defined by mA = 1 +KA. Then, just as in
Theorem 10.6, we see that each element of M(A) is uniquely expressible in
452
19.1 The category 0 of L(A)-modules
453
the form zzmA for some и e 1I(A ). Also, as in Theorem 10.7, we have
M(A)= ф.Ш),
fO if and only i f /х < Л
dimM(A)M = ^(A-^).
This shows that M(A) e 0. The finite set of weights giving an upper bound
for all weights can be taken in this case to have just one element A.
Lemma 19.1 (i) If V e О and U is a submodule of V then U e О and V/U eO.
(ii) If Vj, V2 e О then V1®V2e(9 and V10V2e 0.
Proof (i) We have V=®AefftVA. The argument of Theorem 10.9 shows
that If = U П V2 and U = фл//- UK. It follows that U e О. Moreover we have
(V/U)2 = VJU2 and V/U=Q,H.(V/U)2. It follows that V/U tO.
(ii) We have (Vj ® V2)A = (VX)A ® (V2)A and V, ® V2 = фЛеЯ. (Vj ® V2)A. It
follows that V1 ф V e О.
Now consider Vj ® V2. We have
^=$(4,, V2= ф (v2)Aj
A^tf*	A2eZ7*
thus
V1®V2= Ф ((V1)A1®(V2)AJ.
Ai,A2
Now (Vj)Ai ® (V2)Aj C (Vj ® V2)Ai+A2. Hence Vj ® V2 = фл //. (Vj ® V2)A
where
©®v2)A= £ ((V1)A1®(V2)A2).
Ai,A2
Aj +A2=A
Now there exist e H*, i = 1,... , ,$j, such that Aj -< for some i. Also
there exist Т]. e H*, j = 1,... , s2 such that A2 -< for some j. Thus A =
Aj + A2 -< & + for some pair (z, j). We have
(^ + t77)-A = (^-Aj) + (t7;-A2).
The expressions (£, + 17J — A, — Aj, — A2 are all non-negative integral
combinations of the fundamental roots. Thus for given z, j the (£; + — A
has only finitely many such decompositions. It follows that for each
A with (Vj®V2)at^O there exist only finitely many pairs Aj,A2 with
Aj + A2 = A, (Vj)Aj # O, (V2)A2 7^ O. It follows from this that Vj 0 V2 e 0.
454 Some representations of symmetrisable Kac-Moody algebras
Now each L(A)-module V eO admits a character ch V. We recall from
Section 12.1 that ch V is the function from H* to Z defined by
(ch V)(A) = dim VA.
We also recall from Section 12.1 the definition of the ring St of functions
from H* to Z. A function f : H* -+ Z lies in St if there exists a finite set
Aj,... ,XseH* such that
Supp/c5(AJU • • • U5(As)
where 5(A) = Supp(chM(A)). It follows from the definition of category 0
that ch V e St for all V e 0.
In Proposition 12.4 we obtained a formula for the character of a Verma
module for a finite dimensional semisimple Lie algebra. We now generalise
this result to Verma modules for Kac-Moody algebras. We recall that the
function ел : H* Z was defined by ел(А) = 1 and eA(/x) = 0 if А p.. The
characteristic functions eA lie in St and any function f e St can be written in
the form
f= E /(А)ел
Лей*
where the sum may be infinite.
Proposition 19.2 Let M(A) be a Verma module for the Kac-Moody algebra
L(A). Then
ch M(A) = ——E--------
П
аеФ+
where ma is the multiplicity of a.
Proof We use the fact that the map и um, is a bijection between II (/V)
and M(A). This bijection maps the weight space II (/V ) to the weight
space M(A)a_m.
For each аеФ+ we have dim (N) ,, = ma. Let e_ai, l<i<ma, be a
basis of (N) We choose an order on these basis elements for all a, i. We
then obtain a PBW-basis of II (/V ) consisting of all products
ПП^
a z=l
19.1 The category 0 of L(A)-modules
455
with na i e Z and na i > 0. Thus the weight space II (V ) has a basis con-
sisting of the above elements which satisfy
(m„ \
E na,i I “ = M-
2=1	/
This shows that the character of II (/V) is
сЬП(Г)= П (1 + е_а + е2_а + ---Г
аеФ+
since the number of times e appears on the right-hand side is the number
of sets (иа of non-negative integers such that
(m„ \
E n«,i I
2 = 1	/
Hence the character of M(A) is
chM(A) = eA П (l + e_a + et + -
аеФ+
Now the element 1 + e_a + e2_a -I-e St has inverse 1 — e_a e St. Thus we
have
п (ГЛ .)-
аеФ+
Of course in the special case when /.(A) is finite dimensional this formula
reduces to that obtained in Proposition 12.4. In the general case there are two
differences - the roots need not have multiplicity 1 and the product over the
positive roots can be an infinite product.
Now the Verma module M(A) for /.(A) has a unique maximal submodule
J(A), just as in the proof of Theorem 10.9. We define
L(A) = M(A)/J(A).
Then L(A) is an irreducible L(A)-module in the category 0.
Proposition 19.3 The modules /.(A) for XeH* are the only irreducible
modules in category 0.
Proof Let V be an irreducible L (A) -module with V e 0. The definition of 0
shows that V has a maximal weight A under the partial ordering -<. Let uA e V
be a weight vector with weight A. Then xvK = 0 for xeN and xvK = A(x)ua
for xeH.
456 Some representations of symmetrisable Kac-Moody algebras
We may define a map from the Verma module M(A) into V as follows.
Each element of M(A) has a unique expression of form итл for и e II (/V).
Let в : M(A) V be defined by в (umf) = uvK for и e II (N). It may then
be shown, just as in the proof of Proposition 10.13, that в is a homomorphism
of lI(L(A))-modules. The image of в is a submodule of V containing vA,
so is the whole of V since V is irreducible. Thus the kernel of в is a
maximal submodule of M(A), so must be ./(A). Thus V is isomorphic to
M(A)/J(A) = L(A).	□
Now in Theorem 12.16 we showed that each Verma module M(A) for a
finite dimensional semisimple Lie algebra L has a finite composition series.
The proof of this result made extensive use of the fact that L is finite dimen-
sional, and the result does not carry over to Verma modules for Kac-Moody
algebras /.(A). Lor example it can be shown that the Verma module M(0)
has no irreducible submodule when /.(A) is infinite dimensional.
We would nevertheless like to define the multiplicity [V : L(A)] of the
irreducible module L(A) in the module V e 0. If V had a finite composition
series [V : L(A)] would be the number of composition factors isomorphic
to L(A) in a given composition series, and this would be independent of
the choice of composition series by the Jordan-Holder theorem. However, V
does not in general have a finite composition series. Even so, Kac found a
way of defining the multiplicity [V : /. (A) |. This makes use of the following
lemma.
Lemma 19.4 Let V e О and A e IT. Then V has a filtration
V= V0D ViD-’-D Vt = O
of finite length by means of a sequence of submodules such that each factor
V;_i/V; either is isomorphic to L(p.) for some p. > A or has the property that
(Vi-JV^ = Ofor allp^X.
Proof The definition of О shows that V has only finitely many weights /z
with pc >- A. Thus
A) = E dim E
is finite. We shall prove the lemma by induction on a(V, A). If a(V, A) = 0
then V = Vo D Vj = О is the required filtration. So suppose a(V, A) > 0. Then
V has a weight /z with /z > A. We may choose a maximal weight /z with
pc >- A. Let v.j e V be a weight vector with weight /z. Then xv.j = 0 for x e N
and xv= /z(x)rlz for xeH. Let U = II(L(A))v be the submodule of V
19.1 The category 0 of L(A)-modules	457
generated by v . We then have a map в : M(/z) U defined by в {um^ =
uv^ for и e II (N), and в is a homomorphism of lI(L(A))-modules as before,
as shown in the proof of Proposition 10.13. Moreover в is surjective. Thus
U is isomorphic to a factor module of the Verma module M(/z), and so has
a unique maximal submodule U. We also have
Now consider the filtration
V D V D U D O.
We have a(U, A) < a(V, A) and afV/U, A) < a(V, A), since the weight /x >  A
appears in U/U. Thus by induction we obtain filtrations for the modules
U e О and V/ U e О of the required kind, and these may be combined to give
the required filtration of V.	□
Lemma 19.5 Let V e О and A e H*. Consider filtrations of the type given in
Lemma 19.4 with respect to A. LetgsH' satisfy p >- h. Then the number of
factors L(p) in such a filtration is independent of the choice of filtration and
also of the choice of A.
Proof. We first observe that a filtration with respect to A is also a filtration
with respect to p when p > A. Also the multiplicity of L(p) in such a filtration
is the same whether it is regarded as a filtration with respect to A or p. Thus
to prove the lemma it will be sufficient to take two filtrations with respect to
p and show that L(p) has the same multiplicity in each.
The following variant of the proof of the Jordan-Holder theorem achieves
this. Let
V=V0D ViD-’-D Vh = O	(19.1)
V=V(DV1'D---D^ = O	(19.2)
be two such filtrations of lengths lr, l2. We shall use induction on min (/x, Z2).
Suppose first that min (/x, Z2) = 1. Then either V is irreducible and the two
filtrations are identical, or p is not a weight of V and L(p) does not appear
in either filtration.
Thus suppose min (Zb Z2) > 1. We suppose first that V) = Vf We then con-
sider the two filtrations
Vi D • • • D = О
v;d-dv;i = o
458 Some representations of symmetrisable Kac-Moody algebras
of Vj. By induction they give the same multiplicity for I.(p), and the filtrations
for V are obtained by adding the additional factor V/Vx which is the same
for both.
We may therefore suppose that Vj f Vf Suppose first that one contains the
other, say G Vf Then V/Vx is not irreducible and so p is not a weight of
V/Vv Thus neither V/Vx nor V/V[ is isomorphic to I.(p). Let
Vj D LTj D • • • D Um = О
be a filtration of Vj of the required type with respect to p. We then consider
the filtrations
VD V1Dt/1D---Dt/m = O	(19.3)
Vd Vj'd Vi D Ux D • • • D Um = O.	(19.4)
These are filtrations of V of the required type with respect to p. I.(p.) has
the same multiplicity in filtrations (19.1), (19.3) since they have the same
leading term Vj. Similarly I.(p.) has the same multiplicity in filtrations (19.2),
(19.4). So L(p) has the same multiplicity in filtrations (19.3), (19.4) since
none of V/V-t, V/Vf Vj'/Vj is isomorphic to L(p). Thus L(p) has the same
multiplicity in filtrations (19.1), (19.2) as required.
We may therefore assume that neither of Vj, V)' is contained in the other.
Let U = Vj П V[ and choose a filtration of U of the required kind with respect
to p. This has form
U D LTj D • • • D Um = O.
We then consider the filtrations
VDV1Dt/Dt/1D---Dt/m = O	(19-5)
Vd Vj'd Uj U1 D • • • D Um = O.	(19.6)
These are filtrations of V of the required type with respect to p. This is clear
since
Vj/tZ-CLj + VO/V),	v'/и =(V,+ ¥')/¥,.
Now l.(p) has the same multiplicity in filtrations (19.1), (19.5) and the
same multiplicity in filtrations (19.2), (19.6) since the leading terms are the
same. It is therefore sufficient to show that L(p) has the same multiplicity in
filtrations (19.5), (19.6). These filtrations differ only in the first two factors.
If Vj + V' = V then we have
v/Vj = Vj7t/, v/Vj' = Vj/t/
19.2 The generalised Casimir operator
459
as required. If V) + V[ V then V/Vx and V/V'i are not irreducible. In this case
p is not a weight of V/Vx or V/Vf so is not a weight of Vx/U. Thus none of
V/ Vi, Vi IU, V/ V[, V[/U is isomorphic to L (p.). This completes the proof. □
Definition The multiplicity of L(p) in a filtration ofVeO of the type con-
sidered in Lemmas 19.4 and 19.5 will be denoted by [V : L(p.)].
Of course this agrees with the previous definition of [V : L(p.)] in the case
when V has a composition series of finite length.
Proposition 19.6 Let V e 0. Then
chv= [v : L(A)]chL(A).
Лей*
Proof. Both sides are functions II Z. We have (ch V)(p.) = dim V and
the right-hand side evaluated at p is
£[V : L(A)]dimL(A)M.
Лей*
We choose a filtration of V with respect to p of the type given in Lemma 19.4.
Each factor either is isomorphic to L(A) for some A >- p or does not contain p as
a weight. The multiplicity of L(A) as a factor is | V7 : L(A)]. Hence we have
= : L(A)1 dimL(A)M
л
summed over all A >- p. We may in fact take the sum over all A e H* since
dimL(A)M = 0 unless A >- p.	□
19.2 The generalised Casimir operator
We recall from Section 11.6 that, if L is a finite dimensional semisimple Lie
algebra, the Casimir element of the centre of the enveloping algebra II (L)
plays an important role in the representation theory of L. If xx,... , xm are
any basis of L and yx,... , ym are the dual basis with respect to the Killing
form the Casimir element is given by
J>;y;elI(L).
We showed in Proposition 11.36 that the Casimir element acts on a Verma
module M(A) for L as scalar multiplication by (A + p, A + p) — (p, p) where
(,) is the Killing form and p is, as usual, the element of II given by p (hf) = 1
for i = 1,... , /.
460 Some representations of symmetrisable Kac-Moody algebras
Now let 1ЛЛ) be a Kac-Moody algebra where A is symmetrisable. We
cannot define an analogous Casimir element )LA',.v, in 1I(L(A)) since the sum
will in general be infinite and make no sense. It was shown by Kac, however,
that it is possible to define an operator с : V -+ V on any L(A)-module V
in category 0 which has properties analogous to the action of the Casimir
element for finite dimensional algebras. In order to define Kac’ operator on V
we recall the formula for the Casimir element of a finite dimensional algebra
given in Proposition 11.35. Let hf ... , /i'; be a basis of H and h",... , h" be
the dual basis of H with respect to the Killing form of L. Choose elements
ea^La, fa&L_a such that [еа/а] = й'а for each аеФ+. Then the Casimir
element of II (L) is given by
+ £ ^ + 2 £ faea.
z=l	аеФ+	аеФ+
Since	this element can also be written
i
^h'^ + ^ + 2 £ faea
z=l	аеФ+
where h'pEH satisfies p(x) = (h' л) for all x e H.
We wish to define an analogous element for the symmetrisable Kac-Moody
algebraL( A). The root space La of L (A) need not be 1 -dimensional, so we choose
a basis e*M, e®,... for La. Instead of using the Killing form we use the standard
invariant bilinear form on L(A). (In the case when L(A) is finite dimensional
this is a scalar multiple of the Killing form.) We recall from Corollary 16.5 that
the pairing La x L_a C given by x, у (x, y) is non-degenerate. Thus we
may choose a corresponding dual basis	,... for I.,. such that
(MH.
We choose a basis hf h2,... of H and let h", h2,... be the dual basis of
H satisfying (/i., 1г'^ = 8^. Since the fundamental coroots hr,... , hn eH are
linearly independent there exists p^H* such that p (h) = 1 for i = 1,... , n.
However, p is not in general uniquely determined by this condition. So we
choose any element peH* satisfying p(/i;) = l for z=l,... ,n. We then
have a corresponding element h,ill such that p(x) = (h'p, x} for all xMI.
We then consider the expression
£^" + 2^ + 2 £
i	аеФ+ i
This element does not make sense as an element of 1I(L(A)) in general since
the sum over a e Ф+ may be infinite. However, if V is an L (A)-module in
19.2 The generalised Casimir operator
461
0 we know that ch V e St and so there exist only finitely many a e Ф+ such
that La V f O. Thus the operator fl : V -+ V given by
fi = E^' + 2h'p + 2 £
i	аеФ+ i
is well defined. It is straightforward to check that this operator fl : V V
does not depend on the choice of dual bases hf h'2,... h", h2,... of H or
on the choice of dual bases e®, for La and L_a. It may, however, depend
upon the choice of p.
Definition The operator 12 : V -+ V for V e О is called the generalised
Casimir operator on V with respect to p.
In the case of a finite dimensional semisimple Lie algebra the Casimir
element lies in the centre of the universal enveloping algebra. We shall prove
an analogous result in the present situation, i.e. that the generalised Casimir
operator commutes with the action on V e О of any element of II(L(A)). We
first need some preliminary results.
Lemma 19.7 Let a, /3, /3 — аеФ. Suppose e^, are dual bases ofLa, L_a
and ep , fp1-1 are dual bases of Lp, L_p. Let x e Lp_a. Then in the vector space
L(A)®L(A) we have
E /4 ® [л-, 4° ]=E [4° > *] ® 4° •
i	i
Proof. We note that both sides lie in the subspace L_a®Lp. We define a
bilinear form on L(A)®L(A), uniquely determined by
(л-jOyj, x2®y2) = (x1, x2) <У1, yf) .
Since the standard invariant form is non-degenerate on I.(A) this bilinear
form will be non-degenerate on L(A)®L(A).
Let a®b tLy®Ls. Then the scalar products of both sides of the required
equation with a®b are zero unless у = a and 3 = —/3. We therefore suppose
у = a and 8 = — /3 and consider the scalar products
^E./«"®4’
462 Some representations of symmetrisable Kac-Moody algebras
We have
^E fa ® [*, 4° ], a ® = E {fa > f) ([*> ea ] > b}
= -Е(/4,4(44И])
i
= — {a, [xb])
since 44/4 are dual bases of La, L_a. Similarly
^E [4° ’ *] ® 4°’ a ®	= E ([4° , *] > a] (4°’ b)
= Y,(fi3)’ 4’
i
= {[xa], b)
= — {a, [xb]).
Thus the two sides of our equation have the same scalar product with each a ®
b ELa®L_p. Since the form is non-degenerate on L(A)®L(A) this shows
the two sides are equal.	□
Corollary 19.8 In the enveloping algebra 1I(L(A)) we have
i	i
Proof. We apply the natural homomorphism from the tensor algebra 7'(7/A))
to 1I(L(A)). The result then follows from Lemma 19.7.	□
Theorem 19.9 Let и e II (L (A)) and V e 0. Then the maps Ll : V V and
и : V V commute.
Proof. The algebra II (L(A)) is generated by e;, / for i = 1,... , n and the
elements of H. If x e 77 then x commutes with each term /44^ in U(4(4))
since this term has weight 0. Thus x : V -+ V commutes with /44^ ' V V
and hence with fl : V -+ V. It is therefore sufficient to show that fl : V -+ V
commutes with e; : V V and / : V V.
19.2 The generalised Casimir operator
463
We consider the element	e;] of 11(7X4)). We have

= E [№’ ei] ea +E/E [eE ei]
j	j
= E [/E ei] ea - E[eo E]
j	j
= Е[/ЕЕЕ’-Е[/ХфЕ
j	j
by Corollary 19.8. If а + а^Ф the second term is interpreted as 0.
We show that
E E/EE : v^v
аеФ+ \ j	/
a/az-
commutes with e; : V V. We have
a^aj	a^o-j
on V. If a — a; 0 Ф then fC) , eJ _ g. Thus we may assume a = /3+ai in
the first term with /3 e Ф+ and get
E EC.^ EE- E E[EE^]EE=o.
/ЗеФ+ _ j	_	аеФ+ j
(3^
Since 11 = 'Ejh'jh'.+2h'p + 2Y,a^	on V it is now sufficient to
show that hl/i" + 2h'p + 2fiei commutes with e; on V. In fact these elements
commute in II(L(A)). For we have
EaX;’7
= ElE’Xl hj+Ehj[hj’e‘l
j	j
464 Some representations of symmetrisable Kac-Moody algebras
Since h), h'2,... and h", h2,... are dual bases of H we have

and

Hence
= 2e;/i'ff. + (at, ai) et.
Secondly we have
IM’ e;] =2a; (h'p) ei=2(h'ai, h'p) ei = 2p (h'a) e; = (at, ai) e;
since ht = ------!—г and p (/i;) = 1, hence
M h'aJ
Thirdly we have
[2fiec СI = 2 [fc СI ei = -2 lec fi\ ei-
We recall that e;,/; were chosen so that (e;, fi) = 1. By Corollary 16.5 this
implies [e;/;] = h'a,. Hence
[2/;e;, e;] = -2/г'а_е; = -2e;/i'a_ -2a; (/<.) e;
= —2e;/i'a,. —2 {at, ai) et.
Thus we have shown:

= 2e;/i'ff. + (at, at) e;
[2hp, e;] = (at, at) e;
[2/;e;, e;] = -2еЛ«, ~2(ac ai} ei-
Hence
+	+	=0.
19.2 The generalised Casimir operator
465
Thus we have shown that П : V V commutes with e; : V V. The
proof that fl : V -+ V commutes with	V is similar. Using the fact
that
= E л] +ЕЛ° И, f]
- V fU)
/ j J a+az-
we deduce as before that
аеФ+
a/az-
commutes with : V -+ V. We also obtain
Zlfh'^f = -2/x, + <“o“M
j
[2h'P’fi] = -{aoaffi
[2fiei,fl\ = 2fih'a.
Hence
YJh'jh"j+2h'p + 2fiei,fi =0.
Thus fl : V V commutes with : V V and the proof is complete. □
We next describe the action of the generalised Casimir operator fl on a
Verma module.
Proposition 19.10 fl acts on the Verma module M(A) as scalar multiplica-
tion by (A + p, A + p) — (p, p).
Proof. Let mA be a highest weight vector of M(A). Then
(-1тл = I ^hlh'l + 2^ + 2	E.C+")'’+
\ j	»e®+ j	/
Еа(^)а(й")+2а(й;) UA.
466 Some representations of symmetrisable Kac-Moody algebras
Now JT A (h'j) A (/i") = (A, A) and A = (A, p}. Hence
ilmA = (<A + p, A + p) - (p, p))mA.
Now each element of M(A) has form итА for some и ell (/V). Thus
fl (итА) = и (flmj = ((A + p, A + p) - (p, р))итА,
by Theorem 19.9. Hence fl acts on M(A) as scalar multiplication by
(A + p,A + p)-(p,p).	□
Corollary 19.11 fl acts on the irreducible L(A)-module L(X) as scalar
multiplication by (A + p, A + p) — (p, p)	□
Note Proposition 19.10 is the analogue of Proposition 11.36 for finite dimen-
sional semisimple Lie algebras. In Proposition 11.36 the invariant form which
appeared was the Killing form whereas in Proposition 19.10 and Corol-
lary 19.11 it is the standard invariant form. The difference is explained by the
fact that the Casimir element in the enveloping algebra of a finite dimensional
semisimple Lie algebra was defined in terms of the Killing form, whereas the
generalised Casimir operator was defined in terms of the standard invariant
form.
19.3 Kac’ character formula
Let X be the set of integral weights At//; that is the set of all A such that
A (/i;) e Z for i = 1,... , n. Let X+ be the subset of dominant integral weights,
that is the set of weights A e X such that A (/i;) > 0 for all i. In this section
we shall prove a formula due to Kac for the character of the irreducible
L(A)-module L(A) when A e X. The reason for the restriction to weights in
X+ lies in the fact that the modules L(A) for A e X are integrable.
Definition An L(A)-module V is called integrable if
V=
Лей*
and if e; : V V and f : V V are locally nilpotent for all i= 1,... , n.
Proposition 19.12 The adjoint module /.(A) is integrable.
Proof The proof of Proposition 7.17 carries over to the present situation.
□
19.3 Kac’ character formula
467
Proposition 19.13 Let V be an integrable IfAfmodule. Then dim VA =
dim for each XeH* and each weW.
Proof Since the Weyl group W of 1ЛЛ) is generated by the elements st it is
sufficient to show that dim VA = dim VS.(A).
We may regard V as a module for the 3-dimensional simple subalgebra
(e;, ht, f) of L(A). Let v e VA and consider the (e;, ht, /;)-submodule gener-
ated by v. The vectors
2	r— 1
v, etv, eivy ... , v
lie in this submodule, where r is the smallest positive integer with e[v = 0.
The vectors ffefv also lie in this submodule, and there are only finitely many
(a, b) for which such a vector is non-zero. Each such vector is a weight vector
in V. However, the relation
ej^f^i + nfr^hi-in-l))
obtained in the proof of Theorem 10.20 shows that the subspace spanned by
all vectors ffefv is an (e;, ht, f)-submodule. Hence every weight vector v e V
lies in a finite dimensional (e;, ht, /;)-submodule which is also an //-module,
i.e. it is an (e;, H, f) -submodule.
Now let U be the subspace of V given by
U is clearly an (e;, H, /;)-submodule of V. The (e;, H, /;)-submodule gen-
erated by each weight vector is finite dimensional, thus U is a sum of finite
dimensional (e;, H, -submodules. Now (e;, ht, ft) is a 3-dimensional sim-
ple Lie algebra of type AP Thus every finite dimensional (e;, ht, /;)-module
is a direct sum of finite dimensional irreducible (e;, ht, /^-modules, by the
complete reducibility theorem, Theorem 12.20. The weight spaces involved
in such a decomposition of an //-invariant (e;, ht, f) -module can be chosen
as weight spaces for H, as in the proof of Theorem 10.20, thus every finite
dimensional //-invariant (e;, hh f) -module is a direct sum of finite dimen-
sional //-invariant irreducible (e;, hh f) -modules. Thus U is a sum of finite
dimensional //-invariant irreducible (e;, hh -submodules, so is a direct
sum of certain of these submodules. However, for each of these irreducible
submodules M we have
dimMA = dimMs.(A)
468 Some representations of symmetrisable Kac-Moody algebras
by Proposition 10.22. It follows that
dim VA = dim VS.(A)
and the required result follows.
Proposition 19.14 Let IfA) be a symmetrisable Kac-Moody algebra and
L(X) be an irreducible L(A)-module in the category 0. Then L(X) is inte-
grable if and only if A is dominant and integral.
Proof Suppose first that L(A) is integrable. Let uA be a highest weight vector
in L(A). Then f.vA = 0 for some r. Consider the vectors
vA, fv,, ..., fA.
Since ejf + nf?-1 (Ii, — (и — 1)) for each n we see that these vectors
span an {et, ht, /;)-submodule of L(A). The highest weight of this finite
dimensional {et, hh /;)-module is A. But the highest weight of any finite
dimensional module for a finite dimensional simple Lie algebra is dominant
and integral. Thus A (/i;) e Z and A (/i;) > 0. Since this holds for all i, A is
dominant and integral.
Now suppose conversely that A (/i;) e Z and A (/i;) > 0 for each i. Then we
have
уЖ)+11,л = 0
as in the proof of Theorem 10.20. Now each element of L(A) has form иил
for some и e L(A). We have
f? (,uvs) = E (T) ((adfi)k u) 
Now (ad/;)*1 и = 0 for A sufficiently large since L(A) is integrable, by Propo-
sition 19.12. Also ,//v Z = 0 for N — к sufficiently large, as shown above.
Thus ff (uvA) = 0 for N sufficiently large, and so f : L(A) L(A) is locally
nilpotent. The fact that e; : L(A)^L(A) is locally nilpotent follows from
the fact that L(A) lies in category O. Thus L(A) is integrable.	□
As before we write
X+ = {XeH*-, A (/i;) e Z, A (/i;) > 0 for each;}.
19.3 Kac’ character formula
469
We now turn to Kac’ character formula for ch L(A) when A e X. We recall
from Proposition 19.2 that the character of the corresponding Verma module
M(A) is given by
chM(A) = ^
where Д = П«ф (1 — e-a)m“ and ma is the multiplicity of a.
We begin with a lemma.
Lemma 19.15 Let X++ = {A e X ; A (/i;) > Qfor all z}. Suppose eX++, 17 e
X+ satisfy p < (: and (77, if) = ((;, If). Then p=(:.
Proof Since 17 -< £ we have £ —17 = J2"=1 ktat with к e Z and к > 0. Thus
(£, £) - (17, i?) = « + чЛ - v)
= Hki^ + P>a^=Hki (£ +77) (/i;) .
i	i
Now (at, ctj) > 0 and (£ +17) (/z;) > 0. Hence (£, ^) — (77, 77) = 0 implies that
kt = 0 for each i. Thus £ = 17.	□
Theorem 19.16 (Kac’ character formula). Let L(A) be a symmetrisable
Kac-Moody algebra and L(X) be an irreducible L(A)-module with XeX+.
Then
D £(w)ew(A+p)-p
chL(A) = ^—-------------—
аеФ+
(This is an equality in the ring 9T)
Proof. By Proposition 19.6 we have
chM(A)= [M(A) : L(p,)]chL(p,).
jieH*
Now all p for which |/W(A) : L(p,)]^0 satisfy p < k. For L(p) appears as
a factor in some filtration of M(A), so p is a weight of M(A).
We consider the action of the generalised Casimir operator Ll on M(A). By
Proposition 19.10 fl acts on M(A) as scalar multiplication by (A + p, A + p) —
(p, p). Similarly by Corollary 19.11 Ll acts on L(p) as scalar multiplication
by {p+p, p + p) — (p, p). Thus if |/W(A) : L(/z)] 0 we must have
(A + p, A + p) — (p, p) = (p. + p, p. + p) — (p, p)
470 Some representations of symmetrisable Kac-Moody algebras
that is (A + p, A + p) = (pc + p, рс + р). Thus
chM(A) = £[M(A) : L(p)] ch L(p)
м
summed over all p. < A with (p + p, p + p) = (A + p, A + p). If we take a total
ordering on the weights p satisfying p< A and (p + p, p + p) = (A + p, A + p)
which is compatible with the partial ordering -< these equations can be
written
chM(A) = J>ApchL(p)
where (aAp) is an infinite matrix with non-negative integer entries such that
aAA = 1 and aAp = 0 for all entries below the diagonal. Such a matrix (aAg)
can be inverted to give a matrix (bAp) with K,, e Z, bAA = 1 and K,, = 0 for
entries below the diagonal. Thus we have
chL(A) = J>Ap chM(p)
__ p
Thus Дс11£(А) = ЕрЬАрер and ерД chL(A) = £p b^e^.
We consider the action of the Weyl group on the functions which appear
here. Since st transforms at to — at and Ф+ — {aj into itself we have
^(еРд) = «i [ep	П
\	аеФ+— {a,}
= ep-a,	П (1-е_аГ“
аеФ+ —{aj
= -ерД
since st (p) = p — at. Hence
w (e Д) = s(w) ерД for all w e W.
Also by Proposition 19.13 we have
w(ch L(A)) = ch L(A) for all w e W,
since L(A) is integrable. It follows that
W I E bApep+p j = £(W) E bApep+p-
\ M	M
19.3 Kac’ character formula
471
This implies that
where w(p. + p) = v + p.
Suppose p. is a weight for which b,,, 0. Consider the set of all weights v
for which w(p. + p) = v + p for some w e W. All such weights satisfy Ьк„ 0
and we have p < A. Among all such weights v we can choose one for which
the height of A — v is minimal. Then v + p must lie in X. For if there existed
an i for which (r + p) (/i;) < 0 we would have
S;w(jt + p) = S;(v + p) = v + p - (v + p) (h) at
contradicting the minimality of ht (A — v). Hence v + p G X. We also have
(v + p, v + p} = (рь + р, p^ + p) = (A + p, A + p).
Thus we have
А + рбХ^^,	v + p £ X^,	v + p-<A + p
and (v + p, v + p} = (A + p, A + p). By Lemma 19.15 this implies v = X.
Hence every weight p. for which b,,, 0 satisfies p. + p = w(A + p) for some
w e W. But then
b^ = s(w)b^ = s(w).
Hence
epAchL(A)= s(w)
ew(X.+p) •
w+W
If follows that
D £(w)ew(A+p)-p
chL(A) = ^----------------.
(We note that A = е_л chM(A) lies in St.)
Corollary 19.17 (Kac’ denominator formula). For a symmetrisable Kac-
Moody algebra we have
eP П	E £(w)£+p)-
аеФ+	weW
472 Some representations of symmetrisable Kac-Moody algebras
Proof. L(0) is the 1-dimensional trivial module with chL(O) = e0. Hence
A = Ae0= s(w)ew(p)_p
weW
and so epA = E„ li s(w!)e„ipi.
Corollary 19.18 (Alternative form of Kac’ character formula). Let
L(X), XeX+, be an irreducible module for a symmetrisable Kac-Moody
algebra. Then
E £(WK(A+p)
i т (\\ weW
Proof. This follows from Theorem 19.16 and Corollary 19.17.	□
Note Kac’ character formula and denominator formula appear very similar to
Weyl’s character and denominator formulae for finite dimensional semisimple
Lie algebras. However, the nature of Kac’ formulae is in fact rather different,
since they involve in general infinite sums over the elements of W and infinite
products over the positive roots.
Theorem 19.19 Let L(A) be a symmetrisable Kac-Moody algebra and
XeX+. Then L(A) = M(A)/J(A) where ./(A) is the submodule of M(X) gen-
erated by elements ft^^m^ for i = 1,... , n.
Proof. Let K(A) be the submodule of M(A) generated by the elements
f^hi)+1m^ We know that L(A) = M(A)/J(A) where ./(A) is the unique max-
imal submodule of M(A), and wish to show that K(A) = 7(A). Now we have
/;Л(М+1 _ q where vA = 7(A) + mA
as in the proof of Proposition 19.14 (the detailed argument is given in Theo-
rem 10.20). Thus f^('hi'l+1mA e 7(A) and so K(X) c 7(A).
Let V(A) = M(A)/K(A). Then V(A) is an L(A)-module in the category 0, so
chV(A)=52[V(A) : L(^)] chL(^)
/к A
by Proposition 19.6. We also have
chL(pf=^bpvchM(v).
19.3 Kac’ character formula
473
Hence
chKA) = EcAMchW)
/zxA
for certain cAp e Z. By considering the action of the generalised Casimir
operator fl on M(p.) and on V(A) and using Proposition 19.10 we have
chV(A) =	rApch/W(M).
/zxA
(p+p,p+p)=(A+p,A+p)
Now let v'x = K(X) + mK be the highest weight vector of V(A). Then we have
yA(M+1l,A = 0.
It follows, as in the proof of Proposition 19.14, that : V(A)—> V(A) is
locally nilpotent. Since V(A) e 0, et : V(A) V(A) is locally nilpotent.
Hence V(A) is an integrable L (A) -module. Thus
w(ch V(A)) = ch V(A) for all w e W
by Proposition 19.13. We then have
epAchV(A) =	52 cApeM+P
/zx A
(p+p,p+p)=(A+p,A+p)
by Proposition 19.2. It then follows exactly as in the proof of Theorem 19.16
that every weight p. for which cAp^0 satisfies fi + p = w(A + p) for some
w e W, and that then cAp = s(w). Hence
gpAch V(A)= 52 £(w)ew(A+p)
weW
and so ch V(A) = chL(A) by Theorem 19.16. Since L(A) is a factor module of
V(A) this can only happen if V(A) = L(A). Thus K(K) = ./(A) as required. □
We now recall that for finite dimensional semisimple Lie algebras the
partition function L was defined as follows. If A e // LfA) is the number of
ways of writing f as a sum of positive roots, i.e. as the number of sets of
non-negative integers ra, a e Ф+, such that A = ZL, <i. raa.
For Kac-Moody algebras we define the generalised partition function ®
as follows. If A eH* is the number of ways of writing f as a sum of
positive roots, each such root a being taken ma times, i.e. as the number of
sets of non-negative integers ra t for a e Ф+ and 1 < i < ma such that
<=52 Evf-
аеФ+ i=l
474 Some representations of symmetrisable Kac-Moody algebras
We then have an analogue for symmetrisable Kac-Moody algebras of
Kostant’s multiplicity formula Theorem 12.18.
Proposition 19.20 Let L(A) be a symmetrisable Kac-Moody algebra and let
A e x+. Then for each weight p of L(X) we have
dimL(A)M= E s(w)®(w(A + p) - (p + p')').
weW
Proof. By Proposition 19.2 we have
chM(A) = eA П (1 + ^ + е_2а + ---)И“-
аеФ+
By definition of ® we have
П (i+c,+^+---r = E ад^.
»еФ+	(3e2+
Thus chM(A) = eA ®(J3)e_p. It follows that
chL(A) = £ s(w)chM(w(X +p) — p)
weW
= E E	ew(X+p)-p
weW /3eQ+
= E E	^w(A+p)—p—
weW /3eQ+
= EE £(w)®(w(A + P)-(M + P))eM-
/z weW
Hence the multiplicity of p as a weight of L(A) is
E s(w)®(w(A + p) - (p + p)).	□
weW
19.4 Generators and relations for symmetrisable algebras
We recall that the Kac-Moody algebra L(A) was not defined in terms of
generators and relations. The larger algebra L(A) was defined by generators
and relations and its quotient L(A) is given as L(A)/I where I is the largest
ideal of L(A) satisfying IПH = O. It is natural to ask what additional relations
are required to pass from L(A) to L(A). We shall answer this in the case
when the GCM A is symmetrisable.
We first require some preliminary results on enveloping algebras and mod-
ules in category 0.
19.4 Generators and relations for symmetrisable algebras
475
Proposition 19.21 Let О : I. Il be a surjective homomorphism of Lie
algebras with kernel K. Let ф : II {Lj II (Lz) be the corresponding homo-
morphism between enveloping algebras. Then the kernel of ф is KU{Lj.
Proof. Since К is an ideal of L, 7GI(L) is a 2-sided ideal of 1I(L). For |Лх| G
К for k E К, x E L and so kx = xk + |Лх| in 1I(L). Thus 7GI(L) = 1I(L)/C and
7C1I(L) is a 2-sided ideal of 1I(L). Thus 7GI(L) с кегс/л
Conversely we have a homomorphism
a : 1I(L)/^1I(L)^1I(LZ)
induced by ф. We consider the Lie algebra [1I(L)/7GI(L)]. We shall define
a map L' [1I(L)/7GI(L)] as follows. Given x'eL' we choose xx eL with
в {xf) = x'. Then xx G 1I(L) gives rise to xx G [1I(L)/_L1I(L)]. We show that
the map x'^x1 is well defined. Suppose x2eL also satisfies 0(x2) = xz.
Then x2 G [1I(L)/_L1I(L)]. Now d{x1j = d{x2j so xx — x2eK. Hence xx =
Xj — x2 + x2 = x2. Thus our map is well defined and is clearly a Lie algebra
homomorphism. By the universal property of enveloping algebras there is a
homomorphism
/3 : 1I(LZ)^1I(L)/^1I(L)
compatible with our homomorphism of Lie algebras
LZ^[1I(L)/^1I(L)].
It is readily checked that a, /3 are inverse homomorphisms, and thus isomor-
phisms. Hence the homomorphism ф : II{Lj II (Lz) has kernel KU{Lj.
□
The 2-sided ideal L1I(L) of 1I(L) will be denoted by 1I(L)+. We have
H(L) = C1®1I(L)+.
Proposition 19.22 LA(1I(L)+)2 = [LL].
Proof. Since L G 1I(L)+ and, for x, у G L, [xy] = xy — yx we see that
[LL] С LA (1I(L)+)2.
Conversely let L = L/[LL]. We have a natural homomorphism 1I(L)	1I(L)
under which LA(1I(L)+)2 maps to Ln(lI(L)+) . Now L is an abelian Lie
476 Some representations of symmetrisable Kac-Moody algebras
algebra so II (L) is a polynomial algebra. In such a polynomial algebra it is
evident that
Ln(ll(L)+)2 = 0.
It follows that Ln (1I(L)+)2 lies in the kernel of L—> L and so
Ln(U(L)+)2c[LL].	□
Proposition 19.23 Let К be a subalgebra of the Lie algebra L. Then
КГ\КТ1(Ь)+= [KK],
Proof. Since AclI(A)+ we have [AA] c A'II(A')1, using |vv| = xy — yx.
Hence | A'A"| с АП AII(L). To prove the converse we use the PBW basis
theorem. Let {kt} be a basis of K, and extend it to a basis {kh u,} of L. Then all
finite products of the form П^Г‘ И/ with mt >0, nj >0 form a basis of II(L)
and the subset П «'/,> 0 is a basis for 11(A). The monomials П u"/
with mi + Z2 nj > 1 form a basis of 1I(L)+ and those with £ mi + Z2 nj > 2
and > 1 form a basis of A1I(L)+. Now
К П All (L) + с И (А) П All (L)+.
A linear combination of monomials Hkfu"' lies in 11(A) if and only if all
such monomials have п^ = 0. Thus each element of AnAlI(L)+ is a linear
combination of such monomials with all nj =0 and >2. Hence
АП A1I(L)+ c (U(A)j2nA
and so АП A1I(L)+ с [AA] by Proposition 19.22.	□
We next need some further properties of Verma modules. We recall that
for Л e H* the Verma module M(A) for L(A) is given by
M(A) = 1I(L(A))/Aa
where Ал = II (L(A))V +	H(L(A))(x — А (л)). The Verma module M(A)
can also be described as a tensor product. Let В be the subalgebra of L(A)
given by В = N + H.
Lemma 19.24 M(A) is isomorphic to the U(L)-module 1I(L) CvA,
where CuA is the 1-dimensional В-module with N in the kernel and H acting
by the weight A.
19.4 Generators and relations for symmetrisable algebras
477
Proof. There is a bijection
11(B) CnA II (N )®cCr>A
given as follows. Since I. = N фВ we have a bijection
11(B)—>11 (IV-)	11(B).
Thus we have bijections
11(B)CuA (II (N )®c 11(B))®U(B)CiiA
—>1I(1V )®c (11(B) CuA) II (N )®cCuA.
The H(B)-action on II (N~) ®c CvA is given as follows. Let u' e 11(B) and
и e II G'V). Then
и'и = У^а,В;	where at e II (N~), bt e 11(B).
i
We have и' (и ® vA) = (JT A (b;) a;) ® vA. On the other hand we know that each
element of M(A) is expressible uniquely as итА for	Moreover
for u' e 11(B) we have
u' (umf) =	(^;) ai^ mK-
Thus there is a H(B)-module isomorphism between 11(B) CuA and M(A).
□
We may also define a module M(A) for the larger Lie algebra B(A) by
M(A) = 11(B) CuA
where Л e H* and B = N + H.
Lemma 19.25 For XeH* there is an isomorphism of U(L)-modules
11(B) ®U(I)M(A) = M(A).
Proof. We have a sequence of bijections
11(B) ®	= 11(B)	(11(B) ®u(b) CuA)
(11(B)	11(B)) ®u(b) CuA -+ U(B) CuA.
Now we have a natural homomorphism 11(B)	11(B) with kernel К which
acts trivially on 11(B) and on CvA. Thus we have a bijection
11(B) CuA 11(B) CuA = M(A).
The above bijections are isomorphisms of 11(B)-modules.	□
478 Some representations of symmetrisable Kac-Moody algebras
We next require further information about modules in the category 0. The
following definition turns out to be very useful.
Definition Let V be an L(A)-module with V eO. A vector v e V is called
primitive if
(i) v is a weight vector
(ii) there exists a submodule UcV such that v$.U but Nv c U.
Lemma 19.26 A module V eO is generated as an L(A)-module by its prim-
itive vectors.
Proof Let V be the submodule generated by the primitive vectors in V.
Suppose V ф V. Consider the factor module V/V. This factor module lies
in О so contains a weight vector of maximal weight with respect to -<.
Thus Nv = 0. Let v be a weight vector in V such that v v. Then v g V' and
Nv G V. Thus и is a primitive vector not in V, a contradiction.	□
In fact the following stronger result is true.
Proposition 19.27 A module V e О is generated as a [ЦК )-module by its
primitive vectors.
Proof. We first show that if v e V is a weight vector which is not primitive
then re II(/V ) Il(/V) r. For consider the lI(L)-submodule of V generated
by Nv. We have
1I(L)№ = U(AT)H(fl)lI(7V)M;
= II (/V) II (/V) AT	since и is a weight vector
= U(AT)H(7V)+v.
Now let U be the II (/V )-submodule generated by the primitive vectors in
V. We wish to show U = V. We shall assume U/V and obtain a contradic-
tion. For each weight vector v e V we have
U(L)v = lI(AT)lI(tf)lI(7V)v= n(F)U^)u
= II(,V)(CI +Il(/V))r
= U(AT)v+U(AT)lI(7V)+v.
We can deduce from this that V is generated as lI(L)-module by U and
the II (/V )-submodule generated by Il(/V) r for all primitive veV. Since
19.4 Generators and relations for symmetrisable algebras
479
U V there exists a primitive v such that II (N)v (f U. Let v have weight
A. Then there exists a weight vector их еП(Д1)+ with upugU. So urv is not
primitive in V. Thus we have
e II (/V )
Hence U(N)+u1v <fU. So there exists a weight vector w2elI(W)+ with
u2u3v$. U.
Continuing in this way we obtain a sequence of weight vectors
u2, u3,... in П (/V) such that uk ... urv g U for each k. Let the weight of
и; be /z;. Then the weight of uk... urv is A + p^ 4-\~P+- We have
A<A + p,<A + pi +p.2 -<••*.
But V e 0 and so such a sequence of weights must terminate after finitely
many steps. This gives the required contradiction.	□
We now consider the module ф"=1Л/(—a;) in 0.
Proposition 19.28 Every primitive vector in the module ф"=1М(—a;) has
weight —a where {a, a) = 2(p, a).
Proof. Let v be a primitive vector in ф"=1М(—a;) of weight —a. Then
there is a submodule U of ф"=1 M(—a;) such that v g U and Nv G U. Write
v = vr -I--h vn where vt e M(—a;) and let v where v e (фМ(-aff)/U.
We consider the action of the generalised Casimir operator fl on the module
фМ(—a;). By Proposition 19.10
flv; = (<-a; + p, -a; + p) - (p, p)) vt
= ((a;, af) —2 {p, a;)) vt.
I ^ai \
Now p (hj) = 1 so (p, ------ ) = 1. Hence at) = 2{p, ai) and so flu; = 0.
\ (at, a) I
Thus fl acts as 0 on фЙ(—a;). Hence fl acts as 0 on (фЛ/(—a;))/t/ and
Hv= 0. But v has weight —a and so
flu = ({—a + p, —a + p) — {p, p))v
= ({a, a) — 2{p, a))v.
Thus {a, a) = 2{p, a) as required.	□
We shall now start to see the relevance of the preliminary results which
we have obtained. We concentrate on the kernel I of the natural homo-
morphism L(A) L(A). We recall that 1 = 1 where / C;V and
480 Some representations of symmetrisable Kac-Moody algebras
I+cN. We have I =@аев-1а by Lemma 14.12. Since dimL(A)_a. =
dimL(A)_a. = l we have Га=0. Thus = фаее-.а/_а/". Hence each
element of I has form Jj"=1 uifi where и; e ll(/V ) . It is in fact uniquely
expressible in this form since N is the free Lie algebra on	by
Proposition 14.8 and so II (S') is the free associative algebra on /),... , fn
by Proposition 9.10.
Proposition 19.29 Let в : L(A) L(A) be the natural homomorphism with
kernel I = I ф I+. Then there is a homomorphism of L-modules
n
i=l
given by YZiUifi^YZiO (uf)m_a.. The kernel of this homomorphism
is | / / |.
Proof We begin with the L-module
M(A) = U(L) ® Сг?л for XeH*.
11(B)
The module M(A) has highest weight vector тл = 1 ® and, just as for the
Verma module M(A), each element of M(A) is uniquely expressible in the
form итл for ae U(;V j. We take the special case A = 0. Then u^ um0 is a
bijection between II (/V ) and M(0).
Now II (/V ) is freely generated by /),... , fn so
H(H-) = Cl ФП (H-) Л Ф • • • ФИ(ЛТ) fn.
Thus ®"=|b(Z )./, is a II(/V )- submodule of codimension 1 in II(/V ). It
corresponds to the subspace ф”=1 II(N~) /;m0 of codimension 1 in M(O). Let
n
У(О) = фП(Н-)/;то.
i=l
Then J(0) is a H(L)-submodule of M(O). For it is clearly invariant under
II (/V ) and 11(H), but also
e;/;m0 = /;e;m0 + 1грп0 = 0
ejfim0 = fiejmo = O iij^fi.
Thus H(V)+/;mo = O and so H(V)/;m0 = C/;m0. Hence
n(L)/;m0 = H(H-) H(H)lI(V)/;m0
= П(Н-)П(Н)/;т0 = П(Н-)/;т0.
19.4 Generators and relations for symmetrisable algebras
481
Thus J(0) = ®"=1U(L)/;m0, which is a H(L)-submodule of M (0). Now /;m0
has weight — at and the map
is an isomorphism of II (L)-modules. Thus J(0) is isomorphic to ф"=1М (—a;)
as II (L)-modules. It then follows from Lemma 19.25 that
n
i=l
as H(L)-modules, or as II (L)-modules.
We now consider the map
ф :	® J(0)
ii(i)
given by x^ l®xm0. We note that xm0 lies in J(0) since I cN . We
show that ф is a homomorphism of L-modules. To see this let y&L. Then
[y, x] l®[y, x\mQ
= \®(yx-xy)m0
= 1 ®y (хт0) — 1 ® x(ym0) .
Now we have xm0 e J(0) and ym0 e J(0). Thus
[y, x] -+ 0(y) ® (xmf) - 0(x) ® (ym0)
= 0(y)®(xmo) since 0(x) = O,
= у (1®лт0).
This shows that ф is a homomorphism of L-modules. Moreover | / / ] lies
in the kernel of ф. For if x, у e I we have [у, x] 0 as above, since 0(x) =
0(y) = 0. Thus we have a homomorphism of L-modules
n
i=l
with J2"=1 и;/; -+ J2"=1 в (и;) m_a. where и; e II (/V ) .
We determine the kernel К of ф. We know that [Г I ] с К and prove the
reverse inclusion. Let £ utft e К where и; e II (/V ) . Then £ $ («;) m n = 0.
This implies 9(ui)m_a. =0 for each i and then that 0(и;) = О for each i.
Now the homomorphism of Lie algebras в : N~ N gives rise to a homo-
morphism of enveloping algebras II (/V ) II (/V ) with kernel / II(/V )
482 Some representations of symmetrisable Kac-Moody algebras
by Proposition 19.21. Thus h,g/ ll(/V ) and so uifi e (/V ) .
Hence Д'С / П / II (/V ) . However, / П/ II (/V ) = |/ / | by Propo-
sition 19.23. Hence К c [/’I ]. Thus the kernel of our homomorphism is
[/-Г].	□
We now come to our description of I.(A) by generators and relations.
Theorem 19.30 Let I.(A) be a symmetrisable Kac-Moody algebra. Then
L(A) = L(A)/J where J is the ideal ofl.(A) generated by the elements
(ade,)1'"^ and (ad f/)l~Aii fj for all i^j. Thus we obtain a system of
generators and relations for I.(A) by taking generators and relations for
L(A) and adding the further relations
(adef)1^^0’ (ad/;)1-Ay/)=0
for all i j.
Proof Let J be the ideal of AC4) generated by the elements (ade,)1'"
and (ad /',)1'" fj. We have L(A) = L(A)/I and J d by Proposition 16.10.
We wish to show that I = J.
We shall suppose if possible that If J and obtain a contradiction. Let
1 = 1/J. Then 7^0 and I = 1(QI where
aeQ+	aeQ~
since the analogous property holds for I. The automorphism w of I.(A) given
in Proposition 14.5 satisfies a>(I) = I and = J so induces an automor-
phism on I = 1/J. This automorphism satisfies a> (7+) = 7 . Hence I+ = 0 if
and only if I = 0. Since I = I+ ф I and I f 0 we must have 7^0.
We know from Section 16.2 that the Weyl group W acts on the weights
of I.(A)/I and that weights in the same W-orbit have the same multiplicity.
The same argument can be applied to L(A)/J to give a similar result. Since
dim (L(A)/J)a = dim (L(A)//)a + dim
we see that W acts on the weights of 7 and that weights in the same W-orbit
have the same multiplicity. In fact W acts on the weights of 7 since if a e Q~
is a weight of 7 then у X) G Q also, since — at is not a weight of I.
We choose a weight a = J2”=i e Q+ such that 7 ,, f 0 and a has mini-
mal possible height XК Since If ltl. f 0 we have ht (a) > ht a. Since
z ч	,
s\a) = a — 2--------at
\ai> ai/
19.4 Generators and relations for symmetrisable algebras
483
we have (ai,a)<0. Since a = with each kt>0 we deduce that
(a, a) < 0. On the other hand we have
2<P, a) = kt (p, 2o^ = Y/ki(ai,ai)> 0.
i	i
Thus (a, a) <0 and 2{p, a) >0. In particular {a, a) =£2(p, a). Thus the
weights —a for I for which a has minimal height satisfy (a, a) f2ip, a).
We now recall from Proposition 19.29 that / / |/ , I ] is isomorphic as
L-module to a submodule of ф"=1М (—a;). By Proposition 19.28 all primi-
tive vectors in ф"=1 M (—a;) have a weight —a satisfying (a, a) = 2{p, a).
Thus all primitive vectors of / / |/ I ] have weight —a satisfying (a, a) =
2{p, a). Now / / |/ / ] is generated as an /V -module by its primitive vec-
tors, by Proposition 19.27. Thus / / |/ / | is generated as an N -module by
its weight vectors with weight —a satisfying (a, a) = 2(p, a). (Recall that
N-/I-=N-.~)
We claim the same is true of I . Let К be the N -submodule of
I generated by all weight vectors with weight —a satisfying (a, a) =
2{p, a). Then (|Z Z ] 4-A") / |/ / | has the same property in Z / |Z Z |,
thus \l I \ + К = I . Suppose if possible that Kf!. Then I /К is an
N -module whose weights are non-zero elements of Q . Consider the
submodule [I~ /К, I / K\ of I /K. This is an N -module whose weights
have form /3 + y where /3, у are weights of Z /А'. Thus if a is a weight of
/ /А for which |ht a| is minimal then a cannot be a weight of [I~ /К, I /К}.
Thus
[Г/K, Г/К\^Г/К
and this gives K+1/ , / | #Z“, a contradiction. Thus I is generated as
/V -module by its weight vectors with weight —a satisfying (a, a) = 2{p, a).
The same must therefore be true of I . However, we have seen above that
the weights —a of I for which hta is minimal do not satisfy (a, a) =
2{p, a). This implies that the set of weight vectors with weight —a satisfying
(a, a) = 2{p, a) cannot generate I as N -module. This gives the required
contradiction.	□
20
Representations of affine
Kac-Moody algebras
20.1 Macdonald’s identities
We now consider Kac’ denominator formula
ПО” = E £(wK(p)-p
аеФ+	weW
in the special case when L is an affine Kac-Moody algebra.
We assume first that L is an untwisted affine algebra. Then /. = £'(/.")
where L° is a finite dimensional simple Lie algebra with root system Ф0
and Weyl group W°. We recall from Theorems 17.18 and 16.27, and Corol-
lary 18.6 that
Ф = |a + n8 ; a e Ф0, n e Z} U {n8 ; neZ,n^0}
and that a + nS has multiplicity 1 and n8 has multiplicity /. Also
Ф+ = {а + иЗ; a e Ф0, n > 0} U (Ф°)+ U {n8 ; и>0}.
Thus the left-hand side of the denominator formula can be expressed as
П (i-e-on
«е(ф°)+	”>0
аеФ0
We also recall from Remark 17.34 that W = t	where M* is the
lattice given by
M* =
1=1
E 2<1;+ E p^ai
aj long	aj short
for types At, Dt, E6, E7, Es
for Bt, Ct, F4, G2
484
20.1 Macdonald’s identities
485
and t (Л/*) is the set of ta : H* H* for a e M* given by
ta(X) = A + A(c)a- ((A,	+	a)A(c))5.
In calculating the right-hand side of the denominator formula we recall that
H = H°®(Cc + Cd)
Я* = (Я°)*ф(Су + С8)
where (Н°)* is embedded in H* by assuming A(c) = 0, A(<f) = 0 for A e (№)*.
Lemma 20.1 Let XeH*. Then
A = А° + А(с)у + а01А(й?)3
where A0 e
Proof. Let A = A0 + ry + ,«5 where A°e(№)*. Then A (c) = A0 (c) + ry(c) +
s8(c). But we have afe) = 0 for i = 1,... , / hence A°(c) = 0. Also 5(c) = 0
and y(c) = l. Hence r = A(c). Also A(<f) = A°(</) + ry(d) + s8(d). We know
afd) = 0 for i = 1,... , / thus A°(</) = 0. Also y(d) = 0 and 8(d) = a0. Hence
X(d) = aos and s = a^flfd).	□
Of course in the untwisted case we have a0 = 1.
We recall thatpeH* satisfies p (hf) = 1 for i = 0, 1,... , / and p(d) = 0.
In particular we have p(c) = c0 + q 4-h c;.
Definition The number h = a0 + ax-\---ha; is called the Coxeter number
of L. The number IT = c0 + cx -I-h c; is called the dual Coxeter number
of L.
We note that if L = £ (L°) is of untwisted type then the Coxeter number
of L is equal to the Coxeter number of L°.
486
Representations of affine Kac-Moody algebras
The values of h and /iv are given in the following table.
Type of L
К
	i+1	I +1
B,	21	2/-1
Ct	21	I +1
Dt	21 — 2	21-2
E6	12	12
Ej	18	18
Eg	30	30
f4	12	9
g2	6	4
B\	2/-1	21
C\	I +1	21
F\	9	12
G\	4	6
	3	3
c;	2/+1	2/+1
Lemma 20.2 p = p° + hvy where р°е(Н°У satisfies p°(/i;) = l for i =
Proof This follows from Lemma 20.1 since p(c) = IT and p(d) = 0.	□
We now consider the right-hand side of the denominator formula. Let
w e W have form w = w°ta where w° e И'" and a e M*. Then
Hp)-p =w°L(p)-P
= w° ^p + hva — ^(p, a) + - (a, a)hv^ hj — p
= w°(p) — p + hvw°(a) — ^{p, a) + - {a, a)hv
a})
= w°
since wci(y) = у and (у, a) = 0 for all a e M’
= w° (hva + p°) — p° —
2K
8.
20.1 Macdonald’s identities
487
For convenience we shall write, for A e (№) *,
c(A) = (A + p°, A + p°} - (p°, p°}.
We recall from Corollary 19.11 that when A is dominant and integral the
generalised Casimir operator acts on the irreducible module with highest
weight A as scalar multiplication by c(A).
We also write, for A e (//"),
12 £(w)ew(A+p°)-p°
0/\\ iueW°
X (A) = —-------------•
12 £(w)ew(pO)_po
weW0
We recall from Theorem 12.17 that when A is dominant and integral у (A)
is the character of the irreducible L°-module L(A). However, c(A) and y"(A)
are now defined for all A e (№)*. Then we have, writing e(A) instead of ел
for convenience:
E £(w)e(w(p) - P)
weW
= E E £(wo)e(wo(^+po)-po)e(^^s
= E 40)Ф°(р°)-р0)
w°eW°	aeM*	\ ЛП J
= П (I-»-.) E/W)»!25^^
«е(фО)+	V
by Weyl’s denominator formula.
We now put q = e - and equate the left- and right-hand sides of Kac’
denominator formula. We obtain the following result.
Theorem 20.3 (Macdonald’s identity for untwisted affine Kac-Moody alge-
bras).
П (1-Л; П	= E Х°^а)^^
n>Q	аеФ0	aeM*
where
Hli=iZai	for types Ah bh Ё6, Ё7, Es
J2 H- 12 P^ai for Bl,Cl,F4,G2.
a.j long	a, short
488
Representations of affine Kac-Moody algebras
We next wish to state Macdonald’s identities for the twisted affine Kac-
Moody algebras. The left-hand side of the identity is obtained from a knowl-
edge of the real and imaginary roots together with the multiplicities of the
imaginary roots. The real roots are given in Theorem 17.8 and the multiplic-
ities of the imaginary roots in Corollaries 18.10 and 18.15. The right-hand
side of the identity looks the same as before - the only change being that
the appropriate lattice M* must be taken in each case. The appropriate lattice
was described in Remark 17.34.
Theorem 20.4 (Macdonald’s identity for twisted affine Kac-Moody algebras).
(a)	The left-hand side of the identity is given as follows.
в\ П (1 - Л' (1 - П (1 - J П (1 - <72"e-J
n>0	аеФ°	аеФ°
C\ П (1 - Л' 0 - ^2" ') П (1 - J П (1 - <?2ле-J
n>0	аеФ°	аеФ°
p\ П (1 - л4 (1 - 72" 1 )2 П (1 - J П (1 - <?2ле-J
n>0	аеФ°	аеФ°
П (1-<7Л^) П (1-<73л^)
аеФ°	аеФ°
с; па-л'па-^)
?7>0	аефО
(l	(1 -q2ne__а)
п
аефО
С1”?") П ('-74 ' i/2-j (I-72"^ J
аеФ0
(b)	The right-hand side of the identity is
(hva)/2h'
aeM*
20.1 Macdonald’s identities
489
where
M* =
+	for types B\, С/, F^, &2
i=l
+ |Za; + 12 ^ai for type C'f
a, long	a, short
|Zai	for type Af
We now give some examples to illustrate Macdonald’s identity. Suppose
first that L has type AP Then L° has type Ф° = {а1, —aj, /f = 2 and
M* = Za,. Moreover (a,, a A = 2— =2 and p° =	. Let z = e_„ . The left-
hand side of Macdonald’s identity is
n(i-<f)(i-<A)(i
77>O
Z? ___ Z?	у — П _ „7/+1
Now x° (nai) = —2-----~ ”+1	. We also have
1-Z
c(2naf) = ((2и+|)а1, (2и +
= 4и(2и+ 1).
Thus the right-hand side of Macdonald’s identity is
7 — ln _ 7ln + l
Ei______//(2/7+1)
1-Z	4
ne% 1	4
This can be written in the convenient form
1	/ -2/i n(2n+l) _2//+1 //(2/7+1)\	/___I \m m m(m— 1)/2
1	v	£	7 j— 1	, \ Ч Z 4
Multiplying both sides of the identity by 1 — z we obtain:
Proposition 20.5 (Macdonald’s identity for type A1).
П (1 - +) (1 - +-1z) (1 - q^-1) = £
/7>0	meZ
This is a classical identity known as Jacobi’s triple product identity.
490
Representations of affine Kac-Moody algebras
As a second example we suppose L has type Af Then L° has type and
Ф0 = {cej, — cej} as before, but we now have hv = 3 and M' = jZcEp We have
a0 = 2,	= 1, c0 = 1, cx = 2, thus
2c
(ce1,ce1) =— =4 and 8 = 2cE04-CEp
We write e_ao=z. Then e_ai=z~2q. The left-hand side of Macdonald’s
identity is
П (1 -'?") (1 -	(1 -	(1 - <72n+V2) (1 - <?2"-V).
л>0
We have
c (Jnay) = (|и2+ |и) (<*1, <*1) = 9и2 + 6и.
Also
X°(lnai) =
1 — z.2q
Thus the right-hand side of the identity is
E
1 -Z 2q
и(Зи+2)
<7 2
1
1 — z.2q
1
1 — z.2q
1	/ Зл	-Зл+1\
= -——L.d	^ )q 2 •
We multiply both sides of the identity by 1 — z 2q and obtain
Proposition 20.6 (Macdonald’s identity for type A\).
П (1 - <7") (1 - q^-1) (1 - q^z) (1 - qln-lZ-2) (1 -
n>Q
= Е(г3л-^3л+1)Е^-
ne%
This is also a classical identity known as the quintuple product identity.
20.2 Specialisations of Macdonald’s identities
491
20.2 Specialisations of Macdonald’s identities
We can obtain some striking identities, simpler than the original Macdonald
identities, by specialising the latter identities in various ways. One way of
specialising is simply to replace ea by 1 for all a e Ф0. When this is done the
expression у (A) is replaced by d°(Xf where
om = П«е(Ф°)+ (A + p°,a)
} ПМфО)+ (A ’
This is shown in Theorem 12.19. The identities obtained by specialisation in
this way involve Euler’s «^-function
Ф(<1) = C1 - ?) (i - <?2) (1 - ?3) • • •
If we specialise the identity of Theorem 20.3 we obtain the following.
Theorem 20.7 (Macdonald’s ф-function identity).
ф(ч)&шЬ° = £ d° (h”a) .
aeM*
Proof. The left-hand side of the specialised identity is
ф^)1+^\ =ф^)й™ь°.
On the right-hand side (ITa) specialises to d° (IPa).	□
We give some examples of this «^-function identity.
Type Ax
№)3=E (4их + 1) <?"1(2"1+1).
Type A2
Ф(.О)8 = E	|(6их-Зи2 + 1)(-Зих + 6и2 + 1)(Зих + Зи2 + 2)
3wj— Зщг^+Зп^+щ +П2
x q
Type C2
ф(ц)10 = E (12nx — 6и2+ 1) (—6их + 6и2+ 1) (2и2+ 1) (Зих + 1)
(nj ,n2)eZ2
492
Representations of affine Kac-Moody algebras
Type G2
(f>(q)u = 22 (8их — 12и2 +1) (—12их+24и2+1) (Зих — Зи2 +1)
x (12и2 + 5) (—2их + 6и2+ 1) (4их + 3)
We next specialise the identities of Theorem 20.4 for twisted affine
Kac-Moody algebras.
Theorem 20.8 (Macdonald’s twisted ф-function identities).
(a)	The left-hand side of the identity is given as follows.
B\	ф(дГг-^ф^М
C\	f(q)2,+lf(q2)21
F)	Ф(Ф)26Ф (?2)26
G2	f(qff(q3)2
C't	ф(<к} f(q)2j2-3lf(q2)21
A)	f (q^ f(q)-lf(q2)2
(b)	The right-hand side of the identity is
22 (/iva) ?c(/!V“)/MV
aeM*
where M* is as in Theorem 20.4 (b).	□
We give some examples of twisted «^-function identities.
Type
ф p)2 ф^ф (q2)2 = £ (3W1 +1) q^+2f
щ eZ
Type C‘
ф(^5ф(^)5 = 22 5 (8и1-4и2+1)(-8и1 + 8и2+1)
(п^лг)^2
x (8?г1 + 3) (2m2 Н- 1)
20.2 Specialisations of Macdonald’s identities
493
Type q
/ i \4
Type G‘
<^(?)7^(?3)7
'2<И^2)4= E KIOmj-S^+OC-Smj+S^+I)
(лтА лг)1^2
x (5и2 + 3) (5их+2)
5n2—5nin2+^ri2+ni+n2
£	(12их — 18и2+1) (—6их + 12и2+1)
x (—3nr + 9^-2+ 2) (6nx +5) (3nx — 3n2 + 1)
X (2и I 1) q^n2l — 18nln2 + 18n2+nl+3n2
Another possibility to obtain specialised identities in one variable from
Macdonald’s identity is to apply a homomorphism
between rings of formal power series, given by
в (e_ J = q\ в (e_ai) = q\ ... , в (e_J = qs‘
where s0, sr,... , S[ are non-negative integers. Of course under such a spe-
cialisation e_s would be mapped to дао5о+--+ад, so t|lat q wou|d have to be
replaced by this power of q in our earlier description of Macdonald’s identity.
For example in type Ar we obtain the following.
Proposition 20.9 (Macdonald’s 1-variable identity for AJ.
П Q _	Q _ qSo<n-i)+sirij q _ qSon+sPn-i)^
. .	m(m—1) , m(m-\-\A
=	(-i)m?so-4-L+si -4-1.
meZ
We mention some explicit examples of this identity. If (s0, sf) = (1, 1) we
obtain
Ф(д)2
Ф(ч2)
= E(-W
that is
(1 - <?)2 (1 — <?2) (1 -,3)2 (1-0(1 -q5)2 (l-<?6) •••
= l-2? + 2?4-2?9 + 2?16------------.
494
Representations of affine Kac-Moody algebras
This is a classical formula of Gauss.
Next consider the example given by (s0, s-J = (2, 1). Then we obtain
. .	m(3m—1)
meZ
that is
(1-^(1-^) (1_^)(W)(W)(W)...
= 1 - q - q2 + q5 + q1 - qn - qi5 + q22 + q26-.
This is a well known formula of Euler.
Many additional formulae can be obtained by taking different values of
(«о, or different affine Kac-Moody algebras.
20.3 Irreducible modules for affine algebras
We next consider the weights of the irreducible modules /.(A),AeX, for
the affine Kac-Moody algebra /.(A). We recall that X is the set of A e//
with A (h) e Z for i = 0,1,... , / and X+ is the set of A e X with A (ht) > 0
for i = 0, 1,... ,
It is convenient to introduce the fundamental weights w0, Wj,... , <y;. <y;
is the element of X+ defined by
ai(hj)=8ij	oti(d) = Q.
Since the imaginary root 3 satisfies
<5(/iJ=0	8(d) = 1
we see that w0, Wj,... , <y;, <5 form a basis of H*.
If XeH* satisfies
= £owo + £iwi 4---b + c8
then A lies in X if and only if e Z for i = 0, 1,... , I. i: can be any element
of C. Also A e X+ if and only if e Z and > 0 for i = 0, 1,... , I.
Now every weight p of L(A) has form p = A — moao — m1a1-----miai
for certain m;eZ with m;>0. Since a;(c) = 0 for z = 0,1,, Z we have
p(c) = X(c). Thus all the weights p of L(A) have the same value of p(c).
Since c = c0/i0 + c1/i1H-hc;/i; we have A(c) = J2;=o c;A (h) and so, for
A e X+, A(c) is a non-negative integer. The integer A(c) is called the level of
the module L(A).
20.3 Irreducible modules for affine algebras
495
Proposition 20.10 If L(X) has level 0 then \ = £8 for some £eC and
dimL(A) = 1.
Proof. If A(c) = 0 then A (/i;) = 0 for i = 0, 1,... , /. Writing A = “I-h
£;<y; + £8 we see that £, = 0 for i = 0, 1,... , I, hence A = £8. Kac’ character
formula then shows that chL(£<5) = e^s. Thus dimL(£<5) =1.	□
Since the modules L(A) of level 0 are trivial 1-dimensional modules we
shall subsequently concentrate on modules L(A),AeX+, of level greater
than 0.
If g. is a weight of L(A) then so is w(/z) for any w e W, by Proposition 19.13.
We take a weW for which the height of A — w(ji) is minimal and put
v = w(/z). Since sfv) = v — v (/i;) a; the minimality of the height shows that
v {hjf > 0. Thus for any weight /z of L(A) there exists w e W with w(/z) e X+.
We now show that the converse is true also.
Theorem 20.11 Let A e X+ have A(c) > 0. Then gceX is a weight of L(X) if
and only if there exists w e W such that w(ji) e X+ and w(ji) -< A.
Proof. It will be sufficient to show that if g. e X with g. -< A then g. is a
weight of L(A). The proof of this is non-trivial and reminiscent of that of
Proposition 16.23.
Let gc = X — a where a = 'ffkiai and each A;>0. We may assume A;>0
for some i. supp a is the set of i for which kt > 0. We first show that every
connected component of supp a contains an i with A (/i;) > 0. Suppose if
possible there exists a connected component 5 of supp a with A(/i;) = 0 for
all i e 5. We have
L(A)gC II (AT)_a
where ил is a highest weight vector of L(A) and, by the PBW basis theorem,
II (/V ) is spanned by elements of the form
П ^4
j3e®+
where kp >0, ^кр/3 = a, and each /3 involves fundamental roots which all
lie in the same connected component of supp a. (We recall from Proposi-
tion 16.21 that supp/3 is connected.) Now the e_(3 with fundamental roots
in different connected components of supp a commute with one another, so
we may bring the e_(3 with fundamental roots in 5 to the right of the above
product. But for such /3 we have e_^vK = 0.
496
Representations of affine Kac-Moody algebras
For /;ил = 0 for each ieS by Theorem 19.19, since A(/z;) = O. It follows
that
H(AT)_avA = O
and so L(A)M = O, a contradiction. Hence there exists i e 5 with A (/i;) > 0.
Now let Ф be defined by
ф = {у e (J+ ; у < a, A — у is a weight of L(A)}.
The set Ф is finite. Let /3 e Ф be an element of maximal height. Then [3<a.
We aim to show that /3 = a and hence that A —a is a weight of L(A). Let
= with each mt > 0. We have a = £ with mi < к for each i.
Let I = {0, 1,... , /} and J be the subset of I given by J = {i e I ; kt = mJ.
We aim to show that J = I and so that /3 = a. Suppose if possible that
J f I. Consider the non-empty subset of I given by supp a — (supp а П J).
This set splits into connected components. Let M be a connected component
of supp a — (supp аП J). Let ie.M. Then A —/3 is a weight of L(A) but
A — /З — cti is not. Thus (A — /3) (/ij < 0. Also pc (/ij > 0 since pc e X+ and so
(A — a) (/z;) > 0. Thus we have
a (ht) < A (/ij < /3 (/ij .
Let у = (кj — m) Uj. We have kj — mj > 0 for all j e M. We also have
t(M = E (kj~mj)Aij-
j&M
However, у (/z;) = (a —/3) (/z;) since supp(a — /3) = supp a — J and M is a
connected component of supp a — J. Thus у (/ij < 0 for each i e M.
Let AM be the principal minor (Aj) for z, j e M. Let и be the column vector
with entries kt — mt for ieM. Then we have и>0 and Awzz<0. If M has
finite type AM(—u) > 0 would imply — и > 0 or — и = 0. Thus M does not have
finite type. Since M is a subset of I which has affine type we must have M = I
by Lemma 15.13. Thus supp a = I and J = ф. But then, for all i e I, A — /3 is a
weight of L(A) but A — /3 — a; is not. Thus (A — /3) (/z;) < 0 for all i e I. Hence
a (hj) < A (hj) < /3 (hj) for all i e I. We now have и > 0 and Au < 0. Since A
is affine we can deduce Аи = 0. This shows that a (hj) = /3 (hj) for all iel.
Hence a (hj) = A (hj) for all i e I, that is pc (hj) = 0 for each i. But then we
have pc(c) = 0, and so A(c) = 0, a contradiction.	□
Corollary 20.12 If pc is a weight of L(X) then pc —8 is also a weight.
20.3 Irreducible modules for affine algebras
497
Proof Since ft is a weight there exists w eW such that w(jP)eX+.
Then w(gi — 3) = w(ji) — 3 e X+. Since w(ji) — 8<X it follows from Theo-
rem 20.11 that w(ji) — 8 is a weight of L(A). Hence /x — S is also a weight.
□
It follows from this corollary that fx — i8 is a weight for all positive
integers i. On the other hand there exist only finitely many positive integers
i such that /z + i8 -< A.
Definition A weight p. of L(X) is called a maximal weight iffi. + 8 is not a
weight.
Corollary 20.13 For each weight p. of L(X) there are a unique maximal
weight v and a unique non-negative integer i such that g. = v — i8.
Proof. Consider the sequence p., p. + 8, p. + 28,... There exists i such that
gc + iS is a weight of L(A) but /z +(i+1)<5 is not a weight. Let v = gc + i8.
Then v is a maximal weight of L(A) and p. = v — i8.
If gc = v' — i'8 where v' is a maximal weight and i' a non-negative integer we
show v = v' and i = i'. Otherwise we may assume i < i'. Then v' = v + (i' — i) 3
is a weight. By Corollary 20.12 v + 8 is also a weight. Thus v is not a maximal
weight and we have a contradiction.	□
A string of weights of L(A) is a set
v, v — 8, v — 28,...
where v is a maximal weight. Each weight lies in a unique string of weights.
Thus it is natural to consider the set of maximal weights of L(A).
Proposition 20.14 The set of maximal weights of L(X), A e X+, is invariant
under the Weyl group.
Proof. Let w e W. Then gc is a weight if and only if w(/z) is a weight. Thus
if gc is a maximal weight w(/z) is a weight but w(jjl) + 8 = w(gc + S) is not a
weight. Thus w(/z) is a maximal weight.	□
Corollary 20.15 Each maximal weight of L(X), A e X, has form w(p.) where
weW and gc is a dominant maximal weight.
We shall therefore consider the set of dominant maximal weights of L(A).
We shall show that L(A) has only finitely many dominant maximal weights.
498
Representations of affine Kac-Moody algebras
We recall from Section 17.3 that the fundamental alcove A* c (//'') was
defined by
A* = Me (//'') ; A(7i;)>0 for i= 1,... , I ;	A(/i0)< —I
I	ao J
= {A e (//'') ; (A,a;)>0 for i= 1,... , I ;	<A, 0><1}.
Its closure A* is a fundamental region for the action of W on (77j|)*.
We also recall that
Я* = (Я°)*®(Су + С8)
where (H0)* is embedded in H* by assuming A(c) = 0, A(<7) = 0 for A e (//").
By Lemma 20.1 we have, for A e 77*,
A = А° + А(с)у + ац1А((7)3
where A0 e (77°)*. Let Q° a (77°)* be the set of A0 given by A in the root
lattice 2 c 77*.
Proposition 20.16 Let A e X+ have level A(c) = k > 0. Then the map p.^ p. '
gives a bijection between the set of dominant maximal weights of L(X) and
(x° + Q°)rjkAL
Proof Let p be a dominant maximal weight of L(A). Then /z = A —
moao-------miai for certain m;eZ with m;>0. Hence /х° = А° —
(moao 4----h m;a;)° and so p° e A0 + Q°.
Now /z = p° + ky + aj ' p(d)8. Since p e X+ we have p (hj) > 0 for i = 0,
1,... , /. Now у (hj) = 8 (hj) = 0 for i = 1,... , I and so p° (hj)>0 for i =
1,... , /. We also have
(m°> 0) =	= {p, $ ~ aoao) = p(cj — p (hf) = k-p (hf).
Since p (hf) > 0 we have (p°, в] < к. Thus p° ekA*.
Hence p^ p° maps dominant maximal weights of L(A) into (A° + 2°)n
кА*. We wish to show this map is bijective. We first show it is surjective.
Let v e (A° + <2°) ПАА*. Then, since a° = at for i = 1,... , I and
cf = (—ajfd + ajfS)0 = —ajfO
we have
v = A0 + kx a1 -I-hA;a; — koao 'd
20.3 Irreducible modules for affine algebras
499
for certain k0,kr, ... ,kt e Z. Since Q = a1a1-\-\~alal we have
v = A0 + (m — коа^в — (ma1 — kf) ax------ — (тц — kt) a;.
We choose m E Z with m > kfai for i = 0, 1,... , I. Then
r, = A0 + m0 (a^O) -my,-----------m/O!/
where mt = mat — kt for i = 0, 1,... , I. Thus the mt are non-negative integers
for i = 0, 1,... , Let gL = A — moao----пцщ. Then
fi° = A0 + m0 (a® 10) — mx cy —  -mgy = v.
We show that /z e X+. We have /z (/i;) = /z° (/i;) = v (hi) >0 for i = 1,... ,
Also /z (h0) = k — (/z°, (f) = k — (v, 0} > 0. Hence /z e X+ and /z -< A. Thus /z
is a dominant weight of L(A) by Theorem 20.11. Hence we have shown that
v = /jf for some dominant weight /z of L(A). By replacing /z by the maximal
weight in the chain of weights containing /z we may assume that /z is a
dominant maximal weight. Thus our map is surjective.
To show the map is injective let /z, gJ be dominant maximal weights of
L(A) with gf = (gd)°. We have
g = g.° + ky + aQ1g(d)8
g.’= (g.')° + ky + a^1 gd (d)8
hence g. — gd = a®1 (gc(d) — gd(d)) 8. Now A — g-EQ and A — gdEQ hence
g — g! eQ and <2q 1 (g.(d) — g!(d)) 8 eQ. This shows that a®1 (gc(d) — gd (d)) E
Z. Thus g. = g.' + r8 for some reZ. Since /z, g! are both maximal weights
we must have r = 0. Thus g! = /z.	□
Corollary 20.17 The set of dominant maximal weights of L(X), XeX+, is
finite.
Proof. Q° is a lattice in (//"), that is a free abelian subgroup whose rank is
the dimension of (//") . A0 + Q° is a coset of this lattice. On the other hand
the set kA* is bounded. Hence the intersection (A0 + О") П kA* must be finite.
Thus the set of dominant maximal weights is also finite, by Proposition 20.16.
□
We now have a procedure for describing all weights of L(A), A e X. First
determine the finite set (A'' +O' j ПАЛ where k = X(c). For each element v
in this finite set there is a unique dominant maximal weight g. of L(A) with
g = v. This gives the set of all dominant maximal weights. By applying
elements of the Weyl group to these we obtain all maximal weights. Finally
500
Representations of affine Kac-Moody algebras
by subtracting positive integral multiples of 3 from the maximal weights we
obtain all weights of L(A).
We next consider the weights in a string
/z, p, — 8, /z — 28,...
We wish to show that the multiplicities of these weights form an increasing
function as we move down the string, i.e. that /и (;+1jS > for all i > 0.
In order to do this we consider L(A) as а Г-module where T is the subalgebra
of L(A) given by
T = • • • ф L-2g Ф L_3 ф H ф Lg ф Ф • • • .
Thus T is spanned by H and the root spaces for the imaginary roots. The
algebra T has a triangular decomposition
T=T ®H®T
where 7' = ZZ, .,, A T+ = 'y>0LiS. One can define the category 0 of T-
modules in a manner analogous to that in Section 19.1. One can also define
Verma modules for T. If A e II we define
M(A) = П(Т)/П(Т)Г+ + П(Г)(х- A(x)).
xeH
This is the Verma module for T with highest weight A. There is a bijection
II (7) M(A) given by и i.im,. where mA e Л/f A ) is the image of 1 e 1I(T).
We shall investigate properties of Verma modules for T by considering the
expression
ГК = 2УУ e(j\e^
i>0 j
where e^ is a basis for Li3 and e^s is the dual basis for L _i3. Thus
and \ ef3,	= 8jkic by Corollary 16.5.
Although the expression for fl0 is an infinite sum the action of fl0 on any
T-module in category О is well defined, since all but a finite number of the
terms will act as zero.
Lemma 20.18 Let XeH* and Л/f A ) be the associated Verma module for T.
Let и e II(T)m3 where m e Z and m^XQ. Then Пои — иП0 acts on M(A) in the
same way as —2А(с)ти.
20.3 Irreducible modules for affine algebras
501
Proof, и is a linear combination of products of elements, each in Tr3 for some
r with r MO.
First suppose ueTr3. We assume that и is one of the basis elements и =
Then и commutes with all e®, e^i3 except for e^rS. Thus
Он-иП -2(eJ) ej) eJ) - ej) eJ) ejA
i.iou ui.i0 — z i e_rSerS er3 er3 e_r3er3 i
= — 2rcef = —2ruc = —2rX(c)u
on M(A). The same will then apply to any и eTr3.
Next suppose u = u1u2 where
IIqMj — MjIIq = — 2X(c)r1u1
П0и2 — и2П0 = —2X(c)r2u2 on M(A)
Then
PLqU uPLq — £l0u1u2 u^u2VLq
= urQ.0u2 — 2X(c)r1u — u1Xl0u2 — 2X(c)r2u
= — 2A(c) (rx + r2) и	on .W( A).
The required result then follows for arbitrary ueU(T)m3 by taking linear
combinations of such repeated products.	□
Proposition 20.19 Let XeH* satisfy A(c) > 0. Then the Verma module M(X)
for T is irreducible.
Proof. Suppose if possible that M(A) has a proper submodule K. Let и be a
highest weight vector of К. Then v e M(X\_m3 for some m e Z with m > 0.
Thus v = um2 for some и e II (T ) m3- We consider the actions
<>,, : M(A)^M(A) и : M(A)^M(A).
By Lemma 20.18 we have
(Пои — иП0) mA = 2А(с)титл.
Thus Vlov — u (HomA) = 2A(c)mu. Now HomA = O and Hou = O since mA and
v are highest weight vectors in M(A) and К respectively. Thus 2A(c)mu = 0.
But и M0, m > 0, A(c) > 0 and so we have a contradiction. Thus M(A) is
irreducible.	□
We now consider the structure of L(A) as a Г-module.
502
Representations of affine Kac-Moody algebras
Proposition 20.20 Suppose XeX+ with A(c)>0. Then the T-module L(X)
is completely reducible. Its irreducible components are Verma modules
for T.
Proof Let U be the subspace of L(A) given by
C/={veL(A) ; Г+и = 0}.
Let В be a basis of U. We may choose В to be a basis of weight vectors
of U, i.e. so that each element of В lies in a weight space Г(А)М. Suppose
vEli has weight /z. Then Г+и = 0 and xu = /z(x)u for xeH, hence Tv =
Tv. Let M(/z) be the Verma module for T with highest weight /z. Then
we have a homomorphism of Г-modules	Tv given by uv
for ueT . Now /z = A — i8 for some i > 0 hence /x(c) = A(c) > 0. Thus the
Verma module M(p.) for T is irreducible by Proposition 20.19. Hence the
homomorphism M(/z) Tv is an isomorphism and so Tv is a Verma module
for Г. Let V = Tv. We claim that this sum of Г-modules is a direct sum.
For consider
ГиП Tv'.
v'eB
v'j^v
Since the Verma module Tv is irreducible we have TvC\U = Cv. We also
have
/ \
^Tv' nU=£Cv’.
v'eB	v'^v
xy'^v /
Since и ^-’v' we see ?v 's not contained in Tv'. Again, since
Tv is irreducible we have ГиПЕ1//и Tv' = O. Hence V = Q)veBTv. Thus V is
a direct sum of Verma modules for Г.
We wish to show that V = L(A). We suppose if possible that V^L(A).
We consider the Г-module L(A)/V. Since L(A) = ®/zr(A)/z and V = ^/zVlz
we have L(A)/V = ф/х(Г(А)/У)/х. As L(A)/V is assumed to be non-zero we
can find a weight p. of L(A)/ V such that p. + i8 is not a weight for any i > 0.
Then Г+(Г(А)/V)M = O, that is Г+Г(А)М с V.
We now consider the map 12,, : L(A) L(K). Since the action of 120
preserves weight spaces we have 12,, : L(A)M L(A)M. The weight space
20.3 Irreducible modules for affine algebras
503
L(A)M is finite dimensional, so decomposes into a direct sum of generalised
eigenspaces of fl0, given by
ЦА)^ = Ф(ЦА)Д
^eC
where (fl0 — £l/ = 0 on (Г(А)М)^ for some k. Since Г(А)М does not lie
in V there exists feC such that (L(A)M)^ does not lie in V. We choose
v e (L(A)g) with v^V. Then
(H0-^l)% = 0
and Hov e V, since Г+Г(А)М с V. If f ф0 the polynomials (t — g)k and t are
coprime so we could deduce v e V, a contradiction. Hence f = 0 and Tlkv = 0.
Now Г+и^0 since vgV. So there exist m>0 and ие!1(Г+)т3 with
uvф0 and Г+(ии) = 0. Let v' = uv. Then v'ф0 and floif = 0.
Now all the weights v of L(A) satisfy r(c) = A(c). Thus we may apply the
argument of Lemma 20.18 to L(A) and obtain
Пои — иП0 = —2А(с)ти onL(A).
Then
Поии — a(low= — 2X(c)muv
that is
(fl0 + 2A(c)m) v' = u£lov.
It follows that
(Ho + 2A(c)m)2 v' = (H0 + 2A(c)m) (allow) = и (flgu)
and continuing thus we obtain
(H0 + 2A(c)m)Z: v' = и (flgv) =0.
But A(c) > 0 and m > 0, thus the polynomials (t + 2A(c)m)Z: and t are coprime.
Thus (H0 + 2A(c)m)Z: v' = 0 and flou' = O imply i/ = 0, a contradiction.
Thus we have obtained our required contradiction and can deduce that
V = L(A) and L(A) is the direct sum of the irreducible Г-modules Tv for
v e B, each of which is isomorphic to a Verma module for T.	□
Proposition 20.21 Let p. be a weight of L(Xf where AeX and A(c)>0.
Then the multiplicities of the weights pc, pc —8 satisfy m s > m .
504
Representations of affine Kac-Moody algebras
Proof. This follows from Proposition 20.20. We choose a non-zero element
%eL(A)_s. Consider the action of x on the Г-module L(A). This Г-module
is a direct sum of Verma modules for Г. Since x e T~, x acts on each Verma
module for Г injectively. Thus x acts on L(A) injectively. We have a map
V^XV
which is injective, and so
dimL(A)M_s >dimL(A)M
that is т^_8 > as required.	□
Thus the multiplicities form an increasing sequence as we move down a
string of weights for L(A).
20.4 The fundamental modules for L (A J
We now give an example of the situation described in Section 20.3. We
consider the affine Kac-Moody algebra of type AP This has diagram
and Cartan matrix
We consider the irreducible modules L (<y0) , L (w, ) where w(l, w, are the
fundamental weights. By symmetry we need only determine the character of
one of these. We shall consider the module L (<y0).
In type Ax we have 8 = a0 + aq and c = h0 + h1, that is
a0 = l, cq = l, c0=l, c1 = 1.
We recall that
Я* = (Я°)*®(Су + С8)
and that
A = A'' +Afcjy + a,,1 Aft/)?) by Lemma 20.1.
The root lattice Q is given by
<2 = Za0 + Zoq
20.4 The fundamental modules for L (Л,)	505
and we have o^ = — ar, a° = a1. We have (№)* = Ca1 and the lattice
Q° С (Н°У is given by Q° = Ztf . We have в = аг and hg = j- (c — hf) = hr.
The closure of the fundamental alcove is given by
А* = {Ле(Я°)* ; A(^) >0, A (/i0) < 1}
= {Ae«)* ; 0<A(^)<l}.
We have <y0 = У and y° = 0. Thus
(y° + 2°) ПА” = {mar ; meZ,0<2m<l}
= {0}-
Thus by Proposition 20.16 the module L(y) has only one dominant maximal
weight, which must be the highest weight y. The other maximal weights are
the transforms of у under the affine Weyl group W = (s0, sr). We have
•?o (“о) =-“о	•?о(“1) = 2“о + “1
•^i (“о) = “o + 2“i	•Ji(“i) = -“i-
The action of s0, sr on the basis y,a1,8 of H* is given by
•?о('У) = 'У + “1-'5	So (“i) = - “i +25	s0(<5) = <5
Si(7) = 7	s1(a1) = -a1	s1(5) = 5.
The affine Weyl group W is an infinite dihedral group and has a semidirect
product decomposition
W = t(2°) W0 = W0t(20)
where Q° = M* = Za^ and И'" = {1, sj. The translation t for fi e Q° is given
by
(A) = A + A(c)^ - ({A, fi) +1 (fi, fi)X(c)^ 8
which in the present case gives
1та1{У) = У + та1-т28
tmai (a1) = a1-2m8
tmai (<5) =
for m e Z. The stabiliser of у in W is И'" and the maximal weights in L(y)
have the form y + mar — m28 for m eZ. The set of all weights of L(y) is
y + ma1—m28 — k8 for m e Z, k e Z and k > 0. The weights у + ma1 — m28
506
Representations of affine Kac-Moody algebras
Figure 20.1 Maximal weights in Lfy)
have multiplicity 1, and y + ma1 — m28 — k8 has multiplicity depending only
on k (i.e. independent of m). The weights are shown in Figure 20.1.
We shall determine the multiplicities of these weights. We use Kac’ char-
acter formula
. r / ч ZweW £(w)eiu(-y+p)-p
chLM= П..Ф.(1-„)- 
Now
E s(w')ew(y+p)-p = E E s (w°) ew0tyy+p)-P
a>eW	wQeWQ /zeZcq
= E £(w°)Eewo(rai(7+p)-p-
w°eW°	ne%
Now p = p° + 2y by Lemma 20.2 where р°=|а1. Thus p = 2y+|a1 and
y + p = 3y+ jay. Hence
tnai (t + p) = 3? + (зи + j) ay - (Зи2 + и) 5
so tnai (У + P) - P = У + 3nai — (Зи2 + n) 8. Also
sfnai (? + P) = Зу - (Зи + |)ay - (Зи2 + и) 5
so \|/„,b| (у+ p)-p = у - (Зи + l )tt, - (Зи2+ «) b. Thus
. ^'(^,)^ш(у+р)—p	У . (fSnay	(3n+l)al') ^—(3n2+n)d‘
weW	neZ
We write e_a = z and e_s = q^2. Then our expression is
^E(r3"-?+1)?"(M1)/2.
neZ
20.4 The fundamental modules for L (Л,)	507
Now we may factorise this expression by using Macdonald’s identity for type
Af By Proposition 20.6 it is equal to
ey П О - <7") (1 -	(1 - qn-h) (1 -	(1 -
n>0
= e/l - z) П (1 - Л (1 -	(1 - 7”z) (1 - 72n~V2) (1 - q2n-^]
n>0
= еу(!-z) П 0 - q") 0 - q"7--1) - q^z^ (i - qnz) (i - ^z)
n>0
x (1 + q~z -1) (l + ^z)
= £y(i -z) n (1 - qkl2z-^ (1 - qk/2z) n (1 - 7й)
k>0	n>0
(-	2n-l _I \ / ч 2и-1 \
1 + q 2 z 4 (1 + q 2 zj .
We now make use of Macdonald’s identity for type AP By Proposition 20.5
this asserts that
П (1 - 7") (1 - q^z) (1 - qnz'-1) = Е(-1)Л (zn -z'4-1’) q*^ .
n>0	n>0
Putting z' = -z '</- we obtain
П (1-7”) (1 + 7VZ-X) (l + <7Vz) = E (z-nqn2/2 + zn-1q(n-1'>2/2')
n>Q	n>Q
= £Г//2.
ne%
Hence
1^ r / \ _	^Zir lV	p
C W “ (1 - z) IL>0 (1 - qk/2Z-^ (1 - qk/2z) (1 - qkn
gy^LnezZ q /	'2_.n yAy n^l d-o
= ГЪ>о(1-7*/2) = П,>о(1-е-Ы)’
Now
77---71------7 = П (1 + e-k8 + e-2k8 “I-)
1 U>0 U e-k8) k>o
= Y,P(.k)e-k8
k>0
where p(k) is the number of partitions of к. Thus
chL(y) = EEpW ^y-f-na1 — n2- S—k8 •
neZ k>Q
508
Representations of affine Kac-Moody algebras
Hence we have proved
Proposition 20.22 The weights of the fundamental module L(y) for L(A^
are y + na1—n28 — k8 for neZ and k>0. This weight has multipli-
city p(k~).	□
We note in particular that all the maximal weights у + пах—п28 have
multiplicity 1 and that the multiplicity of the weight p — k8 in the string with
maximal weight p depends only upon k and not on p.
20.5 The basic representation
The module L (<y0) for an affine Kac-Moody algebra L(A) gives the so-called
basic representation of I.(A). Since w,, = y we have described the character
of the basic representation of	We shall state without proof some
generalisations of this character formula to other types of affine Kac-Moody
algebras. For simplicity we shall concentrate on those of types Ah Dl and If.
Theorem 20.23 The basic representation I.(y) for the Kac-Moody algebra
IfA) of types A[, D^E^If, Es has the following properties:
(a)	у is the unique dominant maximal weight of If y).
(b)	The set of all maximal weights is
+	for^eg0}.
(c)	The set of all weights is
\y + p — ^fp,p}8 — k8 for peQ°, fceZ, £>0}.
(d)	The character of the basic representation is
(\ 1
П (1	)
k>0	/
where q= e ".
(e)	The multiplicity of the weight у + p — ^fp, p)8 — k8 is pfk), the number
of partitions of к into I colours. We have
1
(ГЪ>о(1-0;
k>0
20.5 The basic representation
509
The proof of this theorem can be found in the book of Kac, Infinite-
Dimensional Lie Algebras, third edition, Chapter 12.
We shall also describe without proof how to obtain a realisation of the basic
representation L(y) of L(A) in types Al,Dl,El. We first make some comments
on differential operators. Let R = C [xx, x2, x3,... ] be the polynomial ring
over C in countably many variables and R = C [[xx, x2, x3,... ]] be the ring
of formal power series in these variables. We shall consider differential
operators on R with values in R. An example is the partial derivative d/dxt or,
more generally, the divided power	We also have finite products
П/ ~рр;(д/дх[)т‘ where m = (m3, m2, m3,...) satisfies the conditions that e
Z, >0, and mt > 0 for only finitely many i. We define Dm : R^ R by
^=nA(w-
. mJ.
i i
We also allow such operators combined with multiplication by elements of
R. Thus
УРтОт : R^R
m
is a differential operator, where l'mcR and the sum over m will in general
be infinite. PmDm is a linear map from A to A. In fact each linear map
from R to R has this form, as we now show.
Proposition 20.24 Each linear map from R to R can be written as PmDm
for a unique set of elements Pm e R.
Proof. Let Mm e R be the monomial Mm = П, У'  The monomials Mm form
a basis for R. We have Dm (^Mk~) = Q unless ki > for each i. We write this
condition as к > m. We write к > m if k>m and к 7^ m. We also have
/ k\
Dm(Mk)=l Mk ifk>m
ym /
where O=R<) and (°) = 1.
Let Д: R R be the linear map given by Д (Mm) = Qm e R. We show Д is
uniquely expressible in the form yPmDm. We have
(k \
}мк_т
mJ
= pk + ypm\ }Mk_m.
1	\wt /
m<k	\ /
510
Representations of affine Kac-Moody algebras
The condition we require on the Pm is that
In particular Po = Qo. Assuming inductively that Pm is uniquely determined
for all m < к we conclude that
„	(k\
Pk=Qk~^2 Pm\	IMk_m
,	\m)
m<k	z
is uniquely determined.	□
Thus the set of differential operators from R to R is the set of all linear
maps from R to R.
We shall now consider certain special kinds of differential operators. Let
A = (A,, A2, A3,...) where A; e C. Here there may be infinitely many non-zero
A;. Define Tx: R^ R by
TJ <XpX2,x3, ...)=/(x^ApX^A2,x3 + A3,...).
Гл is clearly a linear map from R to R. It may be written as a differential
operator by using the Taylor expansion. We have
f (х1 + АрХ2 + А2,х3 + А3,...)
Ami A™2 A™3
= E ^7	• • • (д/дХ1Г (д/dxfT (д/дх3Г • • • / (xp x2, x3,...)
m mx\ m2' m3\
= E ( П АГ‘J Dmf (xpX2,x3,...).
Thus TA = f2m (П,А7') D„. The operator Гл may also be written in the fol-
lowing convenient form. We have
f (x3 + Ap x2 +A2, x3 + A3,...)
ml	\ /	\ m2	\
E -4 (d/dxi')mi) IE “E (d/dxi')m2)... / (xx, x2, x3,...)
mA	I\ mA	/
1	/	\ w2 2	/
= exp (Ajd/dxj) exp (A25/5x2) .../ (xp x2, x3,...)
= exp (A15/5x1 + A25/5x2 + ...)/ (xp x2, x3,...) .
Thus Гл = exp A; (5/5x;)).
20.5 The basic representation
511
Lemma 20.25 Suppose D : R^ R is a linear map which satisfies [x;, O] =
A;O for all i, that is
xtDf— D(xif) = XiDf for all f eR.
Then
0 = 0(1) exp
Proof We shall show Df = D(Y) exp (— JT A; (5/5x;))/ for all monomials
f e R, using induction on the degree of f. If f has degree 0 then f = с e C
and we have
/ d \
O(l)exp I -EA/V“ I c = O(l)c = O(c).
\	cxi j
Assuming the result for a monomial f we prove it for xj\ We have
(d \
-E^y f-
i UXi /
On the other hand
(d \
- E A;— (V) = О(1)Г_Л (V) = 0(1) (x; - A;) T_J
i dxi j
(d \
“EAy /•
i X' /
Thus the lemma is proved.	□
Lemma 20.26 Suppose i): R^R is a linear map which satisfies [д/дх^ О] =
p.fffor all i, that is
(d/dXj) (Df) — D (df/dx2) = faDf for all f eR.
Then
O(l) = cexp EmiXj I for some с e C.
512
Representations of affine Kac-Moody algebras
Proof Consider the element exp (E — /z;x;) Df e R. We have
d (
7-	exp IE ~Pixi Df
=	-Pi exp I E ~Pixi I Df + exp | E ~Pixi | 7“ (Df)
\i /	\ i / °Xi
=	exp ^E — Pixi^ (d/dxi — Hi) Df.
Thus by the assumption of the lemma we have
^exp ^E ~Pixi^ D^ df/dXi = exp ^E — РЛ^ (f/^xi ~ Pt) Df
9 (
= ^7 I exp I E ~Pixi Df I •
Write Д = exp (E — /z;x;)Z>. Then we have Д5//5х; = (д/дх;)(Д/) for each
i and f. In particular we may put / = 1 and obtain (5/5х;)(Д(1)) = 0. Thus
Д(1) = c for some с e C. Hence
Z>(1) = exp ^E Ptx^ Д(1) = c exp ^E Pix^
as required.	□
Proposition 20.27 The set of all differential operators D : R^-R satisfy-
ing the conditions [x;, /)| = A;D and [d/dx^ Z>] = p^Dfor A;, p.: e C forms a
1-dimensional vector space with basis
exp (E Pixi) exP E
Proof This follows from Lemmas 20.25 and 20.26.	□
Definition Differential operators D : R^- R of the form
exp (E РЛ) exp E \ УУ)
for A;, e C are called vertex operators.
Now let L = £ (/.") be an affine Kac-Moody algebra of type A;, Z); or E;.
Let
j<o
20.5 The basic representation
513
Then 7 has a basis F ® ht for i = 1,... I and j < 0. Consider the symmetric
algebra 5 (7). This is isomorphic to the polynomial ring over C in the
variables
Let Q° be the subgroup of H° generated by hx,... , /i;. We shall write Q°
multiplicatively, so that its elements have form h™1 ...h™1 with ... , ml e
Z. Let C [<2°] be the group algebra of Q° over C. Elements of C [<2°] have
form
V A hmi hm‘
,mie^
C [2°] is isomorphic to the algebra of Laurent polynomials over C in the
variables hx,... , /i;.
We now form the tensor product
У = 5(Г-)®С[2°].
V	is isomorphic to the algebra
C [hr,... , hh h^1,... , h^1,
for i = 1,... , / and j < 0.
We define certain maps ha(n): VV out of which vertex operators will
be constructed. For н eZ with и>0, ha(n) is the derivation of V uniquely
determined by the conditions
Г" ® ht n {ht, ha)
F ® /l;	0 for j ф —П
hi^Q.
For n e Z with n < 0, ha (n) : V -+ V is multiplication by (t" ® ha) ® 1.
We now consider the expression
/	/ V'	^а(и) —л\
exP / ---------z exp 5-------------z
\„<o n /	\„>o n /
where z is an indeterminate. We first observe that exp	maps
V	into C [z 1 ] ® V. To see this we observe that each element v e V is a finite
linear combination of monomials
я.=ПтГГК
i,j	i
j<0
where m = (mti, m,} with m e Z, mif e Z, mif > 0.
\ V 1 '	1	lJ	lJ
514
Representations of affine Kac-Moody algebras
Let d,(m) = JT j niij. Then the derivation ha(n), и > 0, transforms Mm into
a linear combination of monomials in which d,(m) is decreased by 1 and
П/ h"4 remains unchanged. Thus a succession of t/1(wi) + 1 derivations ha(n)
for various н > 0 annihilates Mm. Also, for a given monomial Mm,ha(n)
annihilates Mm for all but finitely many n > 0. Thus in the expression
exp
ve V
only finitely many terms —	" act on v and only a finite set of products
of such terms can act on v to give a non-zero element. Thus we have
V^C[z ']0V.
We shall modify this operator in the following way. Let
s:2°x2°^{±l}
be the function defined by
E (hh /l;) = — 1
£(/i;,/iJ = l ifA;> = 0
£ (hh hj) is given by [Ea.Ea.] = s (hh hj) Eai+clj if Atj = -1
s(h+h',h") = s(h,h")s(h', h")
e (/i, h' + h") = e (h, h') e (h, h") .
Given a e Ф0 we define a map Ea e End V by
Ea :P®h^P®s (ha, h) h
where P e S (T ), heQ°. We also define e“ e End V by
ea : P ® h P ® hah
and z" e End (C [z, z 1 ] 0 V) by
z" : (P 0 h)z‘ (P 0 h)z‘+<ha’h^
We now define, for a e Ф0, the operator Ya(z) on V by
Ta(z) = exp £-------exp £	eazasa.
\„<o n /	\„>o n /
20.5 The basic representation
515
-------z
\„<o n
We claim that Ya(z) can be written in the form
i'»(z)=Er»Or-1
jeZ
where Га(у) eEnd V. In order to see this we consider the effect of Ya(z) on
a monomial Mm in V. We have
eazaEaMmez"a®V
where na = mi (ha, hi). Thus
exp|
\n>0 П /	k=0
for some К > 0, and
I___ h \
exP E------—z~n eazasaMm
\„>o n /
к
e E Ez"" / ®'/-
K>0k=0
Thus to obtain * ' on the right-hand side of Ya(z)Mm we must have
na — k + k' = — j — 1. For each value of k there is at most one k' > 0 sat-
isfying this. Since only finitely many k arise, only finitely many k' can
arise for given j. This shows that only finitely many terms —	jn
exP (12n<o — "j are involved in Ta(j) and only finitely many products
of such terms are involved. Thus we have
EG) : V^V
and
E(z) = EE(j)E^
jeZ
with Ta(j) e End V.
Now the vector space V can be regarded as an L-module giving the basic
representation of L. In order to describe the L-action on V we introduce some
further notation. We have defined ha(n) e End V for n > 0 and n < 0. We now
define /i(0) e End V for any h e H°. In contrast to the ha (n) for n 0, which
act non-trivially on 5 (7) and trivially on C [<2°], h(G) acts trivially on
.S' (7 ) and non-trivially on C [2°]. We define, for h e H°, /i(0) : V V by
/1(0) :/’0 ha^P® (ha, h) ha
for PeS(T-),aeQ°.
516
Representations of affine Kac-Moody algebras
Let hf ... , h'f be a basis for H° and hf ... , h" be the dual basis satisfying
(hf h'- 'j = 8tj. We define Do e End V by
1 1
Do = E -/да"(0) + E hf-nyffi).
1=1 2	л>1
since, for v eV, h"(n)v = 0 for all but finitely many n > 0. Do lies in End V
and is readily seen to be independent of the choice of basis of H°.
We now have the definitions necessary to describe the action of L on V
which gives the basic representation.
Theorem 20.28 The vector space V/ = .S'(’/' )0C[(2 "] is a module for the
Kac-Moody algebra L(A) of type A;, I) or giving the basic representation
under the following action L(A)^ End V:
tn®Ha^ Ha(n) for аеФ0, weZ
tn®Ea^Va(n) for аеФ0,neZ
c —> ly
d—> —Do.
The proof of this result can be found in the book of Kac, Infinite-Dimensional
Lie Algebras, third edition, Chapter 14.
The highest weight vector of V is the element 1 ® 1. The first 1 is the
unit element of the symmetric algebra 5 (7) and the second 1 is the unit
element of the lattice Q° written multiplicatively, which is the unit element of
the group algebra C [2°]. This vector 1 ® 1 is annihilated by the generators
e;, i = 0, 1,... , / of L(A). To see this we recall that
ei = 10Ei z = 1,... , I e0 = t®E0
with Eo e L°_e. We have
е;(1®1) = Га.(0)(1®1)	i=l,...,l
which is the coefficient of z 1 in Ta.(z)(l ® 1). Also
e0(l ® 1) = Г_0(1)(1 ® 1)
which is the coefficient of z 2 in T_0(z)(l ® 1). Recalling that
/ \	I X-'	— n I l	— n I a a a
ya(z) = exp E------exP E----------------e z £
V<0 n /	V>0 n /
20.5 The basic representation
517
we first note that e“z"£"(l ® 1) = 1 ® ha. Now negative powers of z in
Ya(z)(1 ® 1) can only arise from derivations ha(n) with n>0. However,
/?,,(н)(1 ®ha) = 0 for all n>0 since any derivation annihilates the unit ele-
ment 1 e .S' (7). Thus we have
Га.(0)(1 ® 1) = 0 for z = l,... ,1
Г_0(1)(1®1) = О.
Hence e;(l ® 1) = 0 for all i = 0, 1,... , I.
We now check how the elements h0, h.^... , 7z; eH act on 1 ® 1. We have
hi = \®Hi for i = 1,... , /. Thus
/i.(l ® 1) = Ha.(0)(1 ® 1) = 1 ® (0, h) 1 = 0.
(We note that in the scalar product (,) the elements of Q° are written addi-
tively so that the unit element will be 0.) We also have
c = h0 + c1h1 + • • • + C[hi.
Thus
/z0(l ® 1) = c(l ® 1)= 1® 1.
Hence we have
/i;(l ® 1) = у (h) (1 ® 1) for i = 0, 1,... , I
since у (h) = 0 for i = 1,... , / and у (7z0) = 1. The highest weight vector
vy = 1 ® 1 is often called the vacuum vector of the basic representation.
The realisation of the basic representation given by the module 5 (7) ®
C [2°] is called the homogeneous realisation. It is one of a number of descrip-
tions of the basic representation.
The basic representation is of great importance in a number of applications
of the theory of affine Kac-Moody algebras in mathematics and physics. For
example applications to the theory of differential equations are described in
Kac’ book, Chapter 14. There are also particularly interesting applications in
the area of mathematical physics. Vertex operators arose in the context of
dual resonance models, which subsequently developed into string theory, and
the representation theory of affine Kac-Moody algebras plays a key role in
string theory. This involves the calculus of vertex operators. The theory of
modular forms also plays a key role.
Readers wishing to learn more about the relations between Kac-Moody
algebras and string theory may wish to study the 30-page introduction to the
book of Frenkel, Lepowsky and Meurman, Vertex Operator Algebras and the
518	Representations of affine Kac-Moody algebras
Monster which also explains the connections with modular forms and sporadic
simple groups such as the Monster. The book by Kac on Vertex Algebras for
Beginners is a useful introduction to the calculus of vertex operators. This
whole area relating mathematics and physics is of great current interest and
seems certain to continue its rapid development.
21
Borcherds Lie algebras
21.1 Definition and examples of Borcherds algebras
A theory of generalised Kac-Moody algebras was introduced by R. Borcherds
in 1988. The purpose for which these algebras were introduced was as part
of Borcherds’ proof of the Conway-Norton conjectures on the representa-
tion theory of the Monster simple group, for which Borcherds was awarded
a Fields Medal in 1998. These generalised Kac-Moody algebras are now
frequently called Borcherds algebras. A detailed discussion of Borcherds
algebras, including proofs of all the assertions, is beyond the scope of this
volume. However, we shall include the definition of Borcherds algebras and
the statements of the main results about their structure and representation
theory, but without detailed proofs. In fact many of the results are quite sim-
ilar to those we have already obtained about Kac-Moody algebras. However,
the theory of Borcherds algebras includes examples which are quite different
from Kac-Moody algebras. The best known such example is the Monster Lie
algebra, which we shall describe in Section 21.3.
We begin with the definition of a Borcherds algebra. A Lie algebra L over
IR is called a Borcherds algebra if it satisfies the following four axioms:
(i)	L has a Z-grading
L = ^Lt
such that dim L; is finite for all i 0, and L is diagonalisable with respect
to Lo. (Note that dimL0 need not be finite.)
(ii)	There exists an automorphism a> : L —> L such that
W;=l
a> (L;) = L_; for all i e Z
a>=— 1	on L0/L0AZ(L).
519
520
Borcherds Lie algebras
(iii)	There is an invariant bilinear form
(,) : LxL^IR
such that
U y)=o	if x e L,, у e L; and	i + j # 0
(wx, wy) = (x,y)	for all x, у e L	
— (x, wx) > 0	for .i t/., with	x^O.
(iv)	LOC[LL].
We observe some consequences of these axioms. In the first place it can be
shown that [L0L0] = 0, that is Lo is abelian.
To describe a further consequence we define, for x, у eL,
(x, y)0 = -(x, шу).
The scalar product (, )0 : L x L IR is called the contravariant bilinear form.
We now restrict the contravariant form to one of the graded components L;
with i 0 and have (, )0 : L; x L; IR. Let .i t A;. Then
(x, x)0 = —(x, wx) > 0 if x
Thus the contravariant form is positive definite on each graded component L;
for i 0 (though not necessarily on Lo).
We now give some examples of Borcherds algebras. We begin with a
symmetric matrix ?I over IR which is either finite or countable. Thus
SI=(aiy) h j&I
with a;; e IR and I either finite or countable. We shall assume that the matrix
lJ
?I satisfies the conditions
a;><0 if
if > 0 then 2a, ;/a;; e Z for all j.
Proposition 21.1 There is a Borcherds algebra L associated to a symmetric
matrix ?I satisfying the above conditions which is defined as follows by
generators and relations.
L is generated by elements et, ft, h^
i, j^I
21.1 Definition and examples of Borcherds algebras
521
subject to relations
leifj\ = hH
lhijek\ = 8ijaikek
lhijfk\ = ~8ijaikfk
(ade;)"e^ = O, (ad/;)" fj = O ifa;;>0, ifij and n= 1 — 2а^/аи
[eiey]=0’ [ftfj] =°	ап<0 and aij = 0.
The Borcherds algebra L defined by generators and relations in this way
is called the universal Borcherds algebra associated with the symmetric
matrix ?I. Its structure as a Borcherds algebra can be described as follows. Its
involutary automorphism <y is given by
w(O = -/o ыШ = ~ео a(hij) = -h ji-
lts invariant bilinear form is uniquely determined by the condition
(e;, fi) = 1 for all i e I.
In particular, if we write ht = hu, then [e;/;] = ht and
К h}] = ([ej;], h}} = (eh [fihj]) = (et, а^) = atj.
Thus
(/i;, hj'j = atj for all i, j e I.
There are many ways of defining an appropriate grading on this Borcherds
algebra. For each i e Z let h, e Z satisfy nt > 0. Then there is a Z-grading on
L uniquely determined by the conditions
S; G Ln., fi E L_n..
Further examples of Borcherds algebras can be obtained from a universal
Borcherds algebra as follows. The axiom [/iy/iH]=0 shows that the subal-
gebra generated by all elements h^, i, j E I, is abelian. If i j then lies
in the centre of L since [/iy, ek] =0 and [/iy, Д] =0 for all kEl. Thus the
subalgebra generated by the /iy for i j lies in the centre. It can be shown
that the centre Z of L satisfies
(/ly ; i,jEl,i^j}<zZ<z(hij-, i,jel}.
522
Borcherds Lie algebras
In fact the Jacobi identity
[M] hk] + [[fjhk] ei] + [[M] fj]=0
shows that
It can be shown that, as a consequence of this, hy = 0 unless aki = akj for all
к el, i.e. unless the zth and jth columns of ?I are identical.
Proposition 21.2 Let L be a universal Borcherds algebra and I be an ideal
of L contained in the centre Z of L. Then L/I retains the structure of a
Borcherds algebra.	□
The Z-grading, involutary automorphism and invariant bilinear form on
L/I are readily obtained from those on L.	□
We now obtain still further Borcherds algebras. Starting from a universal
Borcherds algebra L we factor out an ideal I of L contained in the centre Z
of L. Then L/I is still a Borcherds algebra. We write L = L/I.
An inner derivation of L is one of form x[xy] for some yeL, and
an outer derivation is a derivation which either is zero or is not an inner
derivation Let
L*=Hom(L, R)
and A c L* be an abelian Lie algebra of outer derivations of L. We suppose
also that
| v| e Re;, [/;x] e R/;
for all x e A where et, ft are images of e;, ft under the natural homomorphism
L^L.
Proposition 21.3 Let L be a universal Borcherds algebra and I be an ideal
of L contained in the centre Z of L. Let L = L/I. Let A be an abelian Lie
algebra of outer derivations of L and let L +A be the semidirect product of
L by A whose elements have form x + a with x eL, a e A where
[x+a, у + b] = |wv| + a(y) - b(x).
Suppose that
[e,x\ e Re;, [/;x] e R/;
for all x e A. Then L + A retains the structure of a Borcherds algebra in
which A<z(L + A)o.	□
21.1 Definition and examples of Borcherds algebras
523
The Z-grading, involutary automorphism and invariant bilinear form on
L + A are easily obtained from those of L.	□
We have now constructed a family of Borcherds algebras which includes
all universal Borcherds algebras, all quotients of such by ideals contained in
the centre, and all semidirect products of such quotients by an abelian Lie
algebra of outer derivations with suitable properties.
This turns out to give all possible Borcherds algebras, as is shown by the
next theorem.
Theorem 21.4 Let Lbea Borcherds algebra. Then there is a unique universal
Borcherds algebra Lu and a homomorphism
f : LU^L
(not necessarily unique) such that ker f is an ideal in the centre of Lu, im/
is an ideal of L, and L is the semidirect product of im/ with an abelian
Lie algebra of outer derivations lying in the О-graded component of L and
preserving all subspaces IRe; and IR/.
The homomorphism f preserves the grading, involutary automorphism,
and bilinear form.	□
Now that we have obtained the complete set of Borcherds algebras in this
way, we explore their relationship with symmetrisable Kac-Moody algebras.
It turns out that every symmetrisable Kac-Moody algebra over IR gives rise
to a universal Borcherds algebra, which is the subalgebra of the Kac-Moody
algebra obtained by generators and relations prior to the extension of the
Cartan subalgebra by an abelian Lie algebra of outer derivations.
Theorem 21.5 Let L be a symmetrisable Kac-Moody algebra with GCM
A=(Ay). Thus there exists a diagonal matrix D = diag (</,... , dn) with
each dteZ, dt>0 such that DA is symmetric. Let ?I = (ay) be given by
Then we have and ait = dt. Thus <0 if ifi j and ait is a positive
integer. Moreover 2а^/ай = Ay e Z.
Then the symmetric matrix (ay) satisfies the conditions needed to construct
a Borcherds algebra, and the universal Borcherds algebra with symmetric
matrix ?I coincides with the subalgebra of the Kac-Moody algebra L obtained
by generators and relations prior to the adjunction of the abelian Lie algebra
of outer derivations.	□
524
Borcherds Lie algebras
In this way every symmetrisable Kac-Moody algebra determines a cer-
tain subalgebra which is a universal Borcherds algebra. In fact the main
points of difference between symmetrisable Kac-Moody algebras and univer-
sal Borcherds algebras are that, in a Borcherds algebra:
I may be countably infinite rather than finite
ай may not be positive and need not lie in Z
2ay/a;; is only assumed to lie in Z when ай > 0.
21.2 Representations of Borcherds algebras
We now introduce the root system and Weyl group of a Borcherds algebra.
We suppose first that L is a universal Borcherds algebra. The root lattice
Q of L is the free abelian group with basis a; for i e I. We have a symmetric
bilinear form
given by (a;, a J -+	а^ = а^.
The basis elements at of Q are called the fundamental roots. The set of
fundamental roots is denoted by П. We have a grading
L =
«eg
determined by the conditions
An element a e Q is called a root of L if a0 and La^O. a is called a
positive root if a is a sum of fundamental roots. For any root a either a or
—a is positive. We have
ф = ф+ и Ф
where Ф is the set of roots and Ф+, Ф are the subsets of positive and negative
roots respectively. We say that a e Ф is a real root if (a, a) > 0 and a e Ф
is an imaginary root if (a, a) < 0.
We next introduce the Weyl group W of the universal Borcherds algebra L.
W is the group of isometries of the root lattice Q generated by the reflections
Si corresponding to the real fundamental roots. We have
/ x	(“o	aa
Si (a;) = Uj — 2---—cc = aj—2—ai.
J J \ac au J au
We recall that 2ay/a;; e Z since ай > 0.
21.2 Representations of Borcherds algebras
525
Let H be the abelian subalgebra of L generated by the elements for all
i, j e I. We have a map
Q^H
under which at maps to ht, which is a homomorphism of abelian groups
and preserves the scalar product. However, this map need not necessarily be
injective.
So for we have assumed that L is a universal Borcherds algebra. How-
ever, if L is an arbitrary Borcherds algebra there is an associated universal
Borcherds algebra Lu given by Theorem 21.4. Then the root system of L is
defined to be the root system of Lu and the Weyl group of L is defined to be
the Weyl group of Lu.
This theory of Borcherds algebras is thus very similar to the theory of
Kac-Moody algebras. The main difference is that for Borcherds algebras there
can exist imaginary fundamental roots, and that the Weyl group is generated
by the reflections with respect to the real fundamental roots only.
We now turn to the representation theory of Borcherds algebras. We define
the set X of integral weights by
[	(A, CL.)	1
X = ! A e Q® R; 2-)----- e Z for all at e IIRe |.
I	(at, at)	J
Here IIRe is the set of real fundamental roots. We recall that (at, at) > 0 when
ai e Ще. We define the subset X с X of dominant integral weights by
X+ = {AeX; (A,	>0 for all at e П}.
In a manner very similar to that we have described for Kac-Moody algebras
in Chapter 19 it is possible to define an irreducible module L(A) for the
Borcherds algebra L associated to any dominant integral weight A. L(A) is
called the irreducible L-module with highest weight A.
Now Borcherds proved a character formula for L(A) analogous to Kac’
character formula Theorem 19.16 for Kac-Moody algebras.
Theorem 21.6 (Borcherds’ character formula). Let L be a Borcherds alge-
bra, A a dominant integral weight and L(X) the corresponding irreducible
L-module with highest weight A. Then the character of L(X) is given by
E E(w)w (e(- 1)|ф|е(А + р-Е'РЙ
C U“ e(p) П (1 -e(-a))m“
аеФ+
526
Borcherds Lie algebras
where ma = dimLa, Ф runs over all finite subsets of mutually orthogonal
imaginary fundamental roots, and p is any element of <2®IR such that
of) = | (at, of)
for all real fundamental roots at.
As usual this character is interpreted as
ch L (A) = (dim L (A) J e(jf)
м
where p. e(p.) is an isomorphism between the additive group of weights
and the corresponding multiplicative group.
(In fact there may not exist a vector p e Q ® IR such that (p, af = ^ (a;, of
for all real fundamental roots at of a general Borcherds algebra. But if there
is no such p e Q ® IR, p may still be defined as the homomorphism from Q
to IR taking at to | (ai, of for all i el, and the character formula can be
interpreted accordingly.)	□
In the special case A = 0, L(A) is the trivial 1-dimensional L-module. Then
Borcherds’ character formula reduces to the following identity.
Theorem 21	.7 (Borcherds’ denominator formula).
e(p) П (1-e(-a))m“= E s(w)w (е(Р)Е(-1)|Ф|е(-Елр)) • 1=1
аеФ+	weW	\ Ф	/
By substituting Borcherds’ denominator formula into Theorem 21.6 we obtain
a second form of Borcherds’ character formula.
Theorem 21	.8 (Borcherds’ character formula, second form). With the nota-
tion of Theorem 21.6 we have
chL(A) =
E £(w)w E(-l)We(A + p-Er)
iw-_W	\ Ф
E £(w)wfe(p)E(-l)|4,|e(-EAP)
weW	\ Ф	.
Comments on the proof of Borcherds’ character formula
We shall not give the proof of Borcherds’ character formula in detail, since
the ideas are quite similar to those which arise in the proof of Kac’ character
21.2 Representations of Borcherds algebras
527
formula for Kac-Moody algebras. However we shall say enough to explain
where the additional term
Е(_1)те(_ЕФ)
ф
comes from, where Ф runs over all finite sets of mutually orthogonal imag-
inary fundamental roots. Of course in Kac-Moody algebras there are no
imaginary fundamental roots so the only possible subset Ф is the empty set.
The additional term then becomes e(0), the identity element of e(<2), and
disappears from the formula.
For a Borcherds algebra L we have
n=nReunIm
where IIRe is the set of real fundamental roots and П1т is the set of imaginary
fundamental roots. We also define
Фке = ^(ПКе), Ф1т = Ф-ФКе
to be the sets of real and imaginary roots respectively. We recall from Theo-
rem 16.24 that, in a Kac-Moody algebra,
Ф1+Ш= U «<)
weW
where K = {aeQ+ ; а ф 0, supp a is connected, —a e C} and
C = {A e Q ; (A,	> 0 for all a; e nRe}.
There is an analogous result for Borcherds algebras given as follows.
Theorem 21	.9 The set of positive imaginary roots of a Borcherds algebra is
given by
Ф1+Ш= U «ЧЮ
weW
where К is given by
К = {a e Q+; a^O, —aeC, supp a is connected}
-{joti; jeZ, j>2, а;еП1т}.
Proof Omitted. The idea is generally similar to that of Theorem 16.24. It is
clearly necessary to exclude positive multiples jat of imaginary fundamental
528
Borcherds Lie algebras
roots with j > 2 since these vectors satisfy the conditions required for belong-
ing to K, but cannot be roots since there is no possible root vector giving rise
to such a root.	□
Following closely the proof of Kac’ character formula we obtain
e(p) П (l-e(-a))'"“chL(A) = 22c/ze(p + p)
аеФ+	M
summed over all weights p such that p< A and {p,+p, p,+p) =
(A + p, A + p), where c^eZ and both sides are skew-symmetric under the
action of the Weyl group W. (See the proof of Theorem 19.16.)
We now define a certain partial sum 5 of terms on the right-hand side.
Let 5 = 22 c e(p, + p), summed over all weights p satisfying p<A,
(p + p, p + p) = (A + p, A + p) and (p + p, ctj) >0 for all at e IIRe.
Since p -< A we have
p = A — A;a; for some к eZ,kt> 0, at e П.
Since (p + p, p + p) = (A + p, A + p) we have
((A + p) — (p + p), (A + p) + (p + p)) = 0
that is (22^iai, A + p + 2p) =0. This implies
22 (a;, A) +22((+ ll + 2p') =0.
We can deduce several consequences from this equation. We note first that
(a;, A) >0 since A is dominant. Also for at e IIRe we have
(a;, p + 2p) = (a;, p + p) + (a;, p) = (a;, p + p) + | (a;, a;) > (a;, p + p).
Now (a;, p + p) >0 by definition of S, so (a;, p + 2p) > 0. On the other
hand, for a; e П1т we have
(a;, p + 2p) = (a;, p + a;) = («/, A — 22^)ау) f°r some к’- > 0.
Thus (a;, p + 2p) >0 since (a;, A) >0,	<0 if i^j, and (a;, a;) <0.
We now collect these results together and put them into the equation
22 (a;, A) + 22^ (ao ll + 2p') =0.
The conclusion is that (a;, A) =0 and (a;, p + 2p) =0 for all at in the sum
This in turn implies that each such at e П1т. But then
(a;, p + 2p) = (a;, A) -22^ aj}=~I2k'j (ac aj} 
21.2 Representations of Borcherds algebras
529
Since k'j >0 and	this implies that a^ = 0. In fact k'- = kj if
j ф i and k\ = kt — 1. So if i ф j we have (a;,	= 0 for all ah ctj in the sum
with i^j. In other words,
A — p. = к at
is a linear combination of mutually orthogonal imaginary fundamental roots
all of which are orthogonal to A.
Now we have
i
for all a. e IIRe since (A,	> 0, к > 0 and (a;,	< 0 since at e П1т, a. e IIRe
so ij. Thus p.eC. It follows that fi + peC since
(M + p, 0^ = 1^,	+	aj)
so (jic + p, a^>0. Thus /л + p lies in the fundamental chamber C. Since our
sum
E sXm+p)
/z+A
(МР.М+рМЛ+р,Л+р)
is skew-symmetric under the action of W, this sum must be equal to
E £(w)w(>5)
weW
since 5 is the partial sum including all summands с^е(р. + р) for which p. + p
lies in C.
We shall now determine S. Let the module L(A) have highest weight
vector vA. If A (h) = 0 then /;ил = 0 by the analogue in Borcherds algebras of
the proof of Theorem 10.20. If all at in the sum kjai satisfied A (/i;) = 0 then
we would have /;ил = 0 for all such i and this would imply that A —
could not be a weight of L(A). So if A — is a weight not equal to A
we must have A (/i;) 0 for some i in this sum. Thus (A, a) 0 for some i
in this sum. On the other hand we know that p = A — A;a; where all at in
the sum satisfy (A, a;) =0. This implies that the weight
рь + р = (A + p) -^kjOtj
can only arise from the term e(A) in chL(A) in the formula
e(p) П (1-e(-“))'"“chL(A) = EeMe(M + p)-
аеФ+	M
530
Borcherds Lie algebras
So all terms on the right of this formula which lie in 5 arise from
e(A + p) П (1 -е(-а)Г“
аеФ+
on the left.
We consider which roots a e Ф+ in the formula
e(A + p) П (l-e(-a))m“
аеФ+
can contribute to give p. + p = (A + p) —	All such roots аеФ+ must
be linear combinations of the fundamental roots a; arising in the sum
But such a; are mutually orthogonal imaginary fundamental roots. So sums
of two or more such a; do not have connected support, so cannot be roots
by Theorem 21.9. Also each at e П has ma. = 1 since the corresponding root
space is spanned by e;. Thus a weight g- + p giving a term on the right which
lies in 5 must arise from
e(A+p) П (1-e(-“i)) = e(A+p)E(-1)l'I'le(-EAp)
“.еПы,	Ф
summed over all finite sets 'I' of mutually orthogonal imaginary fundamental
roots. Thus we have
5 = £(_1)|+е(Л + р_£ф)
and so
e(p) П (1-e(-“))m"chL(A)= E £(w)w (EC-O’^eO + P-E^))
аеФ+	weW	у Ф	/
as required.
This argument therefore explains the difference between Kac’ charac-
ter formula and Borcherds’ character formula, and where the extra term in
Borcherds’ character formula comes from.
21.3 The Monster Lie algebra
In this final section we shall show that, although Borcherds algebras have
many properties which seem quite similar to those of Kac-Moody algebras,
they include examples which behave in a very different way from Kac-Moody
algebras. The example we have in mind is the Monster Lie algebra. The
definition and properties of the Monster Lie algebra are closely related to the
properties of a certain modular function j, so we shall begin by describing
the definition and significance of this function.
21.3 The Monster Lie algebra
531
We first recall the action of the group \Л2(1- ) on the upper half plane H.
Let
УЛ2(Ж) = {( d) ’ ~ ^c=l’ a, fe, с,
77={reC ; Im t> 0}.
The group 5Т2(®0 acts on H by
(ab\	ar + b
, It =-----
\c a/	ст + d
since if 1тт>0 we have Im(^±j) >0. In particular the subgroup SL2(Z)
acts on H. Since
we see that PSL2(Z) = SL2(Z) / (±/2) acts on H. PSL2(Z) is called the
modular group.
We denote by H/SL2^Z) the set of orbits. Since Q	the
elements т and т +1 of H lie in the same orbit. Thus each orbit intersects
{теЯ; -|<ReT<i}.
Again we have e SL2(Z) and so the elements т, — 1/т e H lie in the
same orbit. Thus each orbit intersects
{теЯ; |т| > 1}.
In fact we can obtain a fundamental region for the action of SL2(Z) on H by
taking the region
{теЯ; —| <Rer< |t| > 1}
and identifying the points т, t+ 1 for Rer = — | and the points cr, — \/<j for
|cr| = 1. The fundamental region is illustrated in Figure 21.1.
Having made the above identifications we obtain a set intersecting each
orbit in just one point. The set H/SL^TL) of orbits has the structure of a
compact Riemann surface with one point removed. This is a Riemann surface
of genus 0, i.e. a Riemann sphere. When we remove one point from it we
532
Borcherds Lie algebras
Figure 21.1 Fundamental region
obtain a subset which can be identified with C. Thus we have an isomorphism
of Riemann surfaces
7//5L2(Z)^C.
This can be extended to an isomorphism of compact Riemann surfaces by
adding the point ioo on the left and <x on the right. Thus we have an isomor-
phism
(tf/5L2(Z)) U {ioo} S2 = CU {00}
under which ioo maps to 00. Such an isomorphism of Riemann surfaces is
not uniquely determined. However, if j is any such isomorphism any other
must have the form a(j + b) where a, b are constants and a ^0. Such a map
determines a map from H to C constant on orbits. This map will also be
denoted by j. j is a modular function, i.e. a function invariant under the action
of the modular group.
Since t, r+1 lie in the same orbit we have j(t)=j(t+1), thus j is
periodic. This implies that j has a Fourier expansion of form
neZ
We write q = e2mT. Then we have
j(T) = Ec«?"-
neZ
We shall now describe such a function j. In order to do so we first introduce
some modular forms. A function f : H^C is called a modular form of
weight k if
f = (CT +
\ст + а /
21.3 The Monster Lie algebra
533
г ii I ab
tor all ,
\ c a
eSL2(7L). We give two examples of modular forms. The first
is an example of a so-called Eisenstein series. For each positive integer n let
сгз(и) = Ей?3
d\n
summed over all divisors of n, and let
£4 (r) = 1 + 240	°з (и) Qn •
n> 1
Thus
E4(t) = 1 +240<? + 2160<?2H-.
This function is known to be a modular form of weight 4.
Secondly define Д by
д(7) = <?П(1-Л24-
п>1
Then
&(r) = q — 24q2 + 252q3-----.
This is called Dedekind’s Д-function and is known to be a modular form of
weight 12.
We now define j : H C by
This is a modular form of weight 0, i.e. a modular function, and so is constant
on orbits of SL2(Z) on H. We have
j(r) = q-1 + 744 + 196 884? + 21493 760?2 + • • •
and j has a simple pole at r = i<x>, i.e. q = Q.
j gives an isomorphism of Riemann surfaces
j-.H/SL2(Z)^C
which extends to an isomorphism of compact Riemann surfaces
j: (///.S7.;(Z))U{ioc}^.S': = CU{oc}.
Any other such isomorphism has the form a(j + b) where a, b are constants
and a 7^0. In particular there is just one such isomorphism with leading
534
Borcherds Lie algebras
coefficient 1 and constant term 0. We shall call this the canonical isomor-
phism. This is the function
_ 744 = ?-i+ cnqn
Л>1
where cx = 196 884, c2 = 21493 760, etc. All the cn are positive integers.
We are now ready to introduce the Monster Lie algebra. We first define a
countable symmetric matrix ?I. is defined as a block matrix, with blocks of
rows and columns parametrised by the natural numbers N= {0, 1, 2, 3,...}.
Let be the (z, /')-block of ?I. The number of rows in B^ is 1 if i = 0 and c;
if i ф 0 where c; is the coefficient of z/' in the modular function j. Similarly
the number of columns in B^ is 1 if j = 0 and Cj if All the matrix
entries in a given block B^ are equal to one another. These entries are given
as follows.
The single entry in block Boo is 2.
All entries in block BOrl for n 0 are — (и — 1).
All entries in block Bmn for m 0, n 0 are — (m + n).
These conditions determine the matrix ?I. We have
^196884^ ^21493760^
	2	0 •	• 0	-1 •	• -1	-2 • • •
T	0	—2 •	• —2	-3 •	• —3	—4 • • •
196884						
	6	—2 •	• —2	—3 •	• —3	-4 : • •
§1=	|	-1	-3 •	• —3	-4 •	• —4	-5 • • •
21493760						
1	-1	-3 •	• —3	-4 •	• —4	-5 • • •
	—2	-4 •	• —4	-5 •	• -5	-6 • • •
Let L(?I) be the universal Borcherds algebra determined by the countable
matrix as in Proposition 21.1. Let Z be the centre of L(?I) and H be the
subalgebra of L(?I) generated by all elements hij = \eif^\. We know from
Section 21.1 that
It is also clear that Ir — h^eZ where Ir = h,,, h; = hit and z, j are in the same
block, since columns z, j of ?I are then identical. Z is in fact generated by the
21.3 The Monster Lie algebra
535
elements for ij^j and ht — hj where i, j are in the same block. Moreover
Z is an ideal of L(?I).
Let 2R = L(?I)/Z. ЗЛ is called the Monster Lie algebra. We shall now
determine some properties of ЗЛ.
Let the blocks of rows of be Bo, Въ B2,    with |B0| = 1, \Bn| = cn for
n > 1. We choose one i e I out of each block, such that i lies in the block B;.
We then consider the elements /i; for such elements ie/. Thus we have
elements h0, h3, h2,... EH. Then the elements
2h2 + h0— 3h1
2h3 + 2/i0 — 4/ij
2/i4 + 3/i0 — 5/ij
all lie in Z, since the corresponding linear combinations of the rows of are
all zero vectors.
Let ht ht under the natural homomorphism L(?I) ЗЛ. Then we have
2h2 = 3h3 — h0
2h3 = 4/z, — 2h0
2/z4 = 5h3 — 3h0
Let <SSlQ = H/Z. Ж, is the Cartan subalgebra of W, being the image of H
under the natural homomorphism. We see from the above relations that Ж,
is spanned by h0 and h1. Moreover h0 and h3 are linearly independent since
this is true of the first two rows of ?I. Thus h0, h3 form a basis of ЗЛ0 and
we have
dim Ж, = 2.
In fact we find it more convenient to choose the basis b0, b3 of ЗЛ0 given by
/iq + Zij	-ho + hi
bn =--------, bA =------------.
0	2	1	2
Thus Ж, = IF’L, + IRfoj. The scalar product on Ж, is given by
(/i0, h0} = a00 = 2
(hy, h^ = au = —2
h"i) = M= aoi = 0-
536
Borcherds Lie algebras
It follows that
(foo,M=O, (Ь1,Ь1)=О, {b0,b,} = -L
Hence
(mbQ + nbx, m'bQ + n'bx} = — (mn'+nm') .
We now regard the Monster Lie algebra W as a module over its Cartan
subalgebra 2R0. Let m, n e Z and define ЗЛ(т by
= {*e; [box] = mbo, [b1x] = nb1}.
Then one can show that ЗЛ(0 = ЗЛ0 and
2Ч = ф®1(т л) for(m, и) e Z x Z.
Moreover we have
dim = cmn if m ф 0, n ф 0
dim W (ojCI) =2
dim W(m 0) = dim =0 if m ф 0, n ф 0.
(These results follow from the ‘no-ghost’ theorem of Goddard and Thorn in
string theory! A statement and proof of this theorem in an algebraic context
can be found in E. Jurisich, Journal of Pure and Applied Algebra 126 (1998),
233-266).
Thus the graded components W(m of the Monster Lie algebra ЗЛ are as
shown in Table 21.1. In this table V„ is a vector space of dimension cn if
n > 1 and V_j is a vector space of dimension 1.
Table 21.1 Graded components	of the Monster Lie Algebra.
0	0	0	0	о	v4	v8	v12	v16
0	0	0	0	о	v3	v6	v9	v12
0	0	0	0	о	v2	v4	v6	v8
О	О	О	V , О	V	v2	v3	v4
OOOO R2O ООО
V4	V3	V2	Vi	О	V-i	о	о	о
V8	V6	V4	V2	О	О	ООО
V12	v9	V6	V3	О	О	ООО
V16	V12	V8	V4	О	О	ООО
21.3 The Monster Lie algebra
537
We now consider the roots of the Monster Lie algebra W. Since 2R =
L(?l)/Z we recall from Section 21.2 that the root lattice of W is defined to
be the root lattice of L(?I). The fundamental roots of W are the a; for i el.
We have a homomorphism II under which a; maps to /i;. We pointed
out in Section 21.2 that this homomorphism is not in general injective. In the
Monster Lie algebra it is far from injective, as ah aj have the same image if
and only if i, j lie in the same block of I.
We have
(a0, a0)=2
(ai,ai) = —2m if z^O and ieB,n.
Thus Пге = {a0} and IIim = {a; ; z # 0}. Hence the Monster Lie algebra ЗЛ has
just one real fundamental root and countably many imaginary fundamental
roots.
The Weyl group W of ЗЛ is generated by the fundamental reflections
corresponding to the real fundamental roots. Thus W={s0), and so W has
order 2. Thus ЗЛ has an infinite number of fundamental roots while at the
same time having a very small Weyl group isomorphic to the cyclic group of
order 2.
Finally we shall consider Borcherds’ denominator formula for the Monster
Lie algebra ЗЛ. This formula plays an important role in Borcherds’ proof of the
Conway-Norton conjectures. We recall from Theorem 21.7 that Borcherds’
denominator formula is given by
e(p) П	= E £(w)w(ф)Е(-1)|ф1е(-Елр))
аеФ+	weW	\ Ф	/
where p e Q® R is any vector satisfying
(p, at) = j (ai, at) forallieZ.
Now we have
- (n+ V)h1-(n-V)h0
hn =------------------= b0 + nbr.
Thus we can identify a fundamental root at in the block Bn with its image
b0 + nbx in the Cartan subalgebra Ж, of W provided we remember that
there will be cn different such fundamental roots a; with a given image
Ь0 + иЬР
538
Borcherds Lie algebras
We may take p =--------------
since if G Bn we have
(p, ai) = ( ——— ,b0 + nb\= (~b0, bv + nb^n
whereas
(a;, a;) = {Ь0 + пЬъ Ьи + пЬ^ = —2n.
Thus (p, at) = | {at, at).
Hence we shall use this vector p in Borcherds’ denominator formula. Using
the natural homomorphisms
Q^H^m0 = H/Z
a.^ hi
we may interpret Borcherds’ denominator formula in the integral group ring
of e(W0) rather than the integral group ring of e(<2). Bearing this in mind
we define
p= eb°, q = ebl.
Then ep = p! and so the left-hand side of the denominator identity is
/ПС-/'#"
m>0
neZ
since the positive roots a e Ф+ are the elements of Q which map to elements
of '.Ui-, of the form mb0 + nbx with m > 0 and n e Z, and the number of a e Ф+
mapping to mb0 + nb1 is diiniUi,.,, = cmrl.
We now consider the right-hand side of the denominator identity. We recall
that, for a e Q, s(a) = (— I)*1 where a is the sum of k orthogonal imaginary
fundamental roots and £(a) = 0 otherwise. In the case of the Monster Lie
algebra ЗЛ no two imaginary fundamental roots are orthogonal since
{Ьо + тЬ-^ b0 + nbr} = — (m + n).
Thus the elements a e Q contributing to s(pt)ea are a = 0 with s(a) = 1 and
the imaginary simple roots in Q. These map to elements of form b0 + nbr e
Wo. There are cn such roots aeQ mapping to b0 + nb1 and they all give
e(ce) = — 1. Thus
E£(“)e“ = l- HcnP(T-
ae<2	n>0
21.3 The Monster Lie algebra
539
Now the Weyl group W has order 2 and consists of the elements 1 and .v,,.
We have s0(p) = q and s0(q) = p. Thus the right-hand side of the denominator
identity is
/’ 1 (1 - E cnPqn) ~ 9 ' ( 1 ~ E cn1Pn )
\ n>Q /	\ n>Q /
= [E+Ec/| -(Е+Еш
\	77>O	/	\	77>O	/
Thus we have obtained the following result.
Theorem 21.10 Borcherds’ denominator identity for the Monster Lie algebra
ЗЛ asserts that:
p-1n^-pmqnym"=j(p')-j(q')
m>0
neZ
where cn is the coefficient of qn in the modular function j.	□
In fact this identity was proved by Borcherds from first principles and used
subsequently to prove that the fundamental roots of ЗЛ map to the elements
b0-b1,b0 + b1, Ь0 + 2Ьг, b0 + 3b1,...
of Жо.
Further information about results stated without proof in this chapter can
be found in the papers of R. Borcherds ‘Generalised Kac-Moody algebras’,
Journal of Algebra 115 (1988), 501-512, and ‘Monstrous moonshine and
monstrous Lie superalgebras’, Inventiones Mathematiae 109 (1992), 405 НИИ.
Appendix
Summary pages - explanation
There follow a number of summary pages, one for each Lie algebra of finite
or affine type, giving basic properties of the Lie algebra in question. The
information given differs to some extent between the Lie algebras of finite
type and those of affine type.
In the case of the algebras of finite type we give the name of the algebra,
the Dynkin diagram with the labelling we have chosen for its vertices, the
Cartan matrix, the dimension of the Lie algebra, its Coxeter number, the
order of its Weyl group W and the degrees of the basic polynomial invariants
of W. We also give information about its root system. The roots are most
conveniently described in terms of a basis ,... , /3m of mutually orthogonal
basis vectors all of the same length. In several cases it is convenient to choose
m greater than the rank / of the Lie algebra, so that the root system lies in
a proper subspace of the vector space spanned by /3r,... , /3m. In the cases
when there are roots of two different lengths the long roots and short roots
are both described. The extended Dynkin diagram is given and the root lattice
described in terms of the above orthogonal basis. The fundamental weights
are given, as is the index of the root lattice in the weight lattice. Finally the
standard invariant forms on IE and Hg are described, and the constant is
given which converts the standard invariant form on HK into the Killing form.
The labelling given here for the vertices of the Dynkin diagrams of types
E6 and E7 differs from that used in Chapter 8, where it was convenient to
describe the root systems of type E6, Ej or Л8 together in Section 8.7.
In the case of the Lie algebras of affine type we have given two names for
each algebra which we have called the Dynkin name and the Kac name. The
Dynkin name describes the Dynkin diagram of the algebra whereas the Kac
name, introduced at the end of Chapter 18, indicates whether the Lie algebra is
540
Appendix
541
of untwisted or twisted type, and in the case of those of twisted type indicates
the type of the untwisted affine algebra from which it is obtained, together
with the order of the automorphism of which it is the fixed point subalgebra.
This Kac notation is entirely consistent with the notation normally used to
describe the twisted Chevalley groups.
The Dynkin diagram with chosen labelling is given, together with the gen-
eralised Cartan matrix and the integers a0,	... , a; and c0, cb ... , c;. The
central element c, the basic imaginary root 3, and the elements в e (dd£)* and
hg e //( which play an important role in the theory of affine algebras are writ-
ten down explicitly. The Coxeter number and dual Coxeter number are given.
The type of the finite dimensional Lie algebra L° obtained by removing
vertex 0 from the Dynkin diagram is given. The root system Ф is described
in terms of the root system Ф0 of L°. The real and imaginary roots are given
separately and the multiplicities of the imaginary roots are given. (The real
roots all have multiplicity 1.) In order to clarify the action of the affine Weyl
group we describe the lattices Me//;1 and M* c which give rise to
the translations in the affine Weyl group. We also describe the fundamental
alcoves A(Zdd^ and А* с whose closures give fundamental regions
for the action of the affine Weyl group. Then we describe the fundamental
weights in terms of the fundamental weights of L°, and the standard invariant
forms on dd and on H*.
For the affine algebras of types C;',/>2, and A\ we have given two
different descriptions, corresponding to two choices of the vertex of the
Dynkin diagram labelled by 0. (The algebra A\ behaves just like C;' when
/=1.) Both descriptions are useful, as is shown in Section 18.4. The first
description is the conventional description in which the associated finite
dimensional algebra has type C;, and which is discussed in Chapter 17. The
second description is the one used to obtain the realisation of C;' as 2A2l in
Section 18.4. Here the associated finite dimensional algebra has type B;. A
word of caution is necessary in deriving the results appearing in the second
description. In these cases we have c0 = 2. Thus we cannot apply results from
Chapter 17 uncritically to these cases, since c0= 1 is assumed in Chapter 17.
Instead we have the following situation.
i
0 = 8 — ariari = y^aiai
1=1
satisfies (в, 0) = 2a0c0. We also have
1 1 1
he =------(c - coho) =-------E ciht-
a0C0	a0C0 ; = 1
542
Appendix
Under the natural bijection II we have	hg^a01c01e,
d о a0Cg 1 у. In addition we have
(h0, d)=aoc-1, (a0, y)=a-1c0.
The lattices M, M* are given as follows in these cases. M is the lattice gener-
ated by w (/i0) for all w e W°, and M* is the lattice generated by w
for all w e W°. The alcove A is bounded by the affine hyperplane 0(/i) = 1
and the alcove A* is bounded by the affine hyperplane A (/i0) =	.
In fact in this second description в turns out to be 2d, where d, is the
highest short root, and hg is ^hg .
NAME Ai
543
NAME Az
Dynkin diagram with labelling.
О---О----О---о---о----о---о----о
12	3	1-1 I
Cartan matrix
1	2	3	•	•	•	/ - 1
1	/ 2	-1
2-12-1
3	-12-
/-1	•	2	-1
1 \ -1 2 /
Dimension. dim £ = /(/ +2).
Coxeter number. h = I + 1.
Order of the Weyl group. |W| = (/+1)!
Degrees of the basic polynomial invariants of W.
{di,d2,... ,	= {2,3,... ,1+1].
Number of roots.	|Ф| = 1(1 + 1).
The fundamental roots in terms of an orthogonal basis.
ai — Pi Pi> а2 — Рз Рз> •••’ ai~Pi Pi+i-
The root system.
Ф={/3;-/3- i,j=l,...,l + l, i^j}.
The highest root.	Q = /31— /3l+1.
544
Appendix
The extended Dynkin diagram, for / > 2.
The root lattice Q = "Z'-p.
’/+i
Q = EeA ; £eZ, ££ = 0 .
i=l
The fundamental weights.
(G +1 - 0 (^i 4—Ь /3;) - i (J3i+14------h /3;+i))	i = 1, • • • ,
The index of the root lattice in the weight lattice. |X: Q| = I + 1.	X/2
is cyclic.
The standard invariant form on /Л.
The standard invariant form on //i.
The Killing form on /Л.
Ст = у}
b
where h = 2(Z+ 1).
NAME Bi
545
NAME Bz
Dynkin diagram with labelling.
О----О---О----О---О----О---о > о
12	3	1-11
Cartan matrix
1	2	3
1/2	-1
2	-12-1
3	-12
1 — 2 l-l I
\
1 — 2
l-l
1 <
Dimension. dimL = 1(21 + 1).
Coxeter number. h = 2l.
2	-1
-1	2	-1
-2	2 ?
Order of the Weyl group.	| W| = 2l • /!
Degrees of the basic polynomial invariants of W.
{dx,d2,... ,d;} = {2,4, ••• ,2/}.
Number of roots. |Ф|=2/2.
The fundamental roots in terms of an orthogonal basis.
al~ Pl Pl> а2 — Р1 ft, ..., O'/-!— /3/-1 Pl, Ui — Pi.
The root system. Ф = Ф, U Ф, where
Ф1 = {±Pi±Pj ; h ] = I, i^j}
Ф8 = {±/3;; i=l,...,/}.
The highest root. = Д + /32-
The highest short root. ds = .
546
Appendix
The extended Dynkin diagram.
o-----О------О------о------о > <>
The root lattice Q = ^ai-
Q =
i=l
The fundamental weights.
w. =/31 + -..+/3. z = l,...,/-l
Ml = | (&Ч-------h/3;).
The index of the root lattice in the weight lattice. |X: Q\ =2.
The symmetrising matrix D = diag (</;).
di = l	dl = 2.
The standard invariant form on /Л.
The standard invariant form on //i.
The Killing form on /Л.
Ст .уЕ = 7-Ст у)
b
where b = 4l — 2.
NAME Ci
547
NAME Cz
Dynkin diagram with labelling.
О---О-О---о--о---о--о < о
12	3	1-11
Cartan matrix
1	2	3
1/2	-1
2	-12-1
3	-12
1 — 2
l-l
1 <
Dimension. dimL = 1(21 + 1).
Coxeter number. h = 2l.
/—2 l-l I
2	-1
-1	2	—2
-1 2 >
Order of the Weyl group.	| W| = 2l • /!
Degrees of the basic polynomial invariants of W.
{d^d^... , dt} = {2,4, ••• ,21}.
Number of roots. |Ф|=2/2.
The fundamental roots in terms of an orthogonal basis.
ai~ Pi P2, ai — Pi Рз, •••, ai-i~Pi-i Pi, ai~2Pi-
The root system. Ф = Ф, U Ф, where
Ф1 = {±2/3; ; i=l,...,l}
(|,s = {±Pi±Pj ; i,j=l,...,l z# j} •
The highest root.	0! = 2/31.
The highest short root. 0S = + /32.
548
Appendix
The extended Dynkin diagram.
<) > <>--------О-----О------О------о------и < <>
The root lattice Q = ^ai-
0 =	'>	even •
i=l
The fundamental weights
oj  = /3, T • • • T & i == 1,... , I.
The index of the root lattice in the weight lattice. |X: Q\ =2.
The symmetrising matrix /) = diag (</;).
d, = 7. / = 1,... ,/-l d, = l.
The standard invariant form on HK.
The standard invariant form on 77R.
The Killing form on /Л.
Cr ^ = 7 Ст у)
b
where b = 2(l + 1).
NAME I)
549
NAME Dz
Dynkin diagram with labelling.
Cartan matrix
1	2	3	1 — 2	l-l
1^2	-1
2	-12-1
3	-12-
1 — 2
l-l
1 \
Dimension.	dim L
2	-1	-1
-1	2
-1 2 >
= /(2/-l).
Coxeter number. h = 21 — 2.
Order of the Weyl group. H'|=2/ '/!
Degrees of the basic polynomial invariants of W.
{d^dz,... ,d[} = {2,4, ••• ,21 — 2,1}.
Number of roots. |Ф|=2/(/ —1).
The fundamental roots in terms of an orthogonal basis.
al — Pl Pl, а2 — Р1 Рз, •••, al-2~Pl-2 Pl-1,
ai-i — Pi-i Pi, ai — Pi-i~l~ Pi-
550
Appendix
The root system.
ф={±/3;±/3у; z,j = l,...J z#j}-
The highest root. в = /Зх + /32-
The extended Dynkin diagram.
The root lattice Q = ^ai-
q= '> E£even •
i=l
The fundamental weights.
w. = /31 + ... + /3.	z = l,...,/ —2
Ml-1 = 2 (^1 ----^/-2 + ft-1 - Pl)
Ы1 = |	4----Ь Pl-2 + Pl-1 + Pl) 
The index of the root lattice in the weight lattice. |X: Q\ =4.
X/O is cyclic if / is odd and non-cyclic if / is even.
The standard invariant form on /Л.
^hj] = Aij-
The standard invariant form on //i.
The Killing form on /Л.
(r зЕ = т Ст y)
b
where b = 4(Z — 1).
NAME E6
551
NAME E6
Dynkin diagram with labelling.
1	2	3	5	6
О-----------О------------(j>---------о-----------о
4
Cartan matrix
1 2 3 4 5 6	1	2	3	4	5	6 < 2	-1	0	0	0	C? -12-1000 0-1	2-1-1	0 0	0	-1	2	0	0 0	0	-1	0	2	-1 0	0	0	0	-1	2
Dimension.	dimL = 78.
Coxeter number. h=12.
Order of the Weyl group. | W| = 27 • 34 • 5
Degrees of the basic polynomial invariants of W.
{dx,d2,... , </6} = {2, 5, 6, 8, 9, 12}.
Number of roots. |Ф|=72.
The fundamental roots in terms of an orthogonal basis.
/31,/32, /З3, /З4, /35, /36, /37, /38 orthogonal basis.
ai=Pi~Pi> = ft, О'з = /З3—/З4, a'4 = /34 — /З5,
a5 = /34 + /35, а6 = -|(/31 +/32 + /33 + /34 + /35 + /36 + /37 + /38).
552
Appendix
The root system.
Ф={±/3;±/3 = z, j= 1, 2, 3,4,5
8	8
ei=±1>П£/ = i’ £6=£7=£8
i=l	i=l
The highest root.
M (ft+&+&+&-&-&-/з7 - ft) •
The extended Dynkin diagram.
О----О---Q----О----О
The root lattice Q = "Z'-p.
Q =
8	8
; 2£eZ, ^eZ, ££e2Z, z, 7 = 1,... , 8 = 6 = ^8
i=l	i=l
The fundamental weights.
*>!=&-!(&+&+&)
W2 = /31 + /32-|(/36 + /37 + /38)
«3 =/3X+ /32 + /33 - (/36 + /37 + /38)
= 1 (/3 3 + /32 + /З3 + /34 - /35 - /36 - /37 - /38)
= 1 (/3X + /32 + /з3 + /34 + /35) - I (/36 + /37 + /38)
we = _ j (/36 + /37 + /38).
The index of the root lattice in the weight lattice. |X: Q\ = 3.
The standard invariant form on /Л.
The standard invariant form on //i.
The Killing form on HK.
Cr ^ = 7 Ст y)
b
where b = 24.
NAME E7
553
NAME E7
Dynkin diagram with labelling.
1	2	3	4	6	7
О------О------О------Q-------О------О
		5
Cartan matrix		
	1	2	3	4	5	6	7
1	< 2	-1	0	0	0	0 o'
2	-1	2	-10000
3	0	-1	2-1000
4	0	0	-1	2-1-1	0
5	0	0	0-1200
6	0	0	0-1	0	2-1
7	0	0	0	0	0-1	2 ,
Dimension.	dim L = 133.		
Coxeter number. h =		18.
Order of the Weyl group.		|W|=210-34-5-7.
Degrees of the basic polynomial invariants of W.		
	{d\, d2,	.. , </7} = {2, 6,8, 10, 12, 14, 18}.
Number of roots. |Ф| = 126.
The fundamental roots in terms of an orthogonal basis.
Pi, Pi, Рз, P4, Ps, Рб, Pt, Ps orthogonal basis.
al= Pl~ Pl, а1 = Р1~Рз, аЗ=Рз~Рл, a4 = P^~P5, a5 =
ae = P5 + Ре, а1 = -^+^ + ^ + ^ + ^ + Р6 + ^ + ^.
•5-Рв
554
Appendix
The root system.
Ф={±/3;±/3- г, j= 1,2,3,4,5,6 i^j}
U{±(/37 + /38)}
u
ei=±1,П£/ = i’£7=£8
i=l	i=l
The highest root. в = — /37 — /38.
The extended Dynkin diagram
О----О----О----Q---О-----О----О
The root lattice Q = ^ai-
Q =
££e2z,	e7=e8
i=l	i=l
The fundamental weights.
W1 = /31-l(/37 + /38)
«2 = /31 + /32 -(/37 + /38)
«з = /31 +/32 + /33 -|(/37 +/38)
«4 = /31 +/32 + /33 + /34 -2 (/37 + /38)
Со5 = |(/31 + /32 + /33 + /34 + /35 -/36)-(/37 + /38)
= | (/Зх + /32 + /З3 + /34 + /35 + /36) - 1 (/37 + /38)
W7 = — (/37 + /38) .
The index of the root lattice in the weight lattice. |X: Q\ = 2.
The standard invariant form on /Л.
The standard invariant form on 77R.
The Killing form on /Л.
(T )E = 7-(-T У)
b
where b = 36.
NAME Es	555
NAME Eo
Dynkin diagram with labelling.
12	3	4
О-------О------о------о
Cartan matrix				
	1	2	3	4
1	( 2	-1	0	0
2	-1	2	-1	0
3	0	-1	2	-1
4	0	0	-1	2
5	0	0	0	-1
6	0	0	0	0
7	0	0	0	0
8	<0	0	0	0
5	7	8 	1	0	0 6 5	6	7	8 0	0	0	o' 0	0	0	0 0	0	0	0 -10	0	0 2-1-1	0 -12	0	0 -1	0	2-1 0	0-1	2	?
Dimension. dim L = 248.
Coxeter number. h = 30.
Order of the Weyl group. | W| = 214 • 35 • 52 • 7.
Degrees of the basic polynomial invariants of W.
{dr, d2,..., d^ = {2, 8,12,14,18,20,24, 30}.
Number of roots. |Ф|=240.
The fundamental roots in terms of an orthogonal basis.
ai=Pi~Pi> ai = Pi~ /З3, а'з = /33—/З4, a,4 = /34 —/35,
a5=Pi~ а6=Рб~ Ph а7 = /3б+/37,
a8 = - I (Pl + Pl + Рз + & + /?5 + P6 + /?7 + Pg) 
556
Appendix
The root system.
Ф={±/3;±/3 = i, j = 1,2, 3,4,5, 6,7, 8 i^j}
8
£; = ±1, П£/=1
i=l
The highest root. в = Д — /38.
The extended Dynkin diagram
The root lattice Q = ^ai-
Q =
8	8
2£eZ, ^eZ, ££e2Z i,j=l,...,8
1=1	1=1
The fundamental weights.
wi — Pi Ps
W2 = Pi+ Pi-^Ps
w3 — Pi + Pi + Рз~ ^>Ps
^ = Pl+P1 + P. + P,-^
^ = Pl+P1 + P. + P, + P5-5P,
«6 = |(/31 +/32+/33 + /34 + /35 + /36 -/37)-|/38
W7 = i (/3X+ /32+/3з+ /34+ /35+ /36 +/37) - |/38
w8 = — 2/38.
The index of the root lattice in the weight lattice. |X: Q\ = 1.
The standard invariant form on /Л.
MMr
The standard invariant form on //i.
The Killing form on T/R.
Ст = | Ct y}
b
where b = 60.
NAME F4
557
NAME F4
Dynkin diagram with labelling.
	O— 1		QZ 2	=2=0— 3	—О 4
Cartan matrix				
	1	2	3	4
1	< 2	-1	0	0
2	-1	2	-1	0
3	0	—2	2	-1
4	1 0	0	-1	2 /
Dimension. dim L =	= 52.			
Coxeter number. /i=12.
Order of the Weyl group. | W| = 27 • 32.
Degrees of the basic polynomial invariants of W.
{d1,d2,d3,d4} = {2, 6,8, 12}.
Number of roots. |Ф|=48.
The fundamental roots in terms of an orthogonal basis.
a1=(31—(32^	=	/З3, а,з = /33, a4 = | (—Д — (32 — /З3 + /34).
The root system. Ф = Ф, U Ф, where
Ф1={±/3;±/37 ; z,j = l,2,3,4 i^j}
Ф8 = {±/3; z=l,2, 3,4}U
; £/=±1
i=l
The highest root. = /3, + /34.
The highest short root. d,=/34.
The extended Dynkin diagram.
О
О
В > <
о
558
Appendix
The root lattice Q = "Z'-p.
Q =	2£eZ, f.-^eZ z,J=l,2,3,4
i=l
The fundamental weights.
wi = /3j +/34
w2 = Pi +/З2 + 2/З4
Со3 = |(/31+/32 + /3з + 3/34)
«4 = /?4‘
The index of the root lattice in the weight lattice. |X: Q\ = 1.
The symmetrising matrix /) = diag (</;).
rf1 = l, </2=l, ^з=2, fi?4 = 2.
The standard invariant form on /Л.
(hi’hj} = Aijdj-
The standard invariant form on //i.
The Killing form on /Л.
Ст .уЕ = 7-Ст у)
b
where b = 18.
NAME G2	559
NAME G2
Dynkin diagram with labelling.
<i > t>
1	2
Cartan matrix
1	2
2 V3 2 7
Dimension.	dim L = 14.
Coxeter number. h = 6.
Order of the Weyl group. |W| = 12.
Degrees of the basic polynomial invariants of W.
H,rf2} = {2,6}.
Number of roots. |Ф| = 12.
The fundamental roots in terms of an orthogonal basis.
/Зх, /32, /З3 orthogonal basis.
«i = —2/3,+/32 + /33, a2 = /31-/32.
The root system.	Ф = Ф, U Ф, where
Ф1 = {±(-2/31+/32 + /33), ±(J31-2/32+l33), ± (/3X+/32 — 2/33)}
Ф8 = {±(/31-/32), ±(/з2-/з3), ±№-/33)}.
The highest root. вг = — /3, — /32 + 2/33.
The highest short root. d, = — /32 + /З3.
The extended Dynkin diagram.
О-----n ) <>
The root lattice Q = ^at-
Q=	; £eZ, £i + £2 + £3 = 0 .
560
Appendix
The fundamental weights.
wi = “ft -ft+ 2/33
w2 = -ft + ft.
The index of the root lattice in the weight lattice. |X: Q\ = 1.
The symmetrising matrix /) = diag (d).
<4 = 1, <4 = 3.
The standard invariant form on /+.
The standard invariant form on 77R*.
The Killing form on /+.
b
where b = 8.
561
DYNKIN NAME Ax KAC NAME Ax
DYNKIN NAME A1 KAC NAME A1
Dynkin diagram with labelling.
o 1
Generalised Cartan matrix.
0	1
0/2	-2
1 V2	2	>
The integers a0, ar,... , a;.
1 ,, 1
The integers c0,c1,... , C[.
1 1
<>> <tJ
The central element c.
c = h0 + h1.
The basic imaginary root 3.
5 = a0 + a1.
The element hg e IE.
he = h1.
The element в e
в = а1.
The Coxeter number. h = 2.
The dual Coxeter number. If' = 2.
The Lie algebra L°. L° = A1.
The lattice M c 7/^.
M = Zh1.
562
Appendix
The lattice M* c (H°y.
M* = Xa1.
The fundamental alcove A G //'•.
A \h. П : a1(/i)>0, ce1(^)<1}.
The fundamental alcove А* с (Я^)*.
А* = {Ле(Я°)* ; Л(й1)>0, A(/z1)< 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = | a + r8 ; a e Ф0, reZj
Ф1п1 = {£<5; AeZ, к7^0} Multiplicity!.
The fundamental weights <у;еЯ^ i = 0, 1 in terms of the fundamental
weights w,, i = 1, of L°.
a>0 = y, a>1 = a)1 + y.
The standard invariant form on H.
M = Ai} i,j = 0,1
{h0,d) = l, {h1,d)=0
(d, d}=0.
The standard invariant form on H*.
{a^aj} = Aij	=
(a0,y) = l, (ai,y)=0
(% y) = 0.
A{ KAC NAME :Л;
A) KAC NAME 2A2
563
DYNKIN NAME
DYNKIN NAME
(1st description)
Dynkin diagram with labelling.
Generalised Cartan matrix.
1
The integers a0,	... , a;.
The integers c0,c1,... , C[.
The central element c.
The basic imaginary root 3.
The element hg e II-.
The element в e (H'jff*
The Coxeter number. h = 3.
The dual Coxeter number.
The Lie algebra L°. L° = A1.
The lattice M c H^.
c = 2h0 + h1.
8 = a0 + 2a1.
^0 = 2^1’
0 = 2a1.
/iv = 3.
M=
564
Appendix
The lattice M* с (Н°У
The fundamental alcove A G //'•.
A \h. П : a1(/i)>0, 2^(/i) < 1}.
The fundamental alcove А* с (Я^)*
А* = {Ле(Я°)* ; Л(й1)>0, Л(й1)< 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКеs = {a +r<5 ; аеФ0, reZ}
фке,1 = {2“ + (2г+1)'5 ; «еФ0, reZ}
Ф1п1 = {£<5; AeZ, кт^О} Multiplicity!.
The fundamental weights <y; e Hg i = 0, 1in terms of the fundamental
weights w, i = 1,... , I of L°.
<i)0 = 2y, a>1=a>1 + y.
The standard invariant form on H.
{hi’hj} = ajcJ1Aij	i,j = 0A
{h0,d} = \, (h1,d)=Q
(d, d}=0.
The standard invariant form on H*
i,j = Q, 1
(a0,y)=2, (ai,y) = 0
<% t)=o.
565
DYNKIN NAME A\ KAC NAME :Л;
DYNKIN NAME A\ KAC NAME 2A2
(2nd description)
Dynkin diagram with labelling.
Generalised Cartan matrix.
0	1
0/2 -4\
1 \-i 2/
The integers a0, ar,... ,a:
The integers c0, cx,... , c;.
The central element c.
c = h0 + 2/iP
The basic imaginary root 3.
3 = 2a0 + a1.
The element he e II-.
he = h1.
The element в e (H^)*.
0 = a1.
The Coxeter number. h = 3.
The dual Coxeter number. If' = 3.
The Lie algebra L°. L° = A1.
The lattice M c H^.
M = Zh1.
566
Appendix
The lattice M* c (H°y.
M* = \Xa1.
The fundamental alcove A G //'•.
A \h. П : a1(/i)>0, ce1(^)<1}.
The fundamental alcove А* с (Я^)*.
А* = {Ае(Я°)* ; А(й1)>0, 2A(/11)<1}.
The root system Ф in terms of the root system Ф0 of L°.
Фке,з = {|(« + (2г-1Я ; аеФ0, reZ}
ФКс! = {a + 2rA ; аеФ0, reZ}
Ф1т = {k8 ; к e Z, к 0} Multiplicity 1.
The fundamental weights <y; e Hg i = 0, 1in terms of the fundamental
weights w, i = 1,... , I of L°.
a>0 = y, (i)1=a)1+2y.
The standard invariant form on H.
{hi’hj} = ajcJ1Aij i,j = 0A
{h0,d) = 2, {h1,d)=0
(d, d}=0.
The standard invariant form on H*.
i,j = Q, 1
<% t)=o.
DYNKIN NAME At KAC NAME At I >2	567
DYNKIN NAME At KAC NAME At l>2
Dynkin diagram with labelling.
The integers c0, cb ... , c;.
1
568
Appendix
The central element c.
c = h0 + + • • • +	+ ht.
The basic imaginary root 3.
5 = a0 + a1 + --  + al_1 + al.
The element hg e H^.
hg = /ij + /i2 + • • • + ^/-i + ht.
The element в e (Я^)*.
0 = a1 + a24----haH + a;.
The Coxeter number. h = I + 1.
The dual Coxeter number. If' = I + 1.
The Lie algebra L°. L° = Al.
The lattice M c H^.
M = Z/ij + Z/i2 4-----h Z/i;_j + Z/i;.
The lattice M' c (н£)*.
M* = Zaj + Za2 4------h Za(_j + Za;.
The fundamental alcove A G Iff
A = {h e H^_ ; a; (/1) > 0 for i = 1,... , I
(/1) + a2GO + • • • + “/-1W + “/ W < 1}
The fundamental alcove A* c (Я^)*.
Л={Ле (//'’) ; A(/i;)>0 for / = 1,... , I
A (/iJ + A (/i2)H---hA (/i;_0 + A (/i;) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1п1 = {k8 ; к e Z, к 0} Multiplicity /.
The fundamental weights <y; e i = 0, 1in terms of the fundamental
weights w;, i = 1,... , I of L°.
«0 = % «! = «! + % w2 = w2 + % ..., <^ = <4^ + 7, w; = w; + y.
DYNKIN NAME At	KAC NAME At I >2	569
The standard invariant form on H.
^} = Ау	i, j = 0,1,..., I
{h0,d} = l, {ht,d)=Q
(d, d}=0.
The standard invariant form on H*.
(ai,aj} = Aij i,j = 0,l,... ,1
(a0, y) = l, (a;,y)=0
t) = o.
570
Appendix
DYNKIN NAME Bz KAC NAME Bz Z>3
Dynkin diagram with labelling.
Ю--------О--------О-------о--------о-------<> < о
3	/-1	/
Generalised Cartan matrix.
/-1
The integers a0, ar,... , a
2	-1
-1	2	-1
—2	2
The integers c0, cx,... , c;.
1 CK.----- 2	2	2	2	2	2	2	1
--------о---------о-------о----------о---------о------о	> <>
1
The central element с.
с — //,, Т h. Т2/i2 Т2/?з Т * * * Т2/iz_j Т /iz.
DYNKIN NAME Bl KAC NAME Bl l>3	571
The basic imaginary root 3.
5 = a0 + a1+2a2 + 2a3 4-----h2a;_1 + 2a;.
The element he e II'.
hg = h3 T 2h2 T 2h3 T * * * T 2h^_3 T h^.
The element в e (//'').
0 = a1+ 2a2 + 2a3 4----h 2a;_1 + 2a;.
The Coxeter number. h = 2l.
The dual Coxeter number. If' = 21 — 1.
The Lie algebra L°. L° = Bl.
The lattice M c H^.
M = Z/i3 + Z/i2 4---h Z/i;_j + Z/i;.
The lattice M* c (H°)*.
M* = Zttj + Za24-------hZaH +2Za;.
The fundamental alcove A c H^.
.1	//; : a;(/i)>0 fori=l,...,/
a1(/i) + 2a2(/i) + 2a3(/i)d--h2a;_1(/i) + 2a;(/i) < 1}.
The fundamental alcove А* с (Я^)*.
Л = {A e (//'•') ; A(/i;)>0 for i= 1,... , I
A (/ii) + 2A (/i2) +2A (/i3) + • • • + 2A (/i/~i) + A (Zi;) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1п1 = {k8 ; AeZ, k^O} Multiplicity/.
The fundamental weights <y; eHg i = 0,... , I in terms of the fundamental
weights w;, i = 1,... , I of L°.
a>0 = y, a>3 = <b3 + y, w2 = w2+ 2y,... , <y;_! = d>;_!+2y, <y; = w; + y.
572
Appendix
The standard invariant form on H.
{hi’hj} = ajcJ1Aij z,j = O, 1,... ,/
{h0,d} = l,
(d, d} = 0.
The standard invariant form on I/'.
{ai,a^ = ailciAij	i, j = 0,1,... , I
{a0,y) = l, {at, y)=0
t)=o.
DYNKIN NAME B\ KAC NAME 2A2l_1	l>3	573
DYNKIN NAME B\ KAC NAME 2A2l_1	l>3
Dynkin diagram with labelling.
Generalised Cartan matrix.
1	2	3	•	•	•	/-1
-1
2	-1
-1	2	-1
-1	2
2	-1
-1 2
1 \ -1
The integers a0, ar,... , a;.
The integers c0,c1,... ,ct.
2	2	2	2	2
-O-------О--------О--------ii < <>
The central element c.
c — //,, T h, T2/^2 T2/?з T * * * T2/z^_T 1.hi.
574
Appendix
The basic imaginary root 3.
<5 = a0 +	+ 2a2 + 2a3 4---h 2a;_1 + at.
The element hg e //'•.
hg = Ii T2 /z 2 T2 /z3 T * * * T2/z^_3 T27r2.
The element в e (Я^)*.
в = а1+ 2a2 + 2a3 4------h 2a;_1 + a;.
The Coxeter number. h = 'll — 1.
The dual Coxeter number. If' = 21.
The Lie algebra L°. L° = Cl.
The lattice M c H^.
M = Z/ij + Z/i2H---------hZ/i;_1 +2Z/i;.
The lattice M* c (H°)*.
M* = Za1 + Za2 4---------h Za;_1 + Za;.
The fundamental alcove A G Iff.
Л ^-ll.: a;(/i)>0 for/=l,...,/
a1 (h) + 2a2 (/i) + 2a3 (Я) H-h 2a;_1 (/i) + al (/i) < 1}.
The fundamental alcove A* c (Я^)*.
Л = {A e (//'/) ; A(7i;)>0 for i= 1,... , I
A (/ij) + 2A (/i2) + 2A (/i3) + • • • + 2A + 2A (/i;) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе s = {a +r<5 ; аеФ°, reZ}
ФКеi = {a + 2r<5 ; аеФр reZ}.
Ф1т= {2k8 ; AeZ, A^O}	Multiplicity/
U{(2A+1)<5; AeZ}	Multiplicity/—1.
The fundamental weights <y; e	i = 0, 1,...,/ in terms of the fundamental
weights w;, i = 1,... , / of L°.
<у0 = 'у,	+ w2 = w2 + 2y, ...,
ы1 = Mt + 2y.
DYNKIN NAME B\ KAC NAME 2A2l_1	l>3	575
The standard invariant form on II.
{hi,hj} = ajcfAij i, j = 0, 1, ... , I
{h0,d) = l, {hi,d}=G
(d, d}=0.
The standard invariant form on H*.
(a^a^a^c^ i, j = 0, 1,... , I
(a0, y) = l, (a;,y)=0 z = l,...J
t)=o.
576
Appendix
DYNKIN NAME Cz KAC NAME Cz l>2
Dynkin diagram with labelling.
<i > О----О----О-----О----О----О----О----» < ()
0	12	1-2 l-l I
Generalised Cartan matrix
0	1	2	/	—	2	l-l	I
0^2—1
1-2	2	-1
2	—	12-
/—2	•	2	-1
l-l	-1	2	-2
1 \ -1 2 ,
The integers a0,	, a;
1	2	2	2	2	2	2	2	1
<>	> »--o-------------o-о-о-о-а	< о
The integers с0,с1,... , С/
111111111
<1 > О---О------------О-О-О-О-О < О
The central element с.
с = h0 +	+ h2 + • • • +	+ ht.
The basic imaginary root 3.
8 = a0 + 2a1+2a2-\------h2a;_1 + a;.
The element hg e H^.
/ig — hx + h2 + • • • +	+ ht.
DYNKIN NAME Ci KAC NAME Ci l>2	577
The element в e (//'') •
в = '2а1+ 2a2 4----h 2a;_1 + a;.
The Coxeter number.	h = 2l.
The dual Coxeter number. /г = I + 1.
The Lie algebra L°. L° = C[.
The lattice M c .
M = Zh-i + Zh2 4-------h + Zh[.
The lattice M* с
M* = 2Za1 +2Za2H------h2Za;_1 + Za;.
The fundamental alcove A G H^.
.1	//; : a;(/i)>0 for/=l,...,/
2a1(/i) + 2a2(/i)H--h2a;_1(/i) + a;(/i) < 1}.
The fundamental alcove А* с (Я^)*.
Л={Ле(/Л') ; A(/i;)>0 for i= 1,... , I
A (/iJ + A (/i2)H-------hA +A (/i;) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; a e Ф0, r eZj
Ф1т = {kS ; AeZ, k^O} Multiplicity/.
The fundamental weights <y; e H^_ i = 0, 1,...,/ in terms of the fundamental
weights w, i = 1,... , I of L°.
w0 = % Wi = <bi + y, ы2 = ы2 + у, ...,	+ w; = w; + y.
The standard invariant form on Я.
{hi,hj} = ajCj1Aij /,j = 0,l,...,/
{h0,d) = l, {ht,d)=Q
(d, d)=0.
578
Appendix
The standard invariant form on H*.
=	i, j = 0, 1,... , I
(a0,y) = l, (at, y)=0
<% t)=o.
DYNKIN NAME C\ KAC NAME 2ЬШ I >2	579
DYNKIN NAME Czl KAC NAME 2bl+1	l>2
Dynkin diagram with labelling.
O < t)-----------О--------О--------О--------О--------О-------О--------о > о
0	12	1-2 l-l I
Generalised Cartan matrix.
0	1	2
0^2	—2
1-12-1
2	-12
/—2 l-l I
\
1 — 2	.	2	-1
l-l	-1	2	-1
I у	-2	2 )
The integers a0, ax,... , a;.
1111111111
<> < »-О-О-О-о-о-о--<> > <>
The integers с0, сь ... , с;.
122	2222221
О < <>-О-О-О--о-о--о--о > о
The central element с.
с = //,, Т2hr Т2/^2 Т * * * Т2hl_1 Т h^.
The basic imaginary root 3.
580
Appendix
The element he e H^.
he — 2/z^ T2/i2 T * * * T2/z/_i T lp.
The element в e (Я^)*.
0 = a1 + a24----ha;_i + a;.
The Coxeter number. h = I + 1.
The dual Coxeter number. If' = 21.
The Lie algebra L°.	1,° = !^.
The lattice M c H^.
M = 2Z/i1 +2Z/i2H-----h2Z/i;_1 +Z/i;.
The lattice M* с (Я°)*.
M* = Zaj + Za2 4------h Za(_j + Za;.
The fundamental alcove A G Iff
.1	//; : a;(/i)>0 for/=l,...,/
“i W + “2 W + • • • + ai-i W + ai W < 1} •
The fundamental alcove A* c (Я^)*.
Л = {Л e (/C) ; A(/i;)>0 for i= 1,... , I
2Л (/ij) + 2Л (/i2) + • • • + 2Л	+ A (/i;) < 1} •
The root system Ф in terms of the root system Ф0 of L°.
ФКе s = {a +r<5 ; аеФ“, reZ}
ФКед = {a + 2r3 ; аеФ’, reZ}.
Ф1т= {2k8 ; AeZ, к ^0}	Multiplicity/
U{(2A+1)<5; AeZ}	Multiplicity!.
The fundamental weights <y; e f/l i = 0, 1,...,/ in terms of the fundamental
weights w,, i = 1,... , I of L°.
<y0 = 'y, W|=<7j|+2y, <y2 = 0>2 + 2у, ... ,(ol_1=(i)l_1 + 2y, <y; = w; + y.
DYNKIN NAME C\ KAC NAME 2ЬШ I >2	581
The standard invariant form on II.
{hi’hj} = ajcj1Aij z, J = O, 1,/
{h0,d} = l,
(d, d}=0.
The standard invariant form on H*.
{ai,a^ = ailciAij	i, j = 0,1,... , I
(a0,y) = l, (at, y)=0
<% t)=o.
582
Appendix
DYNKIN NAME C't KAC NAME 2A2Z Z>2
(1st description)
Dynkin diagram with labelling.
t> < tl-----О-----О------О-----о-----о------о-----о < о
О 1	2
Generalised Cartan matrix.
О 1	2
О / 2	—2
1-1	2-1
2	-12
1 — 2		2	-1
/-1	-1	2	-2
1 \	-1 * 3 ъ
The integers а0,аъ... ,at.
222	2222221
<i < »-О--О--О---о--о---о--о < о
The integers с0, сь ... , с;.
122	2222222
<i < fl-О-О--О---о--о---о--о < о
The central element с.
с = //,, Т2hr Т2/^2 Т * * * Т2hl_1 Т2h^.
The basic imaginary root 3.
3 = 2a0 + 2аг + 2a2 4---h2a;_1 + a;.
DYNKIN NAME C' KAC NAME 2A2l I >2	583
The element hg e II-.
hg = h1A- h2 + • • • +	+ ht.
The element в e
в = 1а1 + 2a2 4---h 2a;_1 + a;.
The Coxeter number. h = 'll + 1.
The dual Coxeter number.	hv = ll + 1.
The Lie algebra L°. L° = Ct.
The lattice M c H^.
M = Z/ij + Z/i2 4------h Z/i;_j + Z/i;.
The lattice M* с (#£)*.
M* = Zaj + Za2 4-------h Zci^j + |Za;.
The fundamental alcove A G H^.
Л [k-ll.: a;(/i)>0 for/=l,...,/
lax(/i) + 2a2(/i)H---h2a;_1(/i) + a;(/i) < 1}.
The fundamental alcove A* c
Л={Ле(/Л') ; A(7i;)>0 for i= 1,... , I
2Л (/iJ + 2Л (/i2) + • • • + 2Л	+ 2Л (/i;) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
Фке.з = {|(а + (2г-1)3) ; «еФ/, rez}
ФКе1 = {a + rd ; a e Ф°, reZ)
ФКед = {a + 2r3 ; аеФр reZ}
Ф1ш = {£<5; AeZ, к7^0} Multiplicity/.
The fundamental weights <y; e	i = 0, 1,...,/ in terms of the fundamental
weights <7j, i = 1,... , I of L°.
a>0 = y, w1=w1+2y, a>2 = a>2 + ly, ...,
^i-i=^i-i+2y, w; = w; + 2y.
584
Appendix
The standard invariant form on H.
{hi’hj} = ajcJ1Aij z, J = O, 1,/
{h0,d} = 2,
(d, d} = 0.
The standard invariant form on H*.
i,j = 0, 1,... ,1
<% t)=o.
DYNKIN NAME C' KAC NAME 2A2l I >2	585
DYNKIN NAME C't KAC NAME 1 2 * * 5A2l l>2
(2nd description)
Dynkin diagram with labelling.
O > <>-----О----О----О-----О----О----О----<> > о
0	12	 .	
Generalised Cartan matrix.
0	1	2
0^2	-1
1-2	2-1
2	-12
1 — 2	’	2	-1
l-l	-1	2	-1
I у	-2	2 )
The integers a0, аъ ... , at.
122	2222222
o > o-о---о--о--о---о--о---о > о
The integers с0, ср ... , с;.
222	2222221
О > О-О---О--О--О---О--О---<> > t>
The central element c.
c = 2 /zfl T2hr T2h2 T * * * T2h^_^ T h^.
The basic imaginary root 3.
5 = a0 + 2ax + 2a2 4----h 2a;_1 + 2a;.
586
Appendix
The element he e H^.
he = h1 + h2-\----h /i;_j + | ht.
The element в e (Я^)*.
0 = 2a1 + 2a2-\---h2a;_1 +2a;.
The Coxeter number. h = 21 + 1.
The dual Coxeter number. If = 21+1.
The Lie algebra L°.	1? = !^.
The lattice M c H^.
M = Z/ix + Z/i2H----iZ/i^j + |Z/i;.
The lattice M* c (н£)*.
M* = Zaj + Za2 4------h ZaH + Za;.
The fundamental alcove A G lf+
.1	//; : a;(/i)>0 fori=l,...,/
2a1(/i) + 2a2(/i)H-h2a;_1(/i) + 2a;(/i) < 1}.
The fundamental alcove A* c (Я^)*.
Л = {Ле (//'/) ; A(7i;)>0 for i = 1,... , I
2Л (/ij) + 2Л (/i2) + • • • + 2Л + A (/i;) < 1} •
The root system Ф in terms of the root system Ф0 of L°.
Фке8 = {« + rd ; аеФ°, reZ}
Фкед = {a + r3 ; аеФр, reZ}
Фке,1 = {2а+(2г+1)<5 ; аеФ/. reZ}
Ф1ш = {k8 ; к е Z, к 0} Multiplicity /.
The fundamental weights <y; e i = 0, 1in terms of the fundamental
weights i = 1,... , I of L°.
<j)0 = 2y, а>1 = ы1+2у, a>2 = d)2 + 2y, ...,
(+-i=Mi_i+2y, Ml = Mt + y.
DYNKIN NAME C' KAC NAME 2A2l I >2
587
The standard invariant form on II.
{hi’hj) = ajcJ1Aij z,j = O, 1,... ,/
(h0,d) = ±,
(d, d} = 0.
The standard invariant form on H*.
^a^a^qAy i,j = 0, 1,... ,1
(a0,y)=2, (a;,y) = 0
t) = o.
588
Appendix
DYNKIN NAME Z>4
KAC NAME Z>4
Dynkin diagram with labelling.
Generalised Cartan matrix.
0	12	3	4
0/2 0-1 0 o\
1	0	2-100
2-1-1	2 -1 -1
3
4\
00-120
0	0 -1	0	2,
The integers a0, a3,... , a
The integers c0, cx,... , c;.
The central element c.
c — //,, T h. T2/i2 T ^3 T ^4*
The basic imaginary root 3.
<5 = a0 + + 2a2 + a3 + a4.
The element he e H^.
hg = h3 T 2/i2 T ^3 T ^4*
The element в e (Я^)*.
в = а1+2а2 + а3 + а4.
The Coxeter number.	h = 6.
The dual Coxeter number. If' = 6.
DYNKIN NAME /)4 KAC NAME /)4	589
The Lie algebra L°. L° = D4.
The lattice M c 77^.
M = Zhr + Z/i2 + Z/i3 + Z/i4.
The lattice M* c (tiff.
M* = Zaj + Za2 + Za3 + Za4.
The fundamental alcove A G H^.
A= {//://: a;(/i)>0 fori=l,...,4
a3(h) + 2a2(h) + a3(h) + a4(h) < 1}.
The fundamental alcove А* с
Л={Ле (//'') ; A(/i;)>0 for i= 1,... , I
A (/ii) + 2A (/i2) + A (/13) + A (/i4) < 1} •
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1п1 = {k8 ; AeZ, к ^0} Multiplicity 4.
The fundamental weights w, e /L i = 0, 1,... I in terms of the fundamental
weights i = 1,... , I of L°.
a>0 = y, a>1 = a>1 + y, a>2 = a>2 + 2y, a>3 = a)3 + y, a>4 = a>4 + y.
The standard invariant form on II.
^,^) = Ау i, J = 0,1, 2, 3,4
<Я0,б/> = 1, <Я;,с/>=0	i=l,2,3,4
(d, d}=0.
The standard invariant form on H*.
(ai,aj} = Aij	i, j = 0, 1,2, 3,4
(a0,y') = l, (a;,y)=0	i=l,2,3,4
(% y)=0.
590
Appendix
DYNKIN NAME Д KAC NAME Д l>5
Dynkin diagram with labelling.
Generalised Cartan matrix.
—o—
1-3
Ю----O
0	12	3
0 / 2	-1
1	2 -1
2-1-1	2 -1
3	-12
1-3 1 — 2 l-l
1-3
1 — 2
l-l
I
The integers a0, ax,... , at.
The integers c0, cb ... , c;.
DYNKIN NAME Dt KAC NAME Dt l>5	591
The central element c.
c = /1,, T li T 2/i2 T 2/i3 T * * * T 2/i^_2 T N i T hf
The basic imaginary root 3.
<5 = a0 +	+ 2a2 + 2a3 4----h 2a;_2 + a;_j + a;.
The element he e IE.
he = hx T 2/e T 2/i3 T * * * T 2/^2 T h[_3 T hf
The element в e	*.
в = «j + 2a2 + 2a3 4-----h 2a;_2 + a;_i + «/•
The Coxeter number. h = 'll — 2.
The dual Coxeter number. h4 = 21 — 2.
The Lie algebra L°. L° = D..
The lattice M c H^.
M = Z/ij + Z/i2 4----h UNit-t + Z/i;.
The lattice M* c (H°)*.
M* = Za1 + Za2 4-----h Za;_j + Za;.
The fundamental alcove A c H^.
A= {//://: a;(/i)>0 fori=l,...,/
a1(/i) + 2a2(/i) + 2a3(/i)H--hla^fh) + a^h) + a^h) < 1}.
The fundamental alcove А* с (Я^)*.
Л={Ле (//'’) ; A(/i;)>0 for i = 1,... , I
A (/ii) + 2A (/i2) + 2A (/i3) + • • • + 2A (/i;_2) d- (^1-1) + (^1) < 1} •
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZj
Ф1т = {АЗ; AeZ, к7^0} Multiplicity/.
The fundamental weights <y; e H^, i = 0, 1,... , / in terms of the fundamen-
tal weights i= 1,... , I of L°.
a>0 = y, a>t=a>t + y, a>2 = <b2 + 2y, ...,
w, 2 = <!>,.• 2 +2y, Wt_t = <bt_t + y, Wt = <bt + y.
592
Appendix
The standard invariant form on H.
i, 7 = 0,1,...,/
{h0,d} = l, {hi,d)=O
(d, d} = 0.
The standard invariant form on //'.
{ai^aj} = Aij /,7 = 0,1,...,/
(a0,y) = l, (at, y)=0
t)=o.
DYNKIN NAME E6
KAC NAME E6
593
DYNKIN NAME
E6
KAC NAME E6
Dynkin diagram with labelling.
Generalised Cartan matrix.
0
1
2
3
4
5
6
0
2
0
0
0
-1
0
1	2	3	4	5	6
0	0	0	-1	0	0\
2 -1	0	0	0	0
-12-1000
0-1	2-1-1	0
00-1200
0	0-1	0	2-1
0	0	0	0 -1	2,
The integers a0, ar,... , at.
1	2	3	2	1
О-------О---------О--------о-------о
2 о
1	о
The integers с0,с1,... , сР
1	2	3	2	1
О-------О---------о-------о---------о
2	о
1 о
The central element с.
с — //,, Т h, Т 2/i2 Т 3 h3 Т 2//4 Т 2/ц Т .
The basic imaginary root 3.
5 = a0 + a1 + 2a2 + 3a3 + 2a4 + 2a5 + a6.
594
Appendix
The element he e H^.
he = h3 + 2 /z 2 T 3/z3 T 2 /z4 T 2 /z 3 T h&.
The element в e (Я^)*.
в = а1 + 2a2 + 3 a3 + 2a4 + 2a5 + a6.
The Coxeter number. h=12.
The dual Coxeter number. h4 =12.
The Lie algebra L°. L° = E6.
The lattice M c
M = Z/ix + Z/i2 + Z/i3 + Z/i4 + Z/i5 + Z/i6.
The lattice M' c (н£)*.
M* = Zo^ + Za2 + Za3 + Za4 + Za5 + Za6.
The fundamental alcove A G H^.
A={E : /7 : at(h)>0 fori=l,...,6
a3(h) + 2a2(h) + 3a3(h) + 2a4(h) + 2a5(h) + a6(h) < 1}.
The fundamental alcove A* c (Я^)*.
Л = {A e (//'/) ; A(7i;)>0 for i = 1,... , 6
A (/ij) + 2A (/i2) + ЗА (/i3) + 2A (/i4) + 2A (Тл5) + A (/i6) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1т = {АЗ; AeZ, k^O} Multiplicity 6.
The fundamental weights <y; e H^, i = 0,... , 6 in terms of the fundamental
weights w;, i = 1,... , 6 of L°.
<у0 = 'у,	+	<y2 = w2 + 2y, (i)3 = a>3 + 3y,
w4 = w4 + 2y, a>5 = w5 + 2y, a>6 = a>6 + y.
DYNKIN NAME E6 KAC NAME E6	595
The standard invariant form on H.
{h0,d} = l, <Х</)=0	z=l,...,6
(d, d}=0.
The standard invariant form on H*.
Kaj} = Aij =
(a0, y) = l, (ai,y')=0	z=l,...,6
<% t)=o.
596
Appendix
DYNKIN NAME Ё7 KAC NAME Ё7
Dynkin diagram with labelling.
1	2	3	4	6	7	0
О--------------О-------------О---------------<j>-------------о-----------о---------------о
5
Generalised Cartan matrix.
0	1	2	3	4	5	6	7
0/2000000 -1\
1	02 -1	00000
2	0-12-10000
3	00-12-1000
4
5
6
0
0
0
0
0
0
7 \-l	0
0-1	2-1-1	0
00-1200
0	0-1	0	2-1
0	0	0	0 -1	2,
The integers a0, ar,... , a
1	2
О-------О
3	4	3	2	1
Ю-------------<j>-----------О-----------о-------------о
2
The integers с0, q,... , с;.
1	2	3	4	3	2	1
О---------О----------О----------<j>--------О--------О----------О
2
The central element с.
с = //,, Т h, Т 2/i2 Т 3/i3 Т 4/г4 Т 2/ц Т 3//7 Т 2//7.
The basic imaginary root 3.
8 = a0 + + 2a2 + 3a3 + 4a4 + 2a5 + 3a6 + 2a7.
DYNKIN NAME E7 KAC NAME E7	597
The element hg e IE.
he = hx T 2//2 T 3/z3 T Ah4 T 2/г3 T 3hg -\- 2.h-.
The element в e (//'') •
в = а1+ 2a2 + 3a3 + 4a4 + 2a5 + 3a6 + 2a7.
The Coxeter number. h=18.
The dual Coxeter number. h': = 18.
The Lie algebra L°. L° = E1.
The lattice M c IE.
M = Z/ij + Z/i2 + Z/i3 + Z/i4 + Z/i5 + Z/i6 + Z/i7.
The lattice M* с (#£)*.
M* = Zaj + Za2 + Za3 + Za4 + Za5 + Za6 + Za7.
The fundamental alcove A G H^.
.1	//; : a;(/i)>0 for г =1,..., 7
a1(/i) + 2a2(/i) + 3a3(/i) + 4a4(/i) + 2a5(/i) + 3a6(/i) + 2a7(/i) < 1}.
The fundamental alcove A* c
Л ={Ле(//'') ; A(7i;)>0 for i= 1,... , 7
A (M + 2A (/i2) + ЗА (/i3) + 4A (/i4) + 2A (/i5) + ЗА (/i6) + 2A (/i7) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1т = {k8 ; AeZ, k^AO} Multiplicity 7.
The fundamental weights <y; e H^, i = 0,... ,7 in terms of the fundamental
weights w;, i = 1,... , 7 of L°.
w0 = y, ы1 = ы1 + у, w2 = w2 + 2y, w3 = <7j3 + 3y, ы4 = ы4+Ау,
w5 = w5 + 2y, a>6 = a>6 + 3y, w7 = w7+2y.
598
Appendix
The standard invariant form on H.
{h0,d) = l, {hi,d}=G z = l,...,7
(d, d) =0.
The standard invariant form on I/'.
(ai,aj} = Aij i,j=0,...,7
(a0, y) = l, (a;,y)=0 z = l,...,7
t)=o.
DYNKIN NAME Es KAC NAME Es
599
DYNKIN NAME Ё8 KAC NAME Ё8
Dynkin diagram with labelling.
0
O
1	2	3	4	5	7	8
O------------О--------------О--------------о-------------<j)-----------о--------------о
6
Generalised Cartan matrix.
0/2
1 -1
2
3
4
5
6
7
0
0
0
0
0
0
1
-1
2
-1
0
0
0
0
0
0
2
0
-1
2
-1
0
0
0
0
0
3	4	5	6	7
0	0	0	0	0
0	0	0	0	0
-10000
2-1000
-1	2-1	0	0
0-1	2-1 -1
00-120
00-102
0000-1
8
0\
0
0
0
0
0
0
-1
2/
0
8 \ 0
The integers a0, ax,... , at.
The integers c0, q,... , c;.
1	2	3	4	5	6	4	2
О---------О---------О---------О---------О---------<j>------О---------О
3
The central element с.
с — hQ Т 2/г^ Т 3/i2 + 4/г3 Т	Т 6/^5 Т 3//7 Т 4/г7 Т 2/г§.
The basic imaginary root 3.
8 = a0 + 2a1 + 3a2 + 4a3 + 5a4 + 6a5 + 3a6 + 4a7 + 2a8.
600
Appendix
The element he e H^.
he = 2/z^ T 3/i2 T4/z3 T 5/i4 T 6h3 T 3h7 T4h7 T 2/z8.
The element в e (Я^)*.
в = 2a3 + 3a2 + 4a3 + 5a4 + 6a5 + 3a6 +4a7 + 2a8.
The Coxeter number. h = 30.
The dual Coxeter number. If' = 30.
The Lie algebra L°. L° = E8.
The lattice M c
M = Z/ij + Z/i2 + Z/i3 + Z/i4 + Z/i5 + Z/i6 + Z/i7 + Z/i8.
The lattice M' c (н£)*.
M* = Zaj + Za2 + Za3 + Za4 + Za5 + Za6 + Za7 + Za8.
The fundamental alcove A G IE.
.1	//; : a;(/i)>0 fori=l,...,8
2a3 (/i) + 3a2(/i) +4a3 (/i) + 5a4(/i) + 6a5(/i)
+3a6(/i)+4a7(/i) + 2a8(/i) < 1}.
The fundamental alcove A* c (Я^)*.
Л={Ле (//') ; A(/i;)>0 for i= 1,... , 8
2Л (/ij + ЗЛ (/i2) +4Л (/i3) + 5 A (/i4) + 6A (/i5)
+ЗЛ (/i6) + 4Л (/i7) + 2Л (/i8) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = {a + r8 ; аеФ0, reZ}
Ф1ш = {£<5; AeZ, к 7^0} Multiplicity 8.
The fundamental weights <y; e H^, i = 0,... ,8 in terms of the fundamental
weights w;, i = 1,... , 8 of L°.
ш0 = у, d)1 = 0)1+2y, ы2 = ы2 + 3у, a)3 = d)3+4y, a>4 = a>4 + 5y,
ш5 = ш5 + 6у, ш6 = ш6 + 3у, ш7 = ш7+4у, ш8 = ш8 + 2у.
DYNKIN NAME Es KAC NAME Es	601
The standard invariant form on H.
М =	^ = 0,-..,8
<й0,<7> = 1,	z=l,...,8
(d, d}=0.
The standard invariant form on H*.
(ai,aj} = Aij i,j = 0, ...,8
(a0,y) = l, (at, y)=0	z=l,...,8
<% t)=o.
602
Appendix
DYNKIN NAME F4 KAC NAME F4
Dynkin diagram with labelling.
О--О--<> > О-о
0	12	3	4
Generalised Cartan matrix.
	0	1	2	3	4
0	< 2	-1	0	0	°\
1	-1	2	-1	0	0
2	0	-1	2	-1	0
3	0	0	—2	2	-1
4	0	0	0	-1	2/
The integers a0,	, a;					
1	2	3	4	2
О------О-------> О-----------О
The integers с0, q,... , с;.
1	2	3	2	1
О-----О------о > <>--------о
The central element с.
с = //,, Т 2/z3 Т 3/i2 Т 2/z3 Т й4.
The basic imaginary root 3.
5 = a0 + 1a1 + 3a2 + 4a3 + 2a4.
The element h0
hg = 2/ix T 3/i2 T 2/i3 T hg.
The element в e *.
в = 2a1 + 3tt2 +4«3 + 2a4.
The Coxeter number. h=12.
The dual Coxeter number. If' = 9.
The Lie algebra L°. L° = Fg.
The lattice M c H^.
M = Z/ij + Z/i2 + Z/i3 + Z/i4.
DYNKIN NAME F4 KAC NAME F4	603
The lattice M* с (Я°)*.
M* = Zcej + Za,2 + 2Za'3 + 2Za4.
The fundamental alcove A G H^.
a;(/i)>0 for г =1,..., 4
2a1(h) + 3a2(h)+4a3(h) + 2a4(h) < 1}.
The fundamental alcove A* G (Я1-1)
А* = {Ае(Я°)* ; A(/i;)>0 for i= 1,... ,4
2 Л (M + ЗА (/i2) + 2A (/i3) + A (/i4) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = | a + r8 ; a e Ф0, reZj
Ф1т = {АЗ; AeZ, A^O} Multiplicity 4.
The fundamental weights <y; e//' z = 0,... , / in terms of the fundamental
weights w;, i = 1,... , / of L°.
w,, = y, ы1 = ы1+2у, ы2 = ы2 + 3у, ы3 = ы3 + 2у, ы4 = ы4 + у.
The standard invariant form on Я.
Khj} = ajc^Aij =
{h0,d) = l, {hi,d}=G
(d, d}=0.
The standard invariant form on Я*.
{oil, a^a^CiAij i,j = Q,...,l
(a0,y) = l, (at, y)=0
(y, y)=0.
604
Appendix
DYNKIN NAME F] KAC NAME 2F6
Dynkin diagram with labelling.
О----О----О < О-----О
0	12	3	4
Generalised Cartan matrix.
		0	1	2	3	4
0	/	2	-1	0	0	°\
1		-1	2	-1	0	0
2		0	-1	2	—2	0
3		0	0	-1	2	-1
4	I	0	0	0	-1	2/
The integers a0, a3,... , a;						
1		2		3	2	1
O—		—o-			ZD—		О
The integers c0, cx,... , c;.
1	2	3	4	2
о-----О------<> < О-------о
The central element с.
с = //,, Т 2 /z । Т 3/i2 + 4/z2 Т 2/?д.
The basic imaginary root 3.
8 = a0 + 2ax + 3a2 + 2a3 + a4.
The element he e H^.
hg= 2//1 T 3 /z 2 + 4/z3 T 2//2.
The element в e (Я^)*.
в = 2а1 + 3tt2 + 2a3 + a4.
The Coxeter number. h = 9.
The dual Coxeter number. h4 =12.
The Lie algebra L°. L° = F4.
The lattice M c H^.
M = Zhl + Zh2 + 2Zh3 + 2Zh4.
DYNKIN NAME F\ KAC NAME 2Ё6	605
The lattice M* c (tiff.
M* = Zaj + Za2 + Za3 + Za4.
The fundamental alcove A G Hg.
.1	//; : a;(/i)>0 for г = 1,..., 4
2a1(h) + 3a2(h) + 2a3(h) + a4(h) < 1}.
The fundamental alcove A* G (flj)* •
Л={Ле (//'') ; A(/i;)>0 for i= 1,... ,4
2Л (^) + 3A(/f2)+4A (7i3) + 2A (/i4) < 1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе8 = {« + ей ; аеФ°, reZ}
ФКеj = {a + 2r<5 ; аеФ°, reZ}
Ф1т= {2k8 ; AeZ, k^O}	Multiplicity 4
U{(2A+1)<5; AeZ}	Multiplicity 2.
The fundamental weights w, e Hg i = 0,... , / in terms of the fundamental
weights w,, i = 1,... , / of L°.
<y0 = y, (1)1=ш1+2у, <y2 = w2 + 3y, w3 = w3+4y, w4 = w4 + 2y.
The standard invariant form on H.
Khj} = ajc^Aij =
{h0,d} = \, {hi,d}=G i=l,...l
(d, d)=0.
The standard invariant form on H*.
(a.i, a^ = aGl <\А^	i,j = 0,...,l
{a0,y) = l, (at, y)=0	i=l,...l
t)=o.
606
Appendix
DYNKIN NAME G2 KAC NAME G2
Dynkin diagram with labelling.
	O— 0	—	1	2
Generalised Cartan matrix.		0	1	2
	0 (	2	-1	0
	1	-1	2 	-1
	2 \	0	-3	2
The integers a0, a2,... , a;.	1		2	3
О------о > о
The integers c0, cb ... , c;.
1	2	1
О-----В—>—D
The central element c.
c = //,, T 2 /z । T /z2.
The basic imaginary root 3.
5 = a0 + 2a1+ 3a2.
The element he e 77j|.
/i0 = 2/i1 + /i2.
The element в e (dd£)*.
G = 2a1 + 3a2.
The Coxeter number. h = 6.
The dual Coxeter number.	hv = 4.
The Lie algebra L°. L° = G2.
The lattice M c dd^.
M = Z/i1 + Z/i2.
DYNKIN NAME G2 KAC NAME G2	607
The lattice M* с (Я°)*.
M* = Zo^ + 3Za2.
The fundamental alcove A c H^.
a;(/i)>0 for i= 1,2
2a1(/i) + 3a2(/i)< 1}.
The fundamental alcove А* с. (Я1').
А* = {Ае(Я°)* ; A(/i;)>0 for z= 1,2
2A(/i1) + A(/i2)<1}.
The root system Ф in terms of the root system Ф0 of L°.
ФКе = |a + r8 ; аеФ°,гег)
Ф1т = {k8 ; AeZ, k^O} Multiplicity 2.
The fundamental weights <y; e//' i = 0,... , I in terms of the fundamental
weights w;, i = 1,... , I of L°.
<У0 = % <y1=w1+2y, <y2 = w2 + y.
The standard invariant form on Я.
{h^hj} = ajcJ1Aij =
(h0, <7) = 1, <Я;,с/>=0
(d, d}=0.
The standard invariant form on H*
(oii, a^ = a^ ciAij	i,j = 0,...,l
(a0, y) = l, (a;,y)=0	i=l,...,l
t) = o.
608
Appendix
DYNKIN NAME G\ KAC NAME 3Z>4
Dynkin diagram with labelling.
	O- 0		 1		S3 2
Generalised Cartan matrix.		0	1	2
	0 (	2	-1	0
	1	-1	2 	-3
	2 \	0	-1	2
The integers a0, ax,... , a;.	1		2	1
О-----о <	о
The integers c0, cb ... , c;.
1	2	3
О-----О—t)
The central element c.
c = //,, T 2/ix T 3//2.
The basic imaginary root 3.
5 = a0 + 2a1 + a2.
The element he e H^.
he = 2h1 + 3h2.
The element в e (Я^)*.
в = 2а1 + а2.
The Coxeter number. h = 4.
The dual Coxeter number. If' = 6.
The Lie algebra L°. L° = G2.
The lattice M c H^.
M = Z/i1 + 3Z/i2.
DYNKIN NAME &2 KAC NAME d).	609
The lattice M* с
M* = Zaj + Za2.
The fundamental alcove A G H^.
a;(/i)>0 for i= 1,2
2a1(/i) + a2(/i)< 1}.
The fundamental alcove A* G (flj)* •
Л= {Ле (/Г) ; A(/z;)>0 for i= 1,2
2Л (/z1) + 3A(/z2)< 1}.
The root system Фо in terms of the root system Ф0 of L°.
ФКе 8= {a + r<5 ; аеФ°, reZ}
ФКеi = {a + 3r<5 ; аеФр reZ}
Ф1ш= {3k8 ; AeZ,	k^O}	Multiplicity 2
U{(3A+1)<5;	AeZ}	Multiplicity!
U{(3A + 2)<5;	AeZ}	Multiplicity!
The fundamental weights <y; e//' z = 0,... , / in terms of the fundamental
weights <7j, i = 1,... , I of L°.
a>0 = y, (o1=(o1+2y, <o2 = <o2 + 3y.
The standard invariant form on II.
{hi’hj} = ajcJ1Aij i,j = 0,l,2
{h0,d} = \,	/ = 1,2
{d, d)=0.
The standard invariant form on H*.
z,j = O, 1,2
= t {ai,y) = 0 z = l,2
(% y)=0.
Notation
Symbol
Meaning
Page of definition
[*y]	Lie product of elements	1
[HK]	Lie product of subspaces	1
[A]	the Lie algebra of an associative	
	algebra A	2
v®v'	tensor product	152
VAv'	exterior product	271
G)	the Killing form on a finite	
	dimensional Lie algebra	39
G)	the standard invariant form on a	
	Kac-Moody algebra	367
(Л	a bilinear form on the loop algebra	418
G)o	a contravariant form	520
{>}	a symmetric scalar product	121
	partial order on weights	185
[V : L(M)]	the multiplicity of L(ji) in V	459
1, • • • у Cli	vector associated with an affine	
	Cartan matrix	386
ad x	the adjoint map	7
A=(Aij)	a Cartan matrix	71
A=(A‘j)	a generalised Cartan matrix (GCM)	319
AJ	a principal minor of A	344
A°	the underlying Cartan matrix of an	
	affine Cartan matrix A	394
A	the fundamental alcove	410
610
Notation
611
Symbol	Meaning	Page of definition
A	the closure of the fundamental	
	alcove	413
A*	the fundamental alcove in the dual	
	space	415
A*	the closure of the fundamental dual	
	alcove	415
81	the set of alcoves	411
В	the subalgebra H®N	177
c	the Casimir element	238
c	the canonical central element	391
co, Cl, ... , cl	vector associated with an affine	
	Cartan matrix	388
c(A)	scalar action of generalised Casimir	
	operator	487
ch V	the character of an L-module V	241
ch V	the character of a module in	
	category 0	459
C	the fundamental chamber	112, 247, 378
c	closure of the fundamental chamber	247, 378
C(V)	the Clifford algebra	282
C(V)+	positive part of the Clifford algebra	283
C(V)-	negative part of the Clifford algebra	283
C [t, r1]	the algebra of Laurent polynomials	417
d	the scaling element	388
dx,... ,dt	degrees of the basic polynomial	
	invariants	222
d°W	the Weyl dimension of an	
	L°-module	491
D=(dt)	a diagonal matrix	110, 390
Do	an endomorphism in the basic	
	representation	516
ea	a root vector	88
et	a fundamental root vector	96
et	a generator of L(A) or L(A)	323, 332
	a characteristic function	242
c(A)	the characteristic function ел	487
Et	a generator of L°	421
/,	a root vector for —	96
612
Notation
Symbol	Meaning	Page of definition
fi	a generator of ISA) or ISA)	323, 332
Ft	a generator of L°	421
FL(X)	the free Lie algebra on a set X	161
sW)	the general linear Lie algebra of	
	degree n over k	5
G	the adjoint group	207
h	the Coxeter number of a simple	
	Lie algebra	252
h	the Coxeter number of an affine	
	algebra	485
lr	the dual Coxeter number of an	
	affine algebra	485
hi	a fundamental coroot	88, 320
ha	the coroot of the root a	89, 397
h'a	the element of H corresponding to	
	a in H	46
ho	the coroot of в	405
ha(n)	an endomorphism of the basic	
	module	513
^(0)	an endomorphism of the basic	
	module	515
ht a	the height of a root a	62
H	a Cartan subalgebra of a Lie algebra	23
H	a Cartan subalgebra of a	
	Kac-Moody algebra	334
H*	the dual space of H	46
HQ	a rational vector space in H	56
нж	a real vector space in H	56
тт<	the dual space of H:	57
Hi	a generator of L°	421
Hi	a hyperplane	112
Hf	the positive side of hyperplane H	112
Hr	the negative side of hyperplane H	112
H	the diagonal subalgebra of L	324
i	the kernel of the map from L(A) to	
	L(A)	105, 331
i+	the positive subspace of I	105, 479
i	the negative subspace of I	105, 479
Notation
613
Symbol	Meaning	Page of definition
J	an orbit	166
ДЛ)	the maximal submodule of M(A)	185, 455
j(T)	the modular j-function	533
К	a set of positive imaginary roots	380
К,	the kernel of the map from 11 (L) to	
	M(A)	178
KA	the set of vectors и with Au > 0	339
	the generalised partition function	473
/	the rank of L	59
/(w)	the length of w	63
L	a Lie algebra	1
Ln	a power of the Lie algebra L	7
L(n)	a power of the Lie algebra L	8
^0,x	the null component of x in L	23
La	a root space of L	36, 333
L(X, R)	the Lie algebra with generators X	
	and relations R	163
L{A)	the simple Lie algebra with Cartan	
	matrix A	99
L{A)	the Kac-Moody algebra with	
	GCM A	331
L{A)'	the derived subalgebra of the	
	Kac-Moody algebra ISA)	335
L(A)	a Lie algebra associated with Cartan	
	matrix A	99
L(A)	a Lie algebra associated with	
	GCM A	323,
L(Ay	the fixed point subalgebra of <j on	
	L(A)	166
LW	the irreducible module with highest	
	weight A	186, 455, 525
La	a root space of L(A)	328
L°	the simple Lie algebra with Cartan	
	matrix A0	416
La	a reflecting hyperplane	246
La,k	an affine hyperplane	409
^0,1	a wall of the fundamental alcove	409
Lo, Lr,... ,Ll	the walls of the fundamental alcove	412
614
Notation
Symbol
Meaning
Page of definition
£	a set of affine hyperplanes	409
	a set of affine hyperplanes in the	
	dual space	414
£(L°)	the loop algebra of L°	417
£(L°)	a central extension of the loop	
	algebra	420
£(L°)	realisation of an untwisted	
	Kac-Moody algebra	420
£(L°)T	realisation of a twisted Kac-Moody	
	algebra	432
mA	a highest weight vector in a Verma	
	module	180, 452
ma	multiplicity of a root a	454
M	a lattice	407
M*	a lattice in the dual space	413
M(A)	a Verma module	178, 452
	the orthogonal subspace of a	
	subspace M	40
w	Monster Lie algebra	535
ni	an automorphism of L(A)	373
n(w)	the number of positive roots made	
	negative by w	63
Na,p	structure constant	89
N	positive subalgebra of L(A)	103, 324
N-	negative subalgebra of L(A)	103, 324
N	positive subalgebra of L(A)	107, 331
N-	negative subalgebra of L(A)	107, 331
W)	normaliser of a subalgebra H	23
0	Bernstein-Gelfand-Gelfand	
	category of modules	452
pQi)	the number of partitions of k	507
Pi(k)	the number of partitions of k into	
	/ colours	508
P(L)	algebra of polynomial functions	
	on L	208
P(L)G	G-invariant polynomial functions	
	on L	210
Notation
615
Symbol	Meaning	Page of definition
Piirf	W-invariant polynomial functions	
	on PI	211
PSL^	the modular group	531
Ш)	the number of partitions of A into	
	positive roots	182
Q	the root lattice	103, 328
Q+	positive part of the root lattice	103, 328
Q-	negative part of the root lattice	103, 328
Q°	root lattice of L°	404
QC^,... ,Xl)	quadratic form	73
sa	a ring of functions on H*	242
si	fundamental reflection	63, 373
	reflection	60
	reflection corresponding to root в	405
^a,k	affine reflection	410
M„(C)	special linear Lie algebra of degree	
	n over C	52
supp a	support of a root a	378
Supp/	support of a function /	241
5(L)	symmetric algebra of L	201
W	G-invariants in the symmetric	
	algebra of L	223
.S'(7/)"'	W-invariants in the symmetric	
	algebra of H	223
	a linear map on H	406
ta	a linear map on H*	413
t(M)	translation subgroup of the affine	
	Weyl group	407
t(M*)	translation subgroup of the affine	
	Weyl group	413
T(L)	tensor algebra of L	152
T(V)	tensor algebra of V	324
T	a subalgebra of L(A)	500
p-	the negative part of P	500
U(L)	universal enveloping algebra of L	153
1I(L)+	the ideal L1I(L) of 11 (L)	475
y*	the dual module of V	306
w0	longest element of the Weyl group	65
616
Notation
Symbol	Meaning	Page of definition
(wo)r	longest element of Wj	170
Wi	the weight of at	267
w	the Weyl group of a semisimple	
	Lie algebra	60
w	the Weyl group of a Kac-Moody	
	algebra	373
w°	the Weyl group of L°	394
w.	a Weyl subgroup of W	170
wrr	the group of сг-stable elements of W	169
X	the weight lattice	190, 466
x+	dominant integral weights	190, 466
x++	strictly dominant integral weights	469
Y«(z)	a vertex operator	514
Z(L)	the centre of the enveloping algebra	226
a^... ,al	fundamental roots of a semisimple	
	Lie algebra	62
аъ... ,an	fundamental roots of a Kac-Moody	
	algebra	377
av	dual root	150
У	the fundamental weight <y0 of an	
	affine algebra	389
ra(j)	component in a vertex operator	515
8	the basic imaginary root of an	
	affine algebra	384
Д	the Weyl denominator	245
Д	the Kac denominator	469
Д	the Dynkin diagram of a semisimple	
	Lie algebra	80
Д(А)	the Dynkin diagram of a	
	Kac-Moody algebra	353
Д(т)	Dedekind’s delta function	533
<Kq)	Euler’s «^-function	491
Ф	the root system of a finite	
	dimensional Lie algebra	36
Ф	the root system of a Kac-Moody	
	algebra	377
ф+	set of positive roots	58, 'ill
Ф	set of negative roots	58, ill
Notation
617
Symbol	Meaning	Page of definition
Фу	the dual root system of Ф	148
Фке	the real roots in Ф	377
Фъп	the imaginary roots in Ф	377
Ф|<е.-,	the short real roots in Ф	395
Фке,1	the long real roots in Ф	395
Фке,1	the intermediate real roots in Ф	395
ф°	the root system of L°	394
ф° S	the short roots in Ф0	400
фО	the long roots in Ф0	400
Хл	a central character	226
Х°(А)	an irreducible character of L°	487
Л(У)	the exterior algebra of V	271
Л;(У)	the i th exterior power of V	271
п	fundamental roots of a semisimple	
	Lie algebra	58
п	fundamental roots of a Kac-Moody	
	algebra	320, 334
п	fundamental roots of a Borcherds	
	algebra	524
Пу	fundamental roots in Фу	148
IT	fundamental coroots of a	
	Kac-Moody algebra	320, 397
п°	fundamental roots in Ф0	394
Пге	real fundamental roots of a	
	Borcherds algebra	525
nim	imaginary fundamental roots of a	
	Borcherds algebra	527
р	sum of the fundamental weights	228
р	an element satisfying p (/;,) = 1	460
р	an element in Borcherds’ character	
	formula	526
6	the element 8 —aoao	404
0,	an automorphism of L	108
01	the highest root	251
0S	the highest short root	251
00	an orbit representative	433
(О	an automorphism of L(A)	333
д)	an automorphism of L(A)	323
618	Notation
Symbol	Meaning	Page of definition
	the fundamental weights of a simple Lie algebra	190
w0,«i, ...	the fundamental weights of an affine Kac-Moody algebra	494
n no	generalised Casimir operator	461 an operator on a Г-module in category 0	500
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Index
abelian Lie algebra, 7, 46
adjoint group, 210
adjoint module, 7, 466
affine algebras - summary of properties, 392
affine Cartan matrix, 358
affine hyperplane, 409
affine Kac-Moody algebra, 386
affine list, 354
affine reflection, 410
affine type, 337, 342, 344
affine Weyl group, 404, 408
alcove, 410, 413, 415
arrow on Dynkin diagram, 80
associated graded algebra, 203
automorphism, 25, 323, 333, 373, 519
basic polynomial invariants, 220
basic representation, 508, 516
bijection between H. H*, 46, 390
Borcherds algebra, 519
Borcherds’ character formula, 525, 526
Borcherds’ denominator formula, 526, 539
canonical central element, 391, 392
Cartan decomposition, 36, 45
Cartan matrix, 71
Cartan subalgebra, 23, 334
Casimir element, 238
category 0, 452
central character, 226, 235, 239
centre of universal enveloping algebra, 226
chain of roots, 50
chamber, 246
character of an L-module, 241
character of a module in O, 454
character of a Verma module, 244, 255, 454
characteristic function, 242
Chevalley group, 120
Chevalley’s theorem, 222
classification of Cartan matrices, 82
classification of Dynkin diagrams, 74
classification of simple Lie algebras, 118
classification of GCMs of affine type, 356
classification of GCMs of finite type, 352
classification of GCMs of indefinite
type, 359
Clifford algebra, 282
closure of fundamental chamber, 247
cocycle, 418
complete reducibility theorem, 262
completely reducible module, 7
composition series of a Verma module, 257
conjugacy, 27
conjugacy of Cartan subalgebras, 34
contraction map, 298, 302
contravariant bilinear form, 520
coroot, 89, 397
Coxeter group, 66, 376
Coxeter number, 252, 485
Dedekind’s Д-function, 533
deletion condition, 64
derivation, 25
differential operator, 509
direct sum of Lie algebras, 43
dominant integral weights, 190
dominant maximal weights, 497, 499
dual Coxeter number, 485
dual module, 306
dual root, 150
dual root system, 148
Dynkin diagram, 72, 353
Dynkin name, 540
Eisenstein series, 533
Engel’s theorem, 20
equivalent Cartan matrices, 81
equivalent GCMs, 336
equivalent representations, 5
Euler’s ф-function, 491
629
630
Index
Euler’s identity, 494
existence theorem for simple Lie
algebras, 117
exterior powers, 272
extraspecial pair of roots, 94
factor algebra, 3
filtered algebra, 202
finite list, 352
finite type, 336, 344, 350
fixed point subalgebra, 166, 172, 432, 445
free associative algebra, 98, 99
free Lie algebra, 161
fundamental alcove, 415
fundamental chamber, 112, 247, 249, 378
fundamental module, 267
fundamental modules for Az, 273
fundamental modules for 5Z, 276, 289
fundamental modules for Cz, 302
fundamental modules for /)/. 280, 292
fundamental modules for E6. 307
fundamental modules for Elt 308
fundamental modules for E3. 310
fundamental modules for F4, 314
fundamental modules for G2, 316
fundamental modules for L (АД 504
fundamental reflection, 62, 373
fundamental region, 413, 415, 532
fundamental root, 334, 377, 395
fundamental system of roots, 58, 61
fundamental weight, 190, 192, 494
Gauss’ identity, 493, 494
general linear Lie algebra, 5
generalised Cartan matrix (GCM), 319
generalised Casimir operator, 461, 465
generalised eigenspace, 16
generalised partition function, 473
generators and relations for a Lie algebra, 99,
163, 482
graded algebra, 202
graph automorphism of a simple Lie
algebra, 165
graph automorphism of an affine algebra, 426
Harish-Chandra homomorphism, 227
height of a root, 62
highest root, 251
highest short root, 251
highest weight vector, 452
homogeneous realisation, 517
homomorphism of Lie algebras, 4
ideal of a Lie algebra, 2
identification theorem for Kac-Moody
algebras, 331
imaginary root, 377, 382, 384
indecomposable Cartan matrix, 83
indecomposable GCM, 336
indecomposable module, 7
indefinite type, 337, 344, 350
inner automorphism, 26
integrable module, 466
intermediate root, 395, 400
invariant bilinear form, 363, 520
invariant polynomial function, 210
irreducible module, 7
irreducible module for a semisimple Lie
algebra, 199
irreducible module in G, 455
isomorphism of Lie algebras, 4
j-function, 533
Jacobi identity, 1
Jacobian determinant, 28
Jacobian matrix, 28
Jordan canonical form, 14
Kac’ character formula, 469, 472
Kac’ denominator formula, 471
Kac-Moody algebra, 331
Kac name for a twisted affine algebra, 451,
540
Killing form, 39
Killing isomorphism, 223
Kostant’s multiplicity formula, 260
lattice, 407, 414
Laurent polynomial, 417
length of element of Weyl group, 63
level of module, 494
Lie algebra, 1
Lie algebra of type Az, 122, 543
Lie algebra of type Sz, 128, 545
Lie algebra of type Cz, 132, 547
Lie algebra of type Dz, 124, 549
Lie algebra of type F6, 140, 551
Lie algebra of type F7, 140, 553
Lie algebra of type F8, 140, 555
Lie algebra of type F4, 138, 557
Lie algebra of type G2, 135, 559
Lie algebra of type Ap 561
Lie algebra of type =2A2, 563, 565
Lie algebra of type Az, I > 2, 567
Lie algebra of type 5Z, 570
Lie algebra of type B\ =2A2Z_15 573
Lie algebra of type Cz, 576
Lie algebra of type Cj = 2£>z+1, 579
Lie algebra of type Cz = 2A2Z, 582, 585
Lie algebra of type Z)4, 588
Lie algebra of type Z>z, I > 5, 590
Index
631
Lie algebra of type £6. 593
Lie algebra of type E7, 596
Lie algebra of type £8. 599
Lie algebra of type F4, 602
Lie algebra of type £J = 2£6. 604
Lie algebra of type G2, 606
Lie algebra of type Gl2 = 3Z>4, 608
Lie monomial, 163
Lie word, 163
Lie’s theorem, 11
locally nilpotent map, 107
long root, 145, 395, 397
longest element of Weyl group, 65
loop algebra, 418
Macdonald’s identities, 487, 488
Macdonald’s identity for Ap 489
Macdonald’s identity for A.'v 490
Macdonald’s ф-function identity, 491
Macdonald’s twisted ф-function
identity, 492
maximal weight, 497
minimal realisation, 320, 396
modular form, 532
modular group, 531
modular j-function, 533
module for a Lie algebra, 5
module for the basic representation, 516
Monster Lie algebra, 535, 539
multiplicity formula, 260, 474
multiplicity of a root, 334, 425, 440, 450, 454
nilpotent Lie algebra, 8
no-ghost theorem, 536
normaliser, 23
null component, 23
orbit, 166
partial order on weights, 185
partition function, 182, 507
Poincare-Birkhoff-Witt (PBW) basis
theorem, 155
polarisation, 215
polynomial functions, 208
positive definite quadratic form, 74
positive imaginary roots, 382, 527
positive system of roots, 58, 61
primitive vector, 478
principal minor, 344
quadratic form, 73
quintuple product identity, 490
rank, 35
real minimal realisation, 334
real root, 377, 394, 443
realisation, 319
realisation of twisted affine algebra, 432, 445
realisation of untwisted affine algebra, 421
reduced expression, 63
reflecting hyperplane, 246
reflection, 60
regular element, 23
representation of a Borcherds algebra, 524
representation of a Kac-Moody
algebra, 452
representation of a Lie algebra, 5
representation of a nilpotent Lie algebra, 18
representation of a semisimple Lie
algebra, 199
representation of a soluble Lie algebra, 13
residue, 418
Riemann surface, 531
root, 36, 334, 377
root lattice, 148, 328
root space, 36, 48, 334
scaling element, 388
semisimple Lie algebra, 9, 42, 85
short root, 145, 395, 396
simple Lie algebra, 9
soluble Lie algebra, 8
soluble radical, 9
special linear Lie algebra, 52
special pair of roots, 94
spin modules, 289, 292
spin representations, 281
standard invariant form, 367
standard list of Dynkin diagrams, 81
standard list of Cartan matrices, 82
Steinberg’s multiplicity formula, 265
string of weights, 497, 504
structure constants, 89
subalgebra, 2
submodule, 7
support of a function, 241
symmetric algebra, 201
symmetric GCM, 345
symmetric tensor, 205
symmetrisable GCM, 346, 348
symmetrisation, 206
tensor algebra, 152, 324
total order on a vector space, 58
triangular decomposition, 104, 107, 326, 331
trichotomy theorem, 337, 350
triple product identity, 489
trivial simple Lie algebra, 9
twisted affine algebras, 429
twisted graph automorphisms, 429
twisted Harish-Chandra homomorphism,
228, 234
632	Index
uniqueness theorem for simple Lie
algebras, 95
universal Borcherds algebra, 521
universal enveloping algebra, 153
universal property of enveloping algebra, 153
universal property of free Lie algebra, 161
untwisted affine Cartan matrix, 417
upper half plane, 531
vacuum vector, 517
Verma module, 178, 182, 184, 452
vertex operator, 512
П’-invariant polynomial function,
211, 220
walls of a chamber, 247
weight, 19
weight lattice, 190
weight space, 19, 102, 184
weight space decomposition, 19
weight vector, 102
Weyl group, 60, 373
Weyl’s character formula, 258
Weyl’s denominator formula, 253
Weyl’s dimension formula, 261, 267