Mathematical
Surveys
and
Monographs
Volume 58
Lectures on
Representation Theory and
Knizhnik-Zamolodchikov
Equations
Pavel i. Etingof
Igor B. Frenkel
Alexander A. Kiriiiov, jr.
American Mathematical Society

Editorial Board
Georgia M. Benkart Tudor Stefan Ratiu, Chair
Michael Renardy
The authors were supported in part by the following NSF grants:
P.E.: DMS#9700477, I.F.: DMS#9700765, A.K.: DMS#9610201.
P.E. was also supported in part by an NSF postdoctoral fellowship.
1991 Mathematics Subject Classification. Primary 81R40, 81R50;
Secondary 17B67, 17B69.
Abstract. This book is devoted to the study of some of the mathematical structures arising in
conformal field theory and their ^-deformations. This field, though relatively young, is an area of
intensive study by both mathematicians and physicists, and has already produced many beautiful
results in mathematics and physics. In the book, we have tried to give a self-contained exposition
of the theory of Knizhnik-Zamolodchikov equations and related topics that requires no previous
knowledge of physics. The book would be useful to everyone interested in mathematical physics,
from graduate students to experts. It can be used as a basis for a one-semester graduate course.
To our wives
Tanya, Marina, and Varya
Library of Congress Cataloging-in-Publication Data
Etingof, P. I. (Pavel I.), 1969-
Lectures on representation theory and Knizhnik-Zamolodchikov equations / Pavel I. Etingof,
Igor B. Frenkel, Alexander A. Kirillov, Jr.
p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 58)
Includes bibliographical references and index.
ISBN 0-8218-0496-0 (hardcover : alk. paper)
1. Broken symmetry (Physics) 2. Quantum groups. 3. Kac-Moody algebras. 4. Mathe-
Mathematical physics. I. Frenkel, Igor. II. Kirillov, Alexander A., 1967- . III. Title. IV. Series:
Mathematical surveys and monographs ; no. 58.
QC174.17.S9E88 1998
530.14'2—dc21 98-2948
CIP
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Contents
Preface xiii
Lecture 1. Introduction 1
1.1. Simple Lie algebras and Lie groups and their generalizations 1'
1.2. Affine Lie algebras 1
1.3. Quantum groups 3
1.4. Knizhnik-Zamolodchikov equations 6
1.5. Quantum affine algebras and quantum Knizhnik-Zamolodchikov
equations 8
1.6. Further generalizations of affine Lie algebras and quantum groups. 11
1.7. Contents of the book 12
Lecture 2. Representations of finite-dimensional and affine Lie algebras 15
2.1. Simple Lie algebras 15
2.2. Cartan matrices of simple Lie algebras 16
2.3. Highest-weight modules over simple Lie algebras and contravariant
forms 17
2.4. Finite-dimensional representations and irreducibility of Verma mod-
modules 18
2.5. The maximal root, the Coxeter numbers, and the Casimir operator 19
2.6. Affine Lie algebras 20
2.7. Verma modules and Weyl modules for affine Lie algebras 22
2.8. Integrable representations of affine Lie algebras 24
2.9. The Virasoro algebra and its action on g-modules 25
2.10. Generating functions and currents 26
Lecture 3. Knizhnik-Zamolodchikov equations 29
3.1. Classification of intertwining operators 29
3.2. Operator KZ equation 30
3.3. Gauge invariance of the intertwining operators 33
3.4. KZ equations for correlation functions 33
3.5. Consistency and g-invariance of the KZ equations 36
3.6. Analyticity of the correlation functions 37
3.7. Correlation functions span the space of solutions of the KZ equa-
equations 39
3.8. Trigonometric form of the KZ equations 42
3.9. Consistent systems of differential equations and the classical Yang-
Baxter equation 44

x CONTENTS
Lecture 4. Solutions of the Knizhnilc-Zamolodchikov equations 49
4.1. The simplest solution of the KZ equations for g = s\i 49
4.2. Simplest level one solution and Gauss hypergeometric function 51
4.3. Integral formulas for level one solutions 53
4.4. Solutions of the KZ equations for s^: arbitrary level 56
4.5. Solutions of the KZ equations for a general simple Lie algebra 60
Lecture 5. Free field realization 63
5.1. Fock modules and vertex operators 63
5.2. Matrix elements of products of vertex operators 66
5.3. Interpretation of the rational part of solutions of the KZ equations
in terms of creation and annihilation operators. 67
5.4. Factorization of solutions of the KZ equations 69
5.5. Free field realization of Verma modules over s\.i 69
5.6. Intertwining operators in the free field realization: level zero 73
5.7. Intertwining operators in the free field realization: positive level 75
5.8. Calculation of the correlation functions 77
Lecture 6. Quantum groups 79
6.1. Hopf algebras and their representations 79
6.2. Definition of quantum groups 81
6.3. Quasitriangular structure and braided tensor categories 84
6.4. Quantum Yang-Baxter equation and representations of braid
groups 87
6.5. Quantum double construction 88
6.6. Quantum double construction for Uq($e) 89
6.7. Quantum Casimir element 92
6.8. Intertwining operators and their commutation relations 93
Lecture 7. Local systems and configuration spaces 97
7.1. Local systems 97
7.2. Cohomology and homology with coefficients in local systems 99
7.3. Configuration spaces and OrlLk-Solomon algebra 101
7.4. Cohomology of configuration spaces with coefficients in local sys-
systems associated with the KZ equations for s^ 103
7.5. Gauss-Manin connection 105
7.6. Relative homology 106
7.7. The case of arbitrary jj 110
Lecture 8. Monodromy of Knizhnik-Zamolodchikov equations 113
8.1. Monodromy of KZ equations and the braid group 113
8.2. Asymptotics of solutions of the KZ equations 115
8.3. Asymptotics of the correlation functions 118
8.4. Monodromy with respect to an infinite base point 119
8.5. Commutation relations for intertwining operators 122
8.6. Equivalence of categories and Drinfeld-Kohno theorem 124
8.7. Geometric approach to equivalence of categories 127
Lecture 9. Quantum affine algebras 131
9.1. Definition of quantum affine algebras 131
9.2. Evaluation representations of quantum affine algebras 132
CONTENTS
9.3. Intertwining operators 135
9.4. Quasitriangular structure in quantum affine algebras 136
9.5. Factorization of the .R-matrix 137
9.6. Evaluation representations and .R-matrix for Uq CB) 139
9.7. Quantum currents 144
9.8. Quantum Sugawara construction in degree zero 145
Lecture 10. Quantum Knizhnik-Zamolodchikov equations 147
10.1. Operator quantum KZ equation 147
10.2. Quantum correlation functions 150
10.3. Quantum KZ equations for correlation functions 151
10.4. A fundamental set of solutions of the quantum KZ equations 151
10.5. Holonomic systems of difference equations 152
10.6. Analyticity of the fundamental solution of the quantum KZ equa-
equations 153
10.7. The noncommutative product formula for the fundamental solution 155
10.8. Classical limit of the quantum KZ equations 155
10.9. Modified quantum KZ equations 156
10.10 Another proof of the quantum KZ equations 157
Lecture 11. Solutions of the quantum Knizhnik-Zamolodchikov equations for
Bl2 161
11.1. (^-analogues of classical special functions 161
11.2. Jackson integral 163
11.3. The g-hypergeometric function 164
11.4. Some second order difference equations 165
11.5. The simplest solutions of the quantum KZ equations and the q-
hypergeometric function 166
11.6. Integral formulas for solutions 168
Lecture 12. Connection matrices for the quantum Knizhnik-Zamolodchikov
equations and elliptic functions 171
12.1. Linear difference equations for functions of one complex variable 171
12.2. Connection relation for the g-hypergeometric equation 174
12.3. The connection matrix for the quantum KZ equations in the sim-
simplest case 175
12.4. The connection matrix and the exchange matrix for intertwining
operators 176
Lecture 13. Current developments and future perspectives 179
13.1. KZ equations: quantum versus classical 179
13.2. Monodromy of the KZ equations, tensor categories, and quantum
groups 180
13.3. Vertex operator algebras, conformal field theory and their g-defor-
mations 182
13.4. Elliptic KZ equations and special values of the central charge 185
13.5. Double loop algebras and quantum affine algebras 186
13.6. Quantum KZ equations and physical models 187
References 189
Index 197

Preface
The last twenty years have seen an especially active interaction between mathe-
mathematics and physics. This interaction has given birth to a number of remarkable new
areas of mathematics and has provided powerful new tools in various fields of the-
theoretical physics. This book is devoted to one of those new areas, which deals with
mathematical structures of conformal field theory and their g-deformations. It arose
from the course of lectures on the classical and quantum Knizhnik-Zamolodchikov
equations, given by one of the authors of this book (I. B. F.) in the Spring of 1992
at Yale University, and inspired by his recent (at that time) joint paper with Nicolai
Reshetikhin. The instructor was lucky to find two enthusiastic graduate students
and future co-authors, who improved and extended the exposition and later gave
related courses at Harvard (P. I. E.) and MIT (A. A. K.).
By that time, all three of us had already been severely afflicted with the "g-
disease", a dangerous mathematical illness whose earnest victim was Euler, but
which was first diagnosed by Richard Askey. Mathematicians working in practically
every field, be it algebra, geometry, analysis, differential equations - you name it
- are vulnerable to its addictive charm. The first symptom of the g-disease is
that one day you realize that most of the results obtained or acquired during your
mathematical life admit a g-deformation. The second stage is indicated by the idea
that the g-case is much more interesting than the classical one. It was at that stage
that we started writing the second part of the book, with the intention to g-deform
all structures of conformal field theory. This turned out to be a difficult task, which
has taken five years, and is still not completed. Luckily, during these years the area
grew up quite significantly, and we were able to use some of the more recent results
and refer the reader to active new developments and promising research problems.
When writing this book, we followed one of the rules we learned from Israel
Gelfand: for every new theory, choose the simplest non-trivial example and write
down everything explicitly for this example. Therefore, all general constructions
and theorems are accompanied by explicit calculations for the Lie algebra sb-
We also wanted to put this new area of mathematics into the general perspec-
perspective of development of representation theory. With this intention we wrote the first
and the last lecture, trying to help the reader navigate in the mist of modern repre-
representation theory and mathematical physics. The bare scheme of the development of
the theory is captured by the diagram at the end of Section 1.6 in the Introduction.
Familiar to many mathematicians working in the area, it is closer to alchemical
formulas than to mathematics. Nevertheless, it was instrumental in the discovery
of some of the structures studied in this book, and it might still be useful again.
This book is written for people who are familiar with the representation theory
of simple Lie algebras, but requires no knowledge of physics. It can be used for
teaching an advanced one-semester graduate course, though the instructor and the

xiv PREFACE
students should work really hard, as it was in the Spring of 1992 at Yale. The
number of lectures exactly corresponds to the number of weeks in one semester at
Yale University. Each lecture is assumed to take two and a half hours and can be
split into two or three weekly classes depending on the strength and enthusiasm
of the participants. Our experience shows that the golden mean is always the
best solution. Also the first and the last lectures should not be taken as seriously
(for some people as lightly) as the rest of the book, giving the students and the
instructor an opportunity to relax.
This theory was created by the efforts of many people, and some results were
circulating as "folklore" for a number of years. We tried to give the references in the
first lecture and especially in the introduction to each of the consecutive lectures,
but we would like to apologize in advance for unintentional omissions.
One of the main purposes of this book was to simplify, extend and provide all
the necessary details of the results in the original article [FRJ. We hope that in our
present exposition we have corrected more misprints and inaccuracies in that paper
than we have added new ones.
The credit for the eventual existence of this book rightfully belongs to our
editor, Sergei Gelfand, who achieved a seemingly impossible goal of persuading the
authors - after a five year delay - to bring this book to a conclusion.
We are grateful for fruitful discussions and useful comments to many people,
including M. Finkelberg, D. Kazhdan, A. Matsuo, N. Reshetikhin, O. Schiffman,
K. Styrkas, A. Varchenko. We thank the National Science Foundation for par-
partial support of this project, and the American Mathematical Society for its final
materialization.
Lecture 1. Introduction
1.1. Simple Lie algebras and Lie groups and their generalizations
The creation of the theory of Lie algebras and Lie groups around the turn of the
century was a momentous event in the development of modern mathematics. The
subsequent discovery and classification of the building blocks of the theory, namely,
simple Lie algebras and simple Lie groups, was no less significant and revealed a
remarkable structure of these "simple" objects. It also pointed to the special place
of complex simple Lie algebras and groups and their compact real forms. The study
of the structure of complex simple Lie algebras and groups is intrinsically related
with the study and classification of their complex finite dimensional representations,
which can also be viewed as unitary representations of their compact real forms.
The structure theory and representation theory of compact and complex simple
Lie algebras and groups are among the most complete and harmonious chapters
of modern mathematics. They have numerous relations with different areas in
mathematics and physics, and provide important examples in various subjects.
The remarkable success of the theory of compact and complex simple Lie algebras
and groups inspired different generalizations. However, the inherent rigidity of the
theory did not allow much freedom, and most of the earlier generalizations were
based on replacement of the complex and real fields by finite, local and global fields.
These generalizations led to deep theories, which are still developing. However,
the absence of the complex field entailed a break with differential geometry and
mathematical physics.
The other two generalizations, which appeared much later and still were based
on the complex field, had as their starting point a certain fundamental structure of
simple Lie algebras that becomes apparent in their classification and is known as
Cartan matrices or Dynkin diagrams. The latter structures contain only minimal
information about simple Lie algebras, but are sufficient to reconstruct the Lie
algebras, via the so-called Serre's relations. Serre's relations were generalized in
two different ways, which led to the new algebras that will be the main subject of
this book. The definition of the two classes of algebras suggests that their theories
are parallel in many respects to the theories of compact and complex Lie algebras
and groups. Besides, these theories turn out to have a very rich structure and
numerous new unexpected connections with various subjects in mathematics and
physics.
1.2. AfHne Lie algebras
The first generalization of simple Lie algebras based on Serre's relations was
obtained independently by V. Kac and R. Moody in 1967. They applied Serre's

xiv PREFACE
students should work really hard, as it was in the Spring of 1992 at Yale. The
number of lectures exactly corresponds to the number of weeks in one semester at
Yale University. Each lecture is assumed to take two and a half hours and can be
split into two or three weekly classes depending on the strength and enthusiasm
of the participants. Our experience shows that the golden mean is always the
best solution. Also the first and the last lectures should not be taken as seriously
(for some people as lightly) as the rest of the book, giving the students and the
instructor an opportunity to relax.
This theory was created by the efforts of many people, and some results were
circulating as "folklore" for a number of years. We tried to give the references in the
first lecture and especially in the introduction to each of the consecutive lectures,
but we would like to apologize in advance for unintentional omissions.
One of the main purposes of this book was to simplify, extend and provide all
the necessary details of the results in the original article [FRJ. We hope that in our
present exposition we have corrected more misprints and inaccuracies in that paper
than we have added new ones.
The credit for the eventual existence of this book rightfully belongs to our
editor, Sergei Gelfand, who achieved a seemingly impossible goal of persuading the
authors - after a five year delay - to bring this book to a conclusion.
We are grateful for fruitful discussions and useful comments to many people,
including M. Finkelberg, D. Kazhdan, A. Matsuo, N. Reshetikhin, 0. Schiffman,
K. Styrkas, A. Varchenko. We thank the National Science Foundation for par-
partial support of this project, and the American Mathematical Society for its final
materialization.
Lecture 1. Introduction
1.1. Simple Lie algebras and Lie groups and their generalizations
The creation of the theory of Lie algebras and Lie groups around the turn of the
century was a momentous event in the development of modern mathematics. The
subsequent discovery and classification of the building blocks of the theory, namely,
simple Lie algebras and simple Lie groups, was no less significant and revealed a
remarkable structure of these "simple" objects. It also pointed to the special place
of complex simple Lie algebras and groups and their compact real forms. The study
of the structure of complex simple Lie algebras and groups is intrinsically related
with the study and classification of their complex finite dimensional representations,
which can also be viewed as unitary representations of their compact real forms.
The structure theory and representation theory of compact and complex simple
Lie algebras and groups are among the most complete and harmonious chapters
of modern mathematics. They have numerous relations with different areas in
mathematics and physics, and provide important examples in various subjects.
The remarkable success of the theory of compact and complex simple Lie algebras
and groups inspired different generalizations. However, the inherent rigidity of the
theory did not allow much freedom, and most of the earlier generalizations were
based on replacement of the complex and real fields by finite, local and global fields.
These generalizations led to deep theories, which are still developing. However,
the absence of the complex field entailed a break with differential geometry and
mathematical physics.
The other two generalizations, which appeared much later and still were based
on the complex field, had as their starting point a certain fundamental structure of
simple Lie algebras that becomes apparent in their classification and is known as
Cartan matrices or Dynkin diagrams. The latter structures contain only minimal
information about simple Lie algebras, but are sufficient to reconstruct the Lie
algebras, via the so-called Serre's relations. Serre's relations were generalized in
two different ways, which led to the new algebras that will be the main subject of
this book. The definition of the two classes of algebras suggests that their theories
are parallel in many respects to the theories of compact and complex Lie algebras
and groups. Besides, these theories turn out to have a very rich structure and
numerous new unexpected connections with various subjects in mathematics and
physics.
1.2. AfHne Lie algebras
The first generalization of simple Lie algebras based on Serre's relations was
obtained independently by V. Kac and R. Moody in 1967. They applied Serre's

LECTURE 1. INTRODUCTION
construction to the case of extended (and more general) Cartan matrices and dis-
discovered a new class of infinite-dimensional Lie algebras, usually called affine Lie
algebras. Their first remarkable observation was the realization of the affine Lie
algebras as loop algebras. We will briefly recall this realization.
Let g be a simple Lie algebra over C, and let A be a Cartan matrix associated
to g. The Serre construction attaches to A certain generators and relations, which
yield a Lie algebra ${A) isomorphic to g. Each Cartan matrix A associated to a
simple Lie algebra has a canonical extension by one row and column to a so-called
extended Cartan matrix A. Then the Kac-Moody algebra g(A) is isomorphic to
a canonical central extension g of the loop algebra associated to g, which we now
describe.
Let Lq be the Lie algebra of all trigonometric polynomial maps from the unit
circle to the Lie algebra g with the pointwise commutation. It is convenient to
identify Ljj with the Lie algebra g ® C[t, f], so that the Lie bracket is given by
A.1)
x,y
P,QeC[t,t~
This algebra has a unique nontrivial central extension, which is described below. We
denote by (,) the unique (up to a factor) invariant bilinear form on g: (x, [y, z\) =
{[x,y],z),x,y,z ? g. The central extension g = ?g® Cc is the Lie algebra with the
following commutation law:
A.2)
±
P'(t)Q(t)dt ¦
[c, i®P]=0.
It is often convenient to extend the Lie algebra g by a derivation. We denote
by 8 = g ffi Cd the Lie algebra containing g as a subalgebra with the commutation
law
A.3)
[d,x®P] = x®tP', a; eg,
[c, d] = 0.
Independently of the discovery of affine Lie algebras, I. Macdonald in 1972
proved an "affine" version of the Weyl denominator formula for characters of irre-
irreducible finite-dimensional representations of g. Inspired by the Macdonald general-
generalization of the Weyl formula, Kac in 1974 showed that this formula can be interpreted
as the denominator formula for highest weight representations of affine Lie algebras.
We will briefly recall a few basic facts about representation theory.
The structure theory of simple Lie algebras and, in particular, Serre's construc-
construction, yield a canonical triangular decomposition of g:
A.4)
g = n+ © fj © n.
where f) is a Cartan subalgebra and n* are nilpotent subalgebras of g. Let Cx be
a one-dimensional representation of the subalgebra b = n+ © fj, which is trivial on
n+ and given by the weight A e f)* on fj. Then the induced representation
A.5)
Mx = Ind?CA
1.3. QUANTUM GROUPS
is called the Verma module with the highest weight A. It has a unique irreducible
quotient Lx, which is finite-dimensional for a certain subset P+ C f)*, called the set
of dominant weights. Since the dual representation of Lx is also finite-dimensional
and irreducible, we can define an involution * on the set of dominant weights via
the isomorphism Lx = Lx..
The Serre construction for affine Lie algebras yields a similar triangular decom-
decomposition
A.6) g = n+©fj©ir
where f) = rjffiCc and we identified f) = fj® 1. Then one can repeat the construction
of representations in the affine case. Let CXik be a one-dimensional representation
of the subalgebra b = n+ ffi fj, which is trivial on n+ and is given by a weight A ? f)*
on f), and in addition c acts by a scalar k ? C. We define the Verma module for
the affine Lie algebras by
A.7)
MKk = Indfc
¦\,k
and the corresponding irreducible quotient Lx,k- The above representations of g
are canonically extended to the representations of g via the Sugawara-Segal con-
construction. The action of the element d defines the grading in MXtk and ix,/c- In
particular, the subspaces corresponding to the highest eigenvalue of d are naturally
identified with Mx and Lx, respectively.
The character of Verma modules Mx for a simple Lie algebra g is given, up to
an exponent ex, by the reciprocal of the Weyl denominator. Similarly, the character
of Verma modules M^ for an affine Lie algebra g is given by the reciprocal of the
Macdonald denominator. The striking feature of the Macdonald generalization is
the replacement of trigonometric functions appearing in the classical case by the
Jacobi elliptic theta-functions in the affine case. For example, for g = sfe the Weyl
denominator, considered as a function on the maximal torus T = R/2ttZ, is given
by 2isinx, whereas the Macdonald denominator for the Lie algebra sfe is given by
\8{x,t), where
A.8
8(x,t) =2
- qn)(l -2qncosx + q
,2n\
where q = e21".
The idea that theta-functions are natural generalizations of exponential and
trigonometric functions was well known and explored in the first half of the 19-th
century. What was so remarkable in Macdonald's generalization is that this idea
was combined with the structure theory of simple Lie algebras. The conceptual al-
algebraic interpretation of Macdonald's generalization led to important developments
in the representation theory of affine Lie algebras. The geometric understanding
of the appearance of theta-functions in representation theory came even later with
the discovery of conformal field theory and the realization of characters as certain
functional integrals over an elliptic curve.
1.3. Quantum groups
The second generalization of simple Lie algebras that we will consider was
obtained independently by V. Drinfeld and M. Jimbo in 1985. They shifted the

LECTURE 1. INTRODUCTION
emphasis from a simple Lie algebra jj by regarding Serre's relations as defining re-
relations for its universal enveloping algebra W(jj), and then presented an astonishing
g-analogue of Serre's relations which preserves all the axioms of a Hopf algebra.
This Hopf algebra, denoted by W,(g), and its dual Hopf algebra were neatly named
"the quantum group associated to jj". The important new feature of quantum
groups is that they are no longer cocommutative, i.e. the comultiplication A is
not equal to the opposite comultiplication Aop = P o A, where P is a permutation
of two factors Uq{$). However, they possess a so-called quasitriangular structure
which serves as a substitute for the cocommutativity of the classical universal en-
enveloping algebras. Namely, there exists an element R in a certain completion of
Uq(g) ®W,(jj) which intertwines comultiplication and its opposite and satisfies some
natural additional relations with respect to comultiplication. One has
RA{x) =
A.9) (A®Id)(R)
(Id® A)(R) = R13R12,
where Ri2 = ? Oi ® &i ® 1, etc. for R = ? a*
i i
imply the famous Yang-Baxter identity
x e W,
® biy ai,bi G W,(g). These properties
A.10)
R12R13R23 =
which was an important motivation for the search for quantum groups.
In order to develop a representation theory of quantum groups by analogy
with the classical case, one needs to replace the triangular decomposition A.4) of
a Lie algebra jj by the corresponding Poincare-Birkhoff-Witt decomposition of its
universal enveloping algebra
A.11)
U{g) = U{n+)
and then consider its g-deformation defined via a g-version of Serre's relations. Then
it is straightforward to define (^-analogues of Verma modules, which we denote
by Mx, and their irreducible quotients L\. When q is a formal or a complex
parameter not equal to a root of unity, the structure of representations and relations
between them is similar to the undeformed case. In particular, the tensor category
C(g,q) of finite-dimensional representations of Uq(Q) with a weight decomposition
is semisimple with the simple objects Lqx,\ € P+. There is however one essential
new feature that appears only in the quantum case, namely a profound relation to
the theory of braid groups.
Let Vi,... , Vn be representations of Ug(g). We define intertwining operators
acting in the tensor product of these representations:
where Pt,t+i is the permutation of the i-th and the (i + l)-th factor. Then for n = 3
the Yang-Baxter identity becomes the braid relation
In particular, if Vi = Vj — • • ¦ — Vn, we obtain a representation of the Artin
braid group of n strands. Moreover, since in general (R12J # 1, this representation
does not factor through a representation of the symmetric group of n elements.
1.3. QUANTUM GROUPS
Another important distinction of the quantum group Uq(g) from its classical
counterpart is that in general the antipode 7 of this Hopf algebra does not satisfy the
involutive property f2 = 1. This implies that we can define the dual representation
(Lx)* at least in two different ways using the antipode 7:
A.14)
veL{,
or its inverse. It is customary to choose the former. As in the classical case, we
have an isomorphism (?')* = L\..
The properties of the "braiding" in the tensor category C(jj, q) can be axioma-
tized, which gives a notion of "braided tensor category". Adding to these axioms
suitably formulated axioms of duality and the Casimir element, we get a notion of
"balanced rigid braided tensor category", also known as "a ribbon category" [RT].
These categories are closely related with the topology of framed knots and links.
The "building blocks" of this category are the spaces of intertwining operators
which for generic q have the same dimension as their classical counterparts. By
duality they are canonically identified with the space of invariants
A-16) H^. = Invw<
The operator R also induces an isomorphism Hx = H"x. Composing the basic
building blocks HXfl, one can obtain spaces of intertwining operators for any number
of representations. In particular,
A.17)
® LV =
Now the action of the operator R on the middle two factors induces another im-
important linear transformation
A.18)
B
^AtAs
This map is closely related to the associativity isomorphisms in the category C(g, q)
obtained by comparison of different arrangements of brackets in A.17). If we fix fi
and v in A.18), then the matrix coefficients of B will depend on 6 elements of P+
and are usually called 6,7-symbols. For g = SI2 the spaces of intertwining operators
A.14) are 0 or 1 dimensional and 6,7-symbols can be viewed as complex numbers,
while in general they are also linear maps. The braid relation A.13) implies a
similar identity for 6j-symbols:
A.19)
acting in the spaces
A.20) 0 H
-B12S23-B12 = B23B12B23,
® H,
¦K
where B12 = B ® 1 and B23 = 1'
This is only one of several identities that determine the structure of the braided
tensor category C(g,q). The rest of the relations can also be deduced from the

6 LECTURE 1. INTRODUCTION
representation theory of the quantum group Uq(g), and they fit remarkably well to
the topological moves in knot theory.
1.4. Knizhnik-Zamolodchikov equations
The discovery of two generalizations of simple Lie algebras and groups shattered
the firm belief in the rigidity of their structure. However, it was soon realized that
the affine Lie algebras and the quantum groups have a deep interrelation, which
gives another indication of the canonical nature of both generalizations.
Though the representation theory of affine Lie algebras can be developed to a
great extent similarly to its finite-dimensional counterpart, it possesses an impor-
important extra structure, discovered by E. Witten and known as Wess-Zumino-Novikov-
Witten (WZNW) conformal field theory. It is only defined for integral values of
k, though many algebraic constructions are extended to an arbitrary k. We will
be mainly interested in the generic case k ? Q. Following the general construc-
construction in conformal field theory developed by G. Moore and N. Seiberg, one can
define a rather unusual tensor category C(g, k) associated with the representations
L\,k, A G P+, for a fixed value k = k — hw of the central element (here hw is the
dual Casimir number for g).
It is possible to describe the tensor category C(g, k) without invoking the full
structure of conformal field theory by considering in addition to the integrable
representations a class of evaluation representations of the affine Lie algebra g with
zero value of the central element. Let V be a representation of g, then it can
be extended to a representation V(z) of Lg by assigning to the formal variable
t a nonzero complex number z. One can then consider a family of intertwining
operators depending on z:
A.21) <&(«) : Lv>k —* ix,*:®iM(z),
where ® is a suitably completed tensor product. For generic values of k, these
operators are in one-to-one correspondence with the finite-dimensional intertwining
operators acting in the top level, i.e. with the elements of Hom(it/,LA <?> L^). To
eliminate the dependence of the coefficients on the variable z, one can require
a certain additional intertwining property of the family of operators $>{z) with
respect to the action of d. We denote the space of such intertwining operators by
Hx . Then, by the remarks above, the dimension of these spaces is equal to the
dimensions of the spaces -ffA/i.
Next we consider the matrix coefficients of products of the intertwining opera-
operators, also called correlation functions:
A.22)
¦••*n(«n)«O>,
where $i(zt) G H^iXt with /j,n = fio = 0, and vo G Lo.fc.^o e ^o,fc are the
vacuum vectors. Strictly speaking the products of the intertwining operators and
the correlation functions A.22) should be viewed as formal power series in the
variables z^. An important and remarkable property of these formal series is their
absolute convergence in the region \zi \ > \zi\ > ... > \zn\ > 0. The simplest way to
prove this property is to derive differential equations for the functions \P(zi,... ,zn)
that take values in the tensor product L\l ® ... ® L\n. These differential equations
were first discovered by V. Knizhnik and A. Zamolodchikov in their study of WZNW
1.4. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
conformal field theory. They compose the following commutative linear system:
A.23) " ""- 9
where hw is the dual Coxeter number of g,Pi is the half sum of positive roots of
g acting on the i-th tensor factor, rti(f-) acts only on i-th and j-th tensor factors
and is the simplest trigonometric solution of the classical Yang-Baxter equation
The intertwining property of the operators $i(z,) immediately implies that the
Knizhnik-Zamolodchikov equation can be reduced to the subspace of invariants
InvB(XAl®...®LAJ.
The Knizhnik-Zamolodchikov equations are- regular over the space Xn of all
ordered sets of n distinct complex numbers. Therefore, they prescribe a flat con-
connection on the trivial vector bundle over this space. This connection gives rise to
a representation of the fundamental group of Xn, which is the pure braid group.
Moreover, one can consider "half monodromies"
A-25) Piii+iAi,i+1*(zi,... ,zn),
where Ai^+y is the counterclockwise analytic continuation from the domain \zi\ >
\zi+i\ to the domain \zt\ < \zi+i\ and P;,i+i is the permutation of the i-th and
(i + l)-th factors ?Ai and ?Ai+1. The resulting function is again a solution of
the Knizhnik-Zamolodchikov equation with the permuted points Zi,zi+i and the
corresponding representations L\i, ?Al+1 ¦ Fixing a basis of solutions correspond-
corresponding to the factorization A.22), one can define the connection matrix between two
solutions. In particular, for n = 4 we obtain a linear map
A.26) ?:©??
One can show that connection matrices for arbitrary n factor into the connection
matrices for n = 4, and they satisfy the relations analogous to the relations A.19)
for 6j-symbols of quantum groups. Furthermore, one can verify the rest of the
relations of braided tensor category and conclude that C(q,k) yields the second
class of examples.
The similarity between the tensor categories C(g, k) and C(g, q) turns out to be
an exact equivalence for q = exp( ,fc"feV,), where m = 1 for simply-laced 0, m = 2
for the Lie algebras with Dynkin diagrams of types Bn, Cn, FA, and m = 3 for G2.
In particular, there is an isomorphism H^v = H^ such that the linear maps B in
A.18) and A.26) become identical. The equivalence of two categories for formal
q was first established by Drinfeld, and more recently D. Kazhdan and G. Lusztig
refined his result by allowing q to be a complex number provided that k + hv ? q+_
One can make the correspondence quite transparent by studying the explicit form
of the solutions of Knizhnik-Zamolodchikov equations. These solutions were first
obtained by V. Schechtman and A. Varchenko, who also found a beautiful geometric
interpretation of the integral form of the solutions as a canonical pairing between
homology and cohomology groups of configuration spaces.

8 LECTURE 1. INTRODUCTION
Even in the simplest case when g = 3B these solutions include all general-
generalized hypergeometric functions studied in the last century. The conformal invari-
ance of the solutions implies that the first nontrivial correlation function corre-
corresponds to the case n = 4 and essentially depends on the cross-ratio z of the four
points zi,Z2,Z3,Z4. Finally, the simplest nontrivial case occurs where the space
Invs (?aj ® ?a2 ® L\3 <g> L\4) is two-dimensional. Then the solution can be ex-
expressed in terms of the Gauss hypergeometric function:
A.27)
where (x)n = x(x + 1)... (x + n — 1). It satisfies the second order linear equation
A.28)
z(l -
+ [c - (o + b +
- ab F = 0
and for Re c > Re b > 0, F can be represented in the form of the integral
A.29) F(o,6,c;z) =
rF)r(c - i
{-t)c-b-1{X-tz)-adt,
The function F is regular and takes value 1 at the origin. The second solution of
A.28) is given by z1~cF(a — c + 1, b — c + 1,2 — c; z). We can also consider power
series solutions of A.28) axound z = 00. Then the solutions with fixed asymptotic
behavior are also given by hypergeometric functions z~aF(a, 1 — c+1,1 — b+a, z)
and z~bF(b, l-c + 6,1 - a + b, z). The 2x2 connection matrix relating the
solutions at z = 0 and z = 00 is composed of F-functions and after an elementary
diagonal transformation becomes a matrix consisting of monomials of sin(-j^N),
where N is a linear combination of the parameters a, b, c.
Thus, the properties of the Gauss hypergeometric function allow us to identify
the 2x2 connection matrix B A.26) with 6,j-symbols B A.18) for the quantum
group W?(sl2). By considering more general hypergeometric functions satisfying
n-th order differential equations one can explicitly verify the isomorphism of the
tensor categories C(sl2,«0 and C(sl2,<?).
The solutions of Knizhnik-Zamolodchikov equations expressed in terms of gen-
generalized hypergeometric functions reflect the existence of special structure in repre-
representations of affine Lie algebras. This structure becomes apparent in the free field
realizations of highest weight modules that was first discovered by M. Wakimoto for
5B and generalized to an arbitrary affine Lie algebra by B. Feigin and E. Frenkel.
Thus, the study of the Knizhnik-Zamolodchikov equations unites the representa-
representation theory of affine Lie algebras and quantum groups, the theory of generalized
hypergeometric functions, and the geometry of configuration spaces.
1.5. Quantum affine algebras and quantum
Knizhnik-Zamolodchikov equations
One can combine two generalizations of simple Lie algebras based on Serre's
relations and define a ^-deformation of the universal enveloping algebra of g, which
we also denote by Uq(g). Adding one extra generator yields a deformation of U(g)
denoted by Uq(g). The algebras Uq(g) and Uq(g) are called quantum affine algebras.
One can develop the representation theory of this new class of algebras following
1.5. QUANTUM AFFINE ALGEBRAS AND QUANTUM KZ EQUATIONS 9
the analogy with the three preceding cases. It turns out that there exists a q-
deformation of highest-weight representations for affine Lie algebra. We denote
by M\ k the g-deformed Verma modules and by L\ k their irreducible quotients.
Furthermore, when g = stn, for any finite-dimensional representation V of W?(g)
there is a natural way to define a family of representations of Uq(g) parametrized by
Cx, which we denote by V(z). For g other than si,,, in order to obtain these families,
one has to consider certain special finite-dimensional representations of Uq (fl) for
which the above extension exists. In the case of quantum affine algebras, one can
"twist" the universal i?-matrix by the action of a one parameter gruop generated
by d. The resulting family R{z),z ^ 0, satisfies the Yang-Baxter equation with the
parameters
and an additional unitarity relation
(i.3i) ^MtH-Mtr1-
One can apply these identities to products of finite-dimensional representations and
recognize the famous Yang-Baxter equations of statistical mechanics.
One can then define the intertwining operators $(z) and the correlation func-
functions \t(zi,... ,zn) as in A.21) and A.22), replacing L\tk,L\(z) by their g-defor-
mations and assuming for simplicity that g = s^. Thus we are led to the question
about an analogue of the Knizhnik-Zamolodchikov equation for the correlation func-
functions. These equations, which turn out to be a commutative system of difference
equations, were derived using the representation theory of quantum affine algebras
by I. Frenkel and N. Reshetikhin:
l «+i
where p = q~2i-k+h ) and \t takes values in the tensor product L9Xi <g>... ® Lqx^.
This system is consistent and can be considered as a quantum analogue of the
Knizhnik-Zamolodchikov equation. In fact, the classical Knizhnik-Zamolodchikov
equations A.23) can be obtained from A.32) by taking the limit as q —> 1 and
using the fact that Rij{f-) = 1 + (q^1 - q)ri:i(f:) + O(q - 1). After the discovery
of quantum Knizhnik-Zamolodchikov equations, it was realized that in the special
case k = 0 they had previously appeared in F. Smirnov's approach to integrable
two-dimensional quantum field theory.
The general consistent systems of difference equations
A.33)
,zn),
where 9 € Cn,At € Matn(C) were studied at the beginning of the century by
G. Birkhoff and his collaborators in the case of one variable and more recently by
K. Aomoto in the case of several variables. It was shown that if Ai are regular
and nonsingular in the asymptotic region \z^\ S> |zi| 2> .. - >¦ \zn\ > 0 and tend
to nonsingular matrices which satisfy certain non-resonance condition as the point
(zi,... , zn) tends to 0, then the system A.33) has a unique fundamental solution in
this region - a set of n regular solutions whose values at the origin are the standard
basis column vectors. For generic values of the parameters p and q the system A-32)

10
LECTURE 1. INTRODUCTION
satisfies the required conditions on A*. It turns out that, as in the nondeformed case,
the product of intertwining operators yields precisely the fundamental solution.
In the case of difference equations, there is no notion of monodromy; however,
similarly to the case of differential equations one can define the connection matrices
that relate the fundamental system of solutions of A.33) in the region
Zl » . . . » Z, » Zi+l » . . . » Zn
to the fundamental system of solutions in the region
As in the classical case, the connection matrices admit factorization to the case
n = 4. The connection matrices are no longer constant, but depend on zt/zi+i and
satisfy the equation B(pz) = B(z), which implies that its components are elliptic
functions in x = log z. It turns out that in the case of g = sh the simplest examples
of elliptic solutions of the Yang-Baxter equation were first discovered by R. Baxter
when he solved the XYZ model of statistical mechanics.
If we again consider the first nontrivial example of solutions of the quantum
Knizhnik-Zamolodchikov equations, i.e. g = 5B, n = 4 and
dim
= 2,
we will discover a g-analogue of the Gauss hypergeometric function first introduced
by E. Heine in 1847. They are known as basic hypergeometric series:
A.34)
where {x} = ^^ and {x}n = {a;}{a;-|-l} ... {x + n-1}. The basic hypergeometric
function satisfies the second order difference equation
A.35)
(z{d + a}{d + b}- {d}{d + c-
a, qb, qc; q, z) = 0,
where d = z-^ and {d + x} = ^-^z—, with q9 denoting the operator which takes a
function /(z) to f{qz). Clearly, when q —> 1 this equation transforms into the usual
hypergeometric equation A.28), and the basic hypergeometric function converges
to the Gauss hypergeometric function.
The basic hypergeometric function has an "integral representation" analogous
to A.29). However, now instead of the usual Riemann integral we need to use the
so-called Jackson integral introduced in the beginning of the century:
A.36)
where we let 0 < q < 1. The right hand side of A.36) is a particular Riemann sum
for /, and it is clear that as q —> 1, this expression tends to the usual integral of /.
In terms of the Jackson integral, we have
A.37)
» f1
dqt
1.6. FURTHER GENERALIZATIONS
where by definition
A.38)
and
A.39)
3=0
— nJ+x
j=0
Proceeding as in the classical case, one can consider the second solution of
the difference equation A.35) with the prescribed behavior at 2 = 0, and find the
2x2 connection matrix between these two solutions and the solutions with fixed
asymptotics at z = 00. The connection matrix has entries equal to products of
F,j-functions, and a diagonal conjugation transforms it into a matrix with entries
equal to products of elliptic theta functions 0(f^%N, r), where N is again a linear
combination of the parameters a, b, c and q = e2lrlT. In particular, one can recover
Baxter's original solution of the XYZ model. Recently, more general integral
representations of solutions of quantum Knizhnik-Zamolodchikov equations were
found that confirmed the remarkable parallel between the classical and quantum
cases.
The replacement of trigonometric functions sin(-^p^N) in the connection ma-
matrices of classical Knizhnik-Zamolodchikov equations by the elliptic theta function
9( jjj JV, r) in the quantum case evokes the parallel with the Macdonald generaliza-
generalization. We can pose two questions naturally arising in our theory: Is there an elliptic
deformation of the quantum group that allows us to reproduce the connection ma-
matrices? Is there a more geometric interpretation of the modular invariance inherent
in elliptic theta functions?
Recently, an answer to the first question for g = s^ was found by Felder,
Varchenko, and Tarasov. However, generalization of their results to other Lie alge-
algebras presents serious technical difficulties, so the question is still open.
1.6. Further generalizations of affhie Lie algebras and quantum groups
Quantum affine algebras, by their construction, are generalizations of both
affine Lie algebras and quantum groups. However, our previous discussion and
the relation between affine Lie algebras and quantum groups suggest two further
generalizations.
The first generalization is hinted at by the parallel between the representation
theory of quantum and classical affine algebras. The analogy suggests that the
connection matrices for the quantum Knizhnik-Zamolodchikov equation might also
be obtained from a certain elliptic generalization of quantum groups. In the case of
g = sln, the candidates for elliptic quantum groups can be defined via the elliptic
solutions of the Yang-Baxter equation introduced by A. Belavin. Moreover, one
can define 6j-symbols which have the same dimensions as their counterparts for
Uq(sln). Unfortunately, their definition is rather ad hoc and has no representation
theoretic interpretation. Recently, other candidates for the elliptic quantum groups
for arbitrary simple g have been introduced. This is presently an intensive area of
research. However, a generalization of braided tensor category is already an impor-
important open problem. The culmination of this direction should be a generalization of

12 LECTURE 1. INTRODUCTION
Drinfeld and Kazhdan-Lusztig equivalence of braided tensor categories associated
to the affine Lie algebra g and the quantum group Uq(%).
The second generalization arises from the comparison of the theory of quantum
affine algebras with the theory of quantum groups associated to finite-dimensional
Lie algebras. This analogy suggests that the whole theory of quantum affine alge-
algebras is only a substructure of a representation theory of a certain central extension
of double loop algebras. The natural candidates for these central extensions have
been recently introduced by P. Etingof and I. Frenkel. However, no suitable rep-
representation theory for these double loop algebras is known. The realization of this
program will provide a geometric explanation of the appearance of the modular
group invariance in the connection matrices for quantum Knizhnik-Zamolodchikov
equations.
Three generalizations of affine Lie algebras and quantum groups add a new
layer to the existing three classes of algebras. They can be depicted by the following
diagram:
Here EqiT(g) denotes an elliptic deformation of the quantum group associated
to g, g denotes a central extension of a double loop algebra. The horizontal lines
indicate the Drinfeld equivalence of affine Lie algebras and quantum groups and
their conjectural analogues, and the inclined lines the ascending generalizations of
the simple Lie algebra g.
1.7. Contents of the book
In Lecture 1 (excluding this subsection), we presented the general overview of
representation theory of finite-dimensional simple Lie algebras, affine Lie algebras,
quantum groups and quantum affine algebras, as well as its relations to classical
and quantum Knizhnik-Zamolodchikov equations and their solutions. Those are
the main subjects of our book. Below we will outline the order of the exposition
and contents of the subsequent lectures. Each of the lectures also contains a short
introduction with historical remarks and further information about the contents.
In the first part of Lecture 2 (Sections 2.1-2.5), we recall basic facts from the
representation theory of finite-dimensional Lie algebras. Then in Sections 2.6-2.10,
we give a brief introduction to the representation theory of affine Lie algebras.
We also include here the definition of the Sugawara construction of the Virasoro
algebra.
In the beginning of Lecture 3 (Section 3.1), we introduce an important class of
intertwining operators for tensor products of highest weight and evaluation repre-
representations of affine Lie algebras. In the rest of the lecture, we deduce the Knizhnik-
Zamolodchikov equations for the correlation functions of the intertwining operators
(Sections 3.2-3.4) and study the basic properties of the equations and their solutions
(Sections 3.5-3.9).
1.7. CONTENTS OP THE BOOK 13
In Lecture 4, we derive explicit integral formulas for the solutions of Knizhnik-
Zamolodchikov equations, starting with the level zero solution obtained in Sec-
Section 4.1. Then in Sections 4.2 and 4.3 we explain the relation of the simplest
level one solution in the case g = sfo to the Gauss hypergeometric function. We
present the general formulas for solutions of Knizhnik-Zamolodchikov equations in
Section 4.4 for g = sb and in Section 4.5 for an arbitrary simple algebra.
We analyze further the solutions of Knizhnik-Zamolodchikov equations in Lec-
Lecture 5. In Section 5.4 we obtain a factorization of the solutions for g = si?, us-
using the vertex operators recalled in Sections 5.1, 5.2 and the /?7-system defined
in Section 5.3. This naturally leads to the free field realizations constructed in
Section 5.5. The explicit expressions for intertwining operators obtained in Sec-
Sections 5.6, 5.7 allow us to recover the formulas for the correlation functions and,
thus, for the solutions of Knizhnik-Zamolodchikov equations in Section 5.8.
In Lecture 6, we review basic facts on quantum groups associated to Kac-
Moody Lie algebras and their representation theory. After recalling the definitions
in Sections 6.1 and 6.2, we describe braided tensor categories and representations
of braid groups associated to representations of quantum groups in Sections 6.3
and 6.4, respectively. The quantum double construction and the quantum Casimir
element are in Sections 6.5-6.7. In the last section, Section 6.8, we present the braid
relations in the form of exchange matrices on the spaces of intertwining operators
for the quantum group. These relations will also arise in studying the monodromy
of the Knizhnik-Zamolodchikov equations.
In Lecture 7, we study the geometry of integration cycles appearing in for-
formulas for explicit solutions of the Knizhnik-Zamolodchikov equation. We recall
the definition of local systems in Section 7.1 and the corresponding homology and
cohomology theories in Sections 7.2 and 7.3. Then in Section 7.4, we realize the
space of solutions of Knizhnik-Zamolodchikov equations for g = sh as the homol-
homology groups of certain families of local systems, and in Section 7.5 we identify the
equations themselves with the Gauss-Manin connection on the space parametriz-
parametrizing these families. In Section 7.6, we give an explicit construction of homology
cycles that yield the solutions of the Knizhnik-Zamolodchikov equation for g = SI2.
We discuss a generalization to an arbitrary simple Lie algebra in the last section,
Section 7.7.
In Lecture 8, we return to studying the solutions of Knizhnik-Zamolodchikov
equations, armed with the theories developed in Lectures 6 and 7. First we explain
the relation between the monodromy of the Knizhnik-Zamolodchikov equations and
the braid group in Section 8.1. Then in Sections 8.2-8.4, after studying asymptotic
solutions, we define the monodromy with respect to an infinite base point. We
show in Section 8.5 that the latter monodromy admits factorization in a natural
decomposition of the space of intertwining operators, hence again yielding the braid
relations in the form of factored exchange matrices. In the last two sections, 8.6
and 8.7, we discuss the comparison of these exchange matrices with the ones for
quantum groups, which leads to the equivalence of two types of braided tensor
categories.
We proceed to a generalization of the representation theory of affine Lie algebras
and the Knizhnik-Zamolodchikov equation to the quantum case in Lecture 9. In
Sections 9.1-9.3 we recall the definition and basic properties of quantum affine
algebras, highest weight and evaluation representations and intertwining operators
between tensor products of these representations. Then in Sections 9.4-9.6, we

14 LECTURE 1. INTRODUCTION
study the iJ-matrices for quantum affine algebras, and in Section 9.7 we use the R-
matrices to construct quantum currents. We conclude the lecture with the quantum
analogue of the Sugawara construction in degree zero in Section 9.8.
In Lecture 10, using the constructions of Lecture 9, we are able to generalize
the results of Lecture 3 to the quantum case. In Sections 10.1-10.3 we derive the
quantum counterpart of the Knizhnik-Zamolodchikov equations and study its basic
properties in Sections 10.4—10.9, including the relations to the classical case (Sec-
(Section 10.8). Since the quantum Knizhnik-Zamolodchikov equations play a central
role in this book, we give a second derivation in the last Section 10.10.
In Lecture 11, we continue the analogy between the classical and quantum
Knizhnik-Zamolodchikov equation and study solutions of the latter in the case of
g = SI2. We start with a review of ^-analogues of classical special functions, and
in particular, the <?-hypergeometric function, in Sections 11.1-11.4. Then in Sec-
Sections 11.5 and 11.6, we express the solutions of quantum Knizhnik-Zamolodchikov
equation in terms of q-hypergeometric function and its generalizations.
In Lecture 12, we are able to generalize some of the results of Lecture 8 con-
concerning the monodromy of the Knizhnik-Zamolodchikov equations to the quantum
case. The proper analogue of the monodromy, the connection matrix, is intro-
introduced in Section 12.1. Then we use a relation for solutions of the g-hypergeometric
equation, reviewed in Section 12.2, to derive in Section 12.3 the connection matrix
for the quantum Knizhnik-Zamolodchikov equations in the simplest case. Finally,
the general properties of connection matrices, including the analogues of the braid
relations for exchange matrices, are studied in Section 12.4.
In the last lecture, Lecture 13, we extend further the remarkable parallel be-
between representation theory of affine Lie algebras and quantum affine algebras,
which includes the analogy between classical and quantum Knizhnik-Zamolodchikov
equations, by formulating a few problems. In Section 13.1 we propose the ques-
question of derivation of solutions of the quantum Knizhnik-Zamolodchikov equation
from a free field realization of quantum affine algebra. In Section 13.2, we discuss
the analogues of braided tensor categories and Drinfeld-Kohno isomorphism in the
quantum case. In Section 13.3 we also formulate the question of finding a g-analog
of the Sugawara construction. The rest of Sections 13.3 and 13.4 is dedicated to
problems that were not studied in this book even in the undeformed case, such as
constructions of g-analogues of vertex operator algebras and conformal field theory.
In Section 13.5 we present problems related to another point of view on quantum
affine algebras, considering it as a substructure in the representation theory of dou-
double affine algebras. We conclude this lecture with Section 13.6, where we discuss the
relation of the quantum Knizhnik-Zamolodchikov equation with integrable models
in statistical mechanics and quantum field theory.
Lecture 2. Representations of
Finite-Dimensional and Affine Lie Algebras
In this lecture we recall the basic facts about the structure and representations
of a simple finite-dimensional Lie algebra g and the corresponding affine Lie algebra
g. The theory of finite-dimensional Lie algebras is considered a classical branch of
mathematics even by Bourbaki standards. We refer the reader to the books [Bour],
[Hu], [GG] for further details. The theory of affine Lie algebras originates from the
discovery, made independently by V. G. Kac and R. Moody, that a simple finite-
dimensional Lie algebra over the ring of Laurent polynomials admits a presentation
in terms of generators and relations, similar to the presentation of a simple finite-
dimensional Lie algebra over C. A comprehensive account of this theory, including
results on Virasoro algebra, can be found in the monographs of these authors [Kac],
IMP].
In this book, all the Lie algebras are always considered over the field of complex
numbers C.
2.1. Simple Lie algebras.
Let g denote a simple finite-dimensional Lie algebra over C, and let f) be a
Cartan subalgebra of g; its dimension is called the rank of g and denoted r. Then
g has the following decomposition in a sum of f)-invariant subspaces:
B.1)
)ar
Here R is a finite subset of the dual space f)* to the Cartan subalgebra, and the
subspace g" is the space of all x € g such that for every h e f), [h,x] = oc{h)x.
It is known that all subspaces Qa are one-dimensional. Elements a € R are called
roots, R is called the root system, and the decomposition B.1) is called the root
decomposition of the Lie algebra g. Note that if a is a root then —a is also a root.
Pick an element h € f) such that Re oc{h) ^ 0 for all a € R. A root a is called
positive if Re a{h) > 0 and negative if Re a{h) < 0. Thus the choice of h € f) yields
a polarization of the root system: R = R+ LJ R-, where R+ is the set of positive
roots and R- is the set of negative roots. Clearly, R- = —R+.
A positive root a is called simple if it cannot be written as a sum of other pos-
positive roots. It is known that the simple positive roots form a basis of ()*, ai,..., aT,
and every positive root can be uniquely represented in the form a = X3i=i njaji
where n, > 0 are integers.
Let n* = ©ct€H± g". Then n* are maximal nilpotent subalgebras of g, and
g = n+ © t) © n~. This decomposition is called a polarization of g, and plays a key
role in representation theory.

14 LECTURE 1. INTRODUCTION
study the .R-matrices for quantum affine algebras, and in Section 9.7 we use the R-
matrices to construct quantum currents. We conclude the lecture with the quantum
analogue of the Sugawara construction in degree zero in Section 9.8.
In Lecture 10, using the constructions of Lecture 9, we are able to generalize
the results of Lecture 3 to the quantum case. In Sections 10.1-10.3 we derive the
quantum counterpart of the Knizhnik-Zamolodchikov equations and study its basic
properties in Sections 10.4-10.9, including the relations to the classical case (Sec-
(Section 10.8). Since the quantum Knizhnik-Zamolodchikov equations play a central
role in this book, we give a second derivation in the last Section 10.10.
In Lecture 11, we continue the analogy between the classical and quantum
Knizhnik-Zamolodchikov equation and study solutions of the latter in the case of
g = sb- We start with a review of ^-analogues of classical special functions, and
in particular, the g-hypergeometric function, in Sections 11.1-11.4. Then in Sec-
Sections 11.5 and 11.6, we express the solutions of quantum Knizhnik-Zamolodchikov
equation in terms of g-hypergeometric function and its generalizations.
In Lecture 12, we are able to generalize some of the results of Lecture 8 con-
concerning the monodromy of the Knizhnik-Zamolodchikov equations to the quantum
case. The proper analogue of the monodromy, the connection matrix, is intro-
introduced in Section 12.1. Then we use a relation for solutions of the g-hypergeometric
equation, reviewed in Section 12.2, to derive in Section 12.3 the connection matrix
for the quantum Knizhnik-Zamolodchikov equations in the simplest case. Finally,
the general properties of connection matrices, including the analogues of the braid
relations for exchange matrices, are studied in Section 12.4.
In the last lecture, Lecture 13, we extend further the remarkable parallel be-
between representation theory of affine Lie algebras and quantum afiine algebras,
which includes the analogy between classical and quantum Knizhnik-Zamolodchikov
equations, by formulating a few problems. In Section 13.1 we propose the ques-
question of derivation of solutions of the quantum Knizhnik-Zamolodchikov equation
from a free field realization of quantum afiine algebra. In Section 13.2, we discuss
the analogues of braided tensor categories and Drinfeld-Kohno isomorphism in the
quantum case. In Section 13.3 we also formulate the question of rinding a ^-analog
of the Sugawara construction. The rest of Sections 13.3 and 13.4 is dedicated to
problems that were not studied in this book even in the undeformed case, such as
constructions of q-analogues of vertex operator algebras and conformal field theory.
In Section 13.5 we present problems related to another point of view on quantum
affine algebras, considering it as a substructure in the representation theory of dou-
double afiine algebras. We conclude this lecture with Section 13.6, where we discuss the
relation of the quantum Knizhnik-Zamolodchikov equation with integrable models
in statistical mechanics and quantum field theory.
Lecture 2. Representations of
Finite-Dimensional and Affine Lie Algebras
In this lecture we recall the basic facts about the structure and representations
of a simple finite-dimensional Lie algebra g and the corresponding affine Lie algebra
g. The theory of finite-dimensional Lie algebras is considered a classical branch of
mathematics even by Bourbaki standards. We refer the reader to the books [Bour],
[Hu], [GG] for further details. The theory of affine Lie algebras originates from the
discovery, made independently by V. G. Kac and R. Moody, that a simple finite-
dimensional Lie algebra over the ring of Laurent polynomials admits a presentation
in terms of generators and relations, similar to the presentation of a simple finite-
dimensional Lie algebra over C. A comprehensive account of this theory, including
results on Virasoro algebra, can be found in the monographs of these authors [Kac],
[MP].
In this book, all the Lie algebras are always considered over the field of complex
numbers C.
2.1. Simple Lie algebras.
Let g denote a simple finite-dimensional Lie algebra over C, and let I) be a
Cartan subalgebra of g; its dimension is called the rank of g and denoted r. Then
g has the following decomposition in a sum of ()-invariant subspaces:
B.1)
8 =
18
Here R is a finite subset of the dual space fj* to the Cartan subalgebra, and the
subspace g" is the space of all x € g such that for every h € f), [h, x] = oc{h)x.
It is known that all subspaces Qa are one-dimensional. Elements a € R are called
roots, R is called the root system, and the decomposition B.1) is called the root
decomposition of the Lie algebra g. Note that if a is a root then —a is also a root.
Pick an element h € f) such that Re a (h) ^ 0 for all a € R. A root a is called
positive if Re a(h) > 0 and negative if Re a(h) < 0. Thus the choice of ft € f) yields
a polarization of the root system: R = R+ LJ R_, where R+ is the set of positive
roots and i?_ is the set of negative roots. Clearly, R- = —R+.
A positive root a is called simple if it cannot be written as a sum of other pos-
positive roots. It is known that the simple positive roots form a basis of fj*, ax,..., aT,
and every positive root can be uniquely represented in the form a = X3j=i niai^
where n3 > 0 are integers.
Let n* = 0a?R± g". Then n* are maximal nilpotent subalgebras of g, and
g = n+ © t) © n~. This decomposition is called a polarization of g, and plays a key
role in representation theory.

LECTURE 2. REPRESENTATIONS OF LIE ALGEBRAS
Let (,) be an invariant symmetric bilinear form on g: {[x,y],z) = (x,[y,z\).
Such a form is unique up to a factor, which will be fixed later (see Section 2.5). It
is clear that (x, y) is proportional to the Killing form Tr(ad x ¦ ad y), since both are
nonzero invariant symmetric forms.
The form (,} gives rise to a natural identification h* —> h, f3 i—> h@ defined by
the relation 0(h) = (hp,h), /3 € fj*,ft e (j. Prom now on, we will frequently use
this identification and make no distinction between I) and h*. This identification
allows us to define a scalar product on h* by (/?, 7) = (h/3,hy). We will use the
same notation (,) for the pairing h ® f)* —» C.
We will also use the following notation, which is standard in the theory of Lie
algebras:
Q = © Zoti C b* - root lattice;
Q+ = ©Z+ai;
Qv = © ZccV C h - dual root lattice, where av =
P = {A € f)*|A(aV) € Z} - weight lattice;
P+ = {A € h*|A((*/) € Z+} - cone of dominant integral weights;
u>i € P+ - fundamental weights: (u)i,Oj) = <5y;
Si : f)* —» h* - simple reflection, defined by Si(X) = A — (a/, A)c*i;
IV - Weyl group, generated by sf, for an element w € W we denote by l(w)
the length of a minimal presentation of w as a product of s;;
W(g) - the universal enveloping algebra of g; it admits a triangular decomposi-
decomposition U(g) = U(r
2.2. Cartan matrices of simple Lie algebras.
The Cartan matrix of g is defined by the formula Oy = ,°'ff (note that
this is independent of the normalization of the inner product). The number By is
always an integer. The Cartan matrix is independent of the choice of a polarization
of the root system and undergoes a conjugation by a permutation matrix when
the numeration of the simple roots is changed. Therefore, the Cartan matrix is
uniquely determined for any simple Lie algebra up to a simultaneous remuneration
of rows and columns.
The entries of the Cartan matrix have the following properties:
(i) an = 2 for all i;
(ii) an < 0 if i ^ j;
(iii) atj = 0 iff an — 0.
A simple Lie algebra is called simply-laced if a*, = 0 or — 1 for i ^ j.
known that in this case all roots a € R have equal length.
The Cartan matrix is nonsingular. Therefore, we can choose a basis hi,.
It is
in f) such that aj(hi) = a^. This is equivalent to letting hi = 1
d € gOt, fi € g"**', 1 < i < r be nonzero vectors such that (e^,/;) =
we have the following relations between e^, fi, hf.
[hi, hj] = 0, [hi, e3} = aijej, [hu Si\ = -aijfj
(ad eiI-"^ = 0, (ad fiI""'1 fj = 0.
These relations easily follow from the properties of the root system.
.. -, hr
= hai ia^a.\- Let
;) = ,a2a.y Then
2.3. HIGHEST-WEIGHT MODULES OVER LIE ALGEBRAS 17
It turns out that these relations generate all other relations between the ele-
elements ei, fi,ht:
Theorem 2.2.1 (J.-P. Serre). The Lie algebra generated by the elements eit fit
hi, 1 < i < r, with relations B.2) is isomorphic to g.
2.3. Highest-weight modules over simple
Lie algebras and contravariant forms.
Let A € f)*. The Verma module M\ is generated by one vector v\ (which is
called the highest-weight vector or the vacuum vector) satisfying the relations
B.3)
= 0, 1 < i < r, hv\ = \(h)v\, h € I).
More precisely, M\ is an induced module: M\ = Indjjen+CA, where Ca is a one-
dimensional space with a basis vector v\ and the action of h © n+ is defined by
B.3).
We say that a module V\ is a highest-weight module if it is a quotient of My,
thus, M\ is the largest highest-weight module. Similarly, let I\ be the sum of all
proper submodules of M\; then I\ is the unique maximal proper submodule of
M\, and the quotient L\ = M\/I\ is irreducible; it is the smallest highest-weight
module. Note that for a generic A, the module M\ itself is already irreducible,
therefore I\ = 0 and L\ = M\. Later we will discuss when L\ is finite-dimensional.
Replacing e^ by fi in B.3), we get a definition of a lowest-weight Verma module.
All the results on the highest-weight modules are valid for lowest-weight modules
with obvious changes.
There exists a unique symmetric bilinear form on M\ possessing the following
contravariance property:
B.4)
Mk.
This form is called the Shapovalov form. It is known that the kernel of the Shapo-
valov form coincides with the maximal submodule I\. Therefore, this form descends
to any quotient of M\, in particular, to L\, where it becomes nondegenerate.
The reason the Shapovalov form is called contravariant is the following. There
exists a canonical automorphism of g given by
B.5)
u(hi) = -hi.
This automorphism is called the Chevalley involution. In terms of u>, the contravari-
contravariance property of a symmetric bilinear form can be written as follows:
B.6)
(xu,w) = — (u,uj(x)w), u,w € M\
which can be regarded as the invariance condition twisted by uj.
It is worth mentioning that both the involution u and the contravariant form
have hermitian analogues, which we won't use.
Finally, let us recall the weight decomposition for g-modules. Let V be a g-
module. We say that a vector u € V has weight fj, € f)* if hu = n(h)u for all
h € h. The subspace of all vectors of weight fj, in V is called a weight subspace and
is denoted VM. We say that V has a weight decomposition if every weight subspace
is finite-dimensional, and V = ® VM. It is easy to see that Verma modules (and,
more generally, all highest-weight modules) have a weight decomposition, and the

18 LECTURE 2. REPRESENTATIONS OF LIE ALGEBRAS
weight subspaces are orthogonal to each other with respect to the contravariant
form.
In this book we only consider representations that have weight decomposition
with finite-dimensional weight subspaces. If V is such a representation, we denote
by V its restricted dual: V* = ©A(V*)*. On the contrary, by Homc(Vi,V2)
we denote the space of all linear maps from V\ to V2 without any restrictions on
weight. We will often use the following elementary fact: if the Verma module M\ is
irreducible then its restricted dual is the lowest-weight Verma module with lowest
weight —A.
Some of the results above can be rewritten using the notion of contragredient
Verma module as follows. For a g-module V, denote by Vc the restricted dual to
V with the action of g twisted by u>, i.e.
(xuc,v) = -(uc,w(x)v), uc € Vc,v € V, xeg
(compare with B.6)). The module Vc has the same dimensions of the weight
subspaces as V. If we take V = M\ then there is a unique (up to a constant)
map M\ —> MA. Combining this map with the pairing M? ® M\ —> C, we get
the Shapovalov form. It can be shown that the image of this map is exactly the
irreducible highest-weight module L\; in particular, this map is an isomorphism iff
M\ is irreducible.
2.4. Finite-dimensional representations
and irreducibility of Verma modules.
We have already mentioned that for generic values of A the Verma module M\
is irreducible. Here is the precise statement.
Theorem 2.4.1. The Verma module M\ is irreducible iff the following condi-
condition is satisfied:
B.7) Foralla&R+, (A + p, av) ? {1, 2,... }.
It will be convenient to use a slightly weaker condition. Namely, we will call a
weight A generic if it satisfies the following condition:
B.8) For all a € R+, (A, av) i Z.
It follows from the previous theorem that if A is generic then L\ = M\.
Another extreme case is when the irreducible quotient L\ is finite-dimensional.
The following fundamental theorem was proved by Chevalley.
Theorem 2.4.2.
1. The module L\ is finite-dimensional iff A € P+.
2. The representations L\, A 6 P+, are pairwise nonisomorphic, and any irre-
irreducible finite-dimensional representation of g is isomorphic to one of them.
3. //A = ^Vriiijj, rti € Z+, then the representation Lx is generated by a
vacuum vector v\ satisfying the defining relations B.3) and /" v\ = 0.
In a similar way, irreducible finite-dimensional representations can be also de-
described as lowest-weight modules: namely, L\ is the irreducible lowest-weight mod-
module with lowest weight wo(A), where wq is the element of maximal length in the
Weyl group. Note also that for finite-dimensional modules, L*x :=: ?_„,„(>).
We will frequently use the following well-known result.
2.5. MAXIMAL ROOT, COXETER NUMBERS, AND CASIMIR OPERATOR 19
Proposition 2.4.3. The category C(g) of finite-dimensional representations
is semisimple, i.e. every module V is isomorphic to a direct sum of simple mod-
modules LA, A € P+. Also, the map $ h-» $(ua) identifies the mulitplicity space
Homs(LA,V),A € P+, with the space (Vx)n+ = {v e VA|e^ = 0} of singular
vectors of weight A in V.
Later on we will generalize these facts to afEne Lie algebras.
2.5. The maximal root, the Coxeter
numbers, and the Casimir operator.
The maximal root of the Lie algebra g is, by definition, the highest weight 9 of
its adjoint representation. It is called maximal because it satisfies (and is uniquely
determined by) the condition: for any root a e R, we have 9 - a € Q+.
From now on we assume that the normalization of the invariant form (,) on g
is chosen so that {9,9) = 2.
Let 9 = J2i Xi<*i- The number h = 1 + J2i Xi is called the Coxeter number of
fl-
Let Xi = Xi^f^-, and let ftv = 1 + Y.Xi = (p,B) + 1. The number ftv
(which is always an integer) is called the dual Coxeter number of g. It will play an
important role further on. For simply-laced Lie algebras, h = hv. In general, they
are different; moreover, the dual Coxeter number for a root system R is not the
same as the Coxeter number for Rv.
Let Xi,xl be dual bases in fl with respect to the form {, ). We define the
Casimir element C € U(g) and the symmetric tensor ft € g ® g by
These elements do not depend on the choice of the basis a;*; thus, if B is an
orthonormal basis in g, then C = J2a&B a2 ,fl = YLa&B a®a.
It is known that C lies in the center of W(g) and thus it acts by multiplication
by a scalar in any highest-weight module. If the highest weight of the module is
A, this scalar is equal to (A, A 4- 2p). Similarly, if a lowest weight of the module is
—H, then C acts in this module by (ft, n + 2p). In particular, the value of C in the
adjoint representation equals 2hv.
Similarly, in a tensor product of two representations of g the action of fl com-
commutes with the action of g; thus, if we have an embedding of highest-weight modules
V\ C Vfj. ® Vv, then fl acts by a constant on the image of this embedding. This
constant can be calculated:
B.10)
which follows from the identity
B.11) n = i(A(C)-C®l-l®C),
where A : W(g) -» W(g) ® W(g) is the comultiplication.

20 LECTURE 2. REPRESENTATIONS OF LIE ALGEBRAS
2.6. Affine Lie algebras.
In this section we briefly recall some of the main results about affine Lie alge-
algebras, referring the reader to [Kac] and [MP] for detailed exposition.
As before, let g be a simple finite-dimensional Lie algebra. Define the loop
algebra
B.12) Lg = g®C[t,t'1]
with the commutator given by [x®P,y®Q] = [x,y]®PQ,x,y € g,P,Q &C[t,t].
This algebra can be considered as the algebra of polynomial g-valued functions on
the circle, which explains the name "loop algebra".
This algebra has a unique nontrivial central extension, which is given by
with the commutator given by [c, x ® P] —0 and
(x,y) f
B.14) [x ® P,y® Q] = [x, y] ® PQ + c ¦ ——— * Q dP.
If we introduce the notation x ® tn = x[n], then B.14) can be rewritten as
follows:
Finally, we can extend this algebra, adding to it an exterior derivation d = t-jg:
we introduce
B.16) g
with the commutator given by B.14) and
[d,c] = Q, [d,x®P) = x®t—P,
or [d, x[n]] = nx[n].
Both jj and jj are called in the literature affine Lie algebras; as a rule, we will
use this name for g.
Similarly to the finite-dimensional case, there is a bilinear nondegenerate in-
invariant symmetric form on g given by
B.17)
(x®P,y®Q) =
(x,y)
f PQ7>
J\t\=i t
(c,d) = l, (c,c) = (d,d) = (c,x <S> P) = {d,x ® P) =0.
The restriction of this form to g is again invariant but degenerate.
Define the subalgebras jj C g, b C 0 by jj - f) © Cc, fj = fj ® Cd. By analogy
with simple finite-dimensional Lie algebras, they are called Cartan subalgebras.
Then we have fj* = f)* © CA0,fj* = f)* ffi CA0 © CS, where Ao,<5 are defined by
Ao(c) = 1, A0(d) = Ao(A) = O,S(d) = l,S(c) = S(h) = 0 for h € {,.
We have the following decomposition of g:
B.18)
2.6. APPINE LIE ALGEBRAS.
where, as before, for every 7 € fj* we denote
07 = {x € 0| [h, x] = -y(h)x for all ft € fj}
and ^ = {7 € fj*|gT ^ 0}. It is easy to see that
B.19) R = {a + nS\ a € R,n & Z or a = 0,neZ\ {0}}
and
B.20)
= tj ® tn, n € Z \ {0}.
Elements of R are called affine roots. Similarly to the finite-dimensional case,
R admits a polarization: R = R+ LJ J?__, where J2+ = {a + nA|n > 0 or n =
0, a € .R+}, R- = —ii+. As before, a positive root is called simple if it cannot be
represented as a nontrivial sum of positive roots. It is easy to see that the roots
ai,... ,ar are still simple. We also have a new simple root: ao = S — 6, where 9 is
the maximal root of g. Since any positive root is a linear combination of ao,... ,ar
with nonnegative integer coefficients, there can be no other simple roots.
We define n± = n± ffi g ® i^q**1]. Clearly, n± is the sum of all positive
(negative) root subspaces. Both n+ and n~ are Lie subalgebras of g, and we have
the polarization
B.21)
jj = n+ ffi fj ffi ft ,
which is fundamental in representation theory. As before, we define Q = ©]=„ Zat
and Q+ = ®[=0 Z+a*. We also define o# = (^l) = haa = c - 6»v and let
P ={A e fi*|AKv) e Z} = P ffiZAo © CS,
B.22) P+={Aefj*|A(a,v)€Z+}
={A = A + fcAo - A<5|A e P+, A; € Z+, A e C, (A, 6»v) < A;}.
The similarity between simple finite-dimensional and affine Lie algebras be-
becomes especially transparent when both types of algebras are unified in more general
class of Kac-Moody algebras.
Let A = (Oy)i<ij<r be a matrix with integer entries satisfying the following
conditions:
(i) an = 2 for all i;
(ii) Oy < 0 if i + j;
(iii) oy = 0 iff an = 0.
We will call such a matrix a generalized Cartan matrix. As we discussed before,
a Cartan matrix of a finite-dimensional simple Lie algebra satisfies these conditions.
Definition 2.6.1. Let ibea generalized Cartan matrix. The Kac-Moody
algebra g(A) is the Lie algebra generated by elements ei,fi,hi,i = 1,..., r, with
the relations B.2).
Many results obtained for finite-dimensional simple Lie algebras can be gen-
generalized to Kac-Moody algebras. In particular, we can define Cartan subalgebra,
triangular decomposition, highest-weight modules, roots etc. The simple roots
ai € f)* of this algebra are defined by the conditions (hi, (*j) = Oy.

22 LECTURE 2. REPRESENTATIONS OP LIE ALGEBRAS
We can apply this theory to the description of affine Lie algebras. Let g be a
finite-dimensional simple Lie algebra of rank r and let g be the corresponding affine
Lie algebra. Recall that we have denned roots ao, ¦ ¦ ¦, ar €t), and we have an invari-
invariant inner product on fj. Define the extended (affine) Cartan matrix dij = ^.' ^y ,
0 < i < r. It is easy to see that it is a generalized Cartan matrix (traditionally, the
indices range from 0 to r rather than from 1 to r + 1). The proof of the following
theorem can be found in [Kac] (Theorem 9.11).
Theorem 2.6.2. The Lie algebra g is the Kac-Moody algebra associated with
the extended Cartan matrix defined above. The generators ei,fi,hi,i = l,...,r,
are the same as for g. The generators e0 € Q~e ® t, f0 € ge <g> i are such that
(eo, /o) = 1, and h0 = [e0, /o] = -9V + c.
This theorem is very important because it identifies the algebraic description
of an affine algebra (by generators and relations) with its geometric description (as
a central extension of the loop algebra). An interplay between these two realiza-
realizations is the main source of rich structural properties of affine algebras and their
representations.
The affine Lie algebra g = g © Cd can also be included in the general picture of
Kac-Moody algebras. Let A be a generalized Cartan matrix, which is symmetriz-
able, that is, there exist numbers dj such that did,, = djaji. This is always so for
Cartan matrices of finite-dimensional and affine Lie algebras, and di = (oti,oti)/2.
Then we can define an inner product on I) by letting (hi, hj) = a^jdj. However,
this inner product may be degenerate. Let Z denote the kernel of this inner product
in t). Define the extended Kac-Moody algebra
B.23) fle(A)
with the commutation relations [z1;z2] = [z, h] = 0, [z, e*] = z((*j)ej, [z,fi] =
—z(ai)fi, where h 6 t),z,zltZ2 € 2*. We will also denote by \)e the extended
Cartan subalgebra f) © Z*. The benefit of this extension is that f)e has a non-
degenerate inner product, given by (hi,hj) = Oij/dj, (h,z) = z(h), {zi,z2) = 0.
In particular, in the case when A is the Cartan matrix of an affine Lie algebra,
the extended Kac-Moody algebra ge(A) defined above is nothing but the algebra
g, defined by B.16).
2.7. Verma modules and Weyl modules for affine Lie algebras.
Let us define highest-weight representations of affine Lie algebras. As before,
we consider the affine Lie algebra g associated with a simple finite-dimensional Lie
algebra g.
Given a weight A € t)*, we can define the Verma module
B.24) Ma = Ind?eft+CA,
where Ca is a one-dimensional representation of f) © n+ with a basis vector va on
which n+ acts by zero and t) acts by hvA = A(h)vA-
As before, we have a weight decomposition
where m e A — Q+.
2.7. VERMA MODULES FOR APFINE LIE ALGEBRAS
We will say that a representation V of g has the highest weight A if it is
generated by a vector va of weight A which is annihilated by n+. Clearly, this is
equivalent to saying that V is a quotient of Ma.
Every weight A can be represented in the form A = A + kA0 — AS, A € I)*. If
we regard Ma as a g-module then its structure is independent of A. We will only
use weights A such that
B.25)
A = A(A) =
(A, A + 2p)
for a reason to be explained later (see Section 2.9). The Verma module correspond-
corresponding to such a weight A will be hereafter denoted by Ma,*. The number k is called
the central charge of the action of the affine algebra or the level of the representa-
representation. More generally, we say that a g-module V is of level k or that V is a module
with central charge k if c acts on V by multiplication by k. The level k = —hv is
called critical, and in this case the representation theory changes drastically; unless
otherwise specified, we assume that k ^ —ftv.
The module M\^ admits a Z-grading:
B.26)
: = eB^A,*[-
where M\:k[—n] is the eigenspace of d with the eigenvalue — n — A(A). It is easy to
see that Ma,*!—n] is naturally a g-module. For instance, JWa,*[0] = M\.
As in the finite-dimensional case, we define I a to be the maximal proper sub-
module in Ma, and construct the irreducible representation La = Ma/Ia- If
A = A(A), La will be denoted LA,*-
The Chevalley involution Cj of g is defined exactly as in Section 2.3, by relations
B.5) and <j(d) = -d. On Lg = g ® C[t, t~l], it coincides with u composed with the
map 11—» i.
Define the contravariant symmetric form (,) on Ma by conditions B.6) (with
lj replaced by u>). As in the finite-dimensional case, one can check that the kernel
of this form is the maximal proper submodule in Ma, and thus it descends to the
irreducible quotient La, where it becomes nondegenerate.
An important class of representations of g comprises the modules induced from
g-modules. Let g+ = g ® C[t] © Cc © Cd C g. We will deal with modules induced
from g+ to g. More precisely, let V be a module over g. Then it is automatically a
module over g+: we let g®<C[i] act by zero and c, d by constants k, —A respectively.
Thus, we can define the induced module Ind?+V\ Clearly, Verma modules belong
to this class:
B.27) Ma,* = Ind?+MA,
where d acts on M\ by multiplication by — A(A).
Another important example are the modules induced from irreducible repre-
representations of g:
B.28) VKk
where d again acts in L\ by multiplication by —A(A). If A is a dominant integral
weight of g then Va,* is called the Weyl module.

24
LECTURE 2. REPRESENTATIONS OP LIE ALGEBRAS
Finally, we can define the contragredient Verma module similarly to the con-
construction in Section 2.3, replacing the involution uj by Cj. These modules will be
denoted M% k. As in the classical case, we have a unique (up to a constant) map
Mx.jt —> AfJi, and this map is an isomorphism iff Ma,* is irreducible. We can also
define contragredient Weyl modules V?k, and again, the canonical map V\,i —> V?ik
is an isomorphism iff V\^ is irreducible.
2.8. Integrable representations of affine Lie algebras.
As in the finite-dimensional case, for generic values of highest weight, Verma
modules are irreducible. Here is a more precise statement (see [KK]).
Theorem 2.8.1. The Verma module M\ is irreducible iff the following condi-
condition is satisfied:
B.29) For every a eR+,Ne {1,2,...}, (A +p,a) ^ —(&,&),
where p = p + ftvAo-
Note that for a such that (a, a) ^ 0, condition B.29) can be rewritten as
(A + p, av) ^ N - compaxe with B.9).
We will call weights A satisfying B.29) generic weights. For a generic weight,
LA = AfA.
Another important theorem describes when a Weyl module (see B.28)) is irre-
irreducible. We will use a weak form of this result, referring the reader to [KK] for a
more precise statement.
Theorem 2.8.2. Let A e h*,A; € C be such that the following condition is
satisfied:
B.30) For alla?R+, k ? Q(av, A) + Q.
Then the induced module V\^ defined by B.28) is irreducible.
In this case, we will say that k is generic with respect to A.
Corollary 2.8.3. If X € P+, andk t? Q, the Weyl module Va,j= is irreducible.
Finally, let us discuss the analogues of finite-dimensional representations. All
nontrivial highest-weight representations of g are infinite-dimensional. However,
if the weight A is a dominant integral weight, the representation L\ resembles
finite-dimensional representations in many respects.
Let us say that a highest-weight representation V of g is integrable if for any
vector u € V and any 0 < i < r there exists a positive integer m such that
/™i+1« = 0. For such modules, the action of the affine Lie algebra can be integrated
to an action of the affine Lie group, which explains the terminology.
The following statement, proof of which can be found in [Kac, Chapter 10], is
an analogue of Theorem 2.4.1 for affine Lie algebras.
Theorem 2.8.4.
1. The module L\ is integrable iff A € P+.
2. The representations L^, A S P+, are pairwise nonisomorphic, and any
highest-weight integrable module is isomorphic to one of them.
2.9. THE VIRASORO ALGEBRA AND ITS ACTION ON jj-MODULES.
3. If A is a dominant integral weight and ni = (A, 0%), then the g-module L\
is generated by the highest-weight vector v\ with defining relations eiV\ = 0,
, f"i+1VA = 0, 0 < i < r.
The structure of integrable highest-weight modules is very rich and deserves a
separate course of lectures. However, most of the time we will assume that the level
k is generic, and work with induced modules, which have a much simpler structure.
The main features of the Knizhnik-Zamolodchikov equations and related objects
can be seen already at this level. The behavior of these structures at integer points
is a much more subtle business which we will discuss only briefly.
Another class of representations of g (not of g) consists of evaluation repre-
representations associated to fl-modules. Let V be a module over g; we assume that V
admits a weight decomposition with finite-dimensional weight subspaces. Then we
can make V into a representation of g as follows: pick a nonzero complex number
z and set
B.31)
x ® P(t) ¦ u = P(z)xu, cu = 0, u
This representation is denoted by V(z) and called an evaluation representation (be-
(because we are evaluating the polynomial P at the point z). If V is finite-dimensional
then V(z) is an integrable jj module.
Note that the action of g on V{z) does not extend to an action of g. Thus one
is led to consider a bigger space z~AV[z, z] = V ® z~AC[z, z] (where z is no
longer a complex number but a formal variable and A is a complex number to be
fixed later) which admits an action of the whole affine algebra 0 by letting d = z?.
For any zq ^ 0 we have the evaluation map ezo : z~AV[z, z] —> V(z0), which is a
g-epimorphism (defined up to a phase factor e-27rmA; n g z).
2.9. The Virasoro algebra and its action on g-modules.
Now we are in a position to explain the special choice of A that we made in
the previous section.
The Witt algebra is the Lie algebra with the basis (Ln, n € Z) and commutation
relations
B.32)
[Lm, Ln] = (m - n)Lm+n.
This algebra is isomorphic to the Lie algebra of complex polynomial vector fields
on the unit circle \t\ = 1 in C: Lny-* -tn+1-§j.-
The Virasoro algebra Vir is the unique nontrivial one-dimensional central ex-
extension of the Witt algebra. The basis of the Virasoro algebra includes the elements
Ln, n € Z, and a new central element K satisfying the new commutation relations
B.33)
[Lm, Ln] = (m - n)Lm+n
12
[K,Ln]=0.
The representation theory of the Virasoro algebra is very rich and has important
applications to theoretical physics. In string theory, the Virasoro algebra arises as
the Lie algebra of infinitesimal quantum gauge symmetries of a string.
The Witt algebra (and hence the Virasoro algebra) acts on the affine algebra
g by derivations:
B.34)
[Lm, x[n}\ = -nx[m + n], [Ln,c] = [K, ¦} = 0.

LECTURE 2. REPRESENTATIONS OP LIE ALGEBRAS
(recall the notation x[n] = x®tn). Note that the action of Lq coincides with the
adjoint action of — d. Therefore, the semidirect product Vir ix g constructed from
B.34) contains g as a subalgebra.
Let V be a highest-weight representation of g of level k; as before, we assume
that k t^ —hv. The following construction, due to in general to Sugawara and in
this particular form to Segal, defines an action of the Virasoro algebra on V. Let
B be an orthonormal basis of g. Define the following operators in V:
B35) s^
where the normal ordered product : a[n]a[fe] : is defined by
f a[nla[fe], k > n,
B.36) :a[n]a[k]:={ ' "
1 J (. o[A]o[n], k < n.
It is easy to see that although the sum is infinite, the Lm are still well defined
operators in V, Indeed, if we take u e V and apply Lm to it, then, because of
the normal ordering, only a finite number of terms in the sum will give a nonzero
contribution.
Theorem 2.9.1. Formulas B.35) define an action of the Virasoro algebra on
V. This action and the original action ofg combine into an action of the semidirect
product Vir ix fl onV.
Proof of this theorem can be found in [Kac, Section 12.8].
Since g is contained in the semidirect product Vir ix g, the theorem implies
that there is a natural way to extend the action of g on V to an action of g by
letting d = —Lq. To describe this extension explicitly, it is enough to prescribe the
eigenvalue of d at the highest-weight vector: dv = — Av. Thus, there must be a
preferred value of A. Let us calculate this value.
Recall that x[n] annihilates v if n > 0. Therefore,
u 2(k + hv) *->
where C is the Casimir element B.9). Thus,
constant A. Under this choice,
1
2(fc + hv)
is a natural choice of the
B.37)
d = -in = -
: a[n]a[-n] : .
Had we made a different choice of A, we would have to insert an additional constant
term in B.37).
2.10. Generating functions and currents.
It is often convenient to use generating functions when working with the affine
Lie algebras and the Virasoro algebra. The calculus of the generating functions is
encoded into the structure of vertex operator algebras (see [FLM]). The simplest
2.10. GENERATING FUNCTIONS AND CURRENTS. 27
examples of generating functions are "currents", a term adopted from quantum
physics.
Let x 6 g. The current associated to x is the formal sum
B.38)
n€Z
The current can be split in two parts:
B-39) J+{Z) = ? X[n] ¦
n<0
J-(z) = - ? x[n]
n>0
Let V be a highest-weight representation of g with central charge k, and let
z ^ 0 be a complex number. Then Jx(z) can be regarded as an operator: V —>
V, where V is the completion of V with respect to the natural Z-grading (this
completion consists of all, finite or infinite, series of homogeneous vectors with
strictly decreasing degrees). Moreover, J^{z) is actually a well defined operator
V ^ V.
Let us now look at products of currents. It is easy to see that the products
J?(z)Jy«)i Jx(z)Jy@, and J?(z)Jy(.O are well defined operators V -+ V. The
only product which is not well defined is J~(z)J+(C). This product makes no
sense as a formal sum, since after expanding it we obtain infinite sums of terms
of the same degree. Indeed, let u be a homogeneous vector of degree p; then the
component of J~(z)J+(Qu of degree p + n is
B.40)
z j 1w~n+j~1
x\j]y[n - j]u.
However, if j is large enough, we have x\j] ¦ u = 0, which implies
B.41) x\j]y[n - j}u = [x\j], y[n - j]]u = {{x, y])[n]u + j(x,yNnfiku.
This shows that all but finitely many terms of the series B.40) in fact lie inside
a two-dimensional subspace of V spanned by u and ([x,y]<8>tn)u. Therefore, we can
talk about the convergence of this series from the analytic point of view. Indeed,
when summing B.40), we will need to calculate two sums:
/ *
3>jo
3>3O
We know from calculus that these sums are convergent if and only if \z\ > |?|. Thus,
we can make sense of the product J~(z)J^(Cl) as an operator V —> V, provided
> ici-
We can now rewrite the commutation relations in jj in terms of currents.
Theorem 2.10.1.
B.42)
k{x,y)
{z-wJ'
The proof can be obtained by straightforward calculation.
z > w.

28 LECTURE 2. REPRESENTATIONS OF LIE ALGEBRAS
Corollary 2.10.2.
We conclude this section with the generating function for the Virasoro algebra.
Denote
B.43)
n€Z
Then formula B.35) can be rewritten as follows:
B.44) H*) = », 1 "
Lecture 3. Knizhnik-Zamolodchikov Equations
In this lecture we will begin to study a special class of intertwining operators
between a highest-weight module and a tensor product of another highest-weight
module and an evaluation representation for affine Lie algebras. These intertwin-
intertwining operators, also called vertex operators, or primary fields, were first studied
by V. G. Knizhnik and A. B. Zamolodchikov [KZ], who derived the celebrated
equations for vacuum expectation values (that is, some special matrix elements) of
products of such operators. A rigorous mathematical formulation of their work (for
g = sl2) was given by A. Tsuchiya and Y. Kanie [TK]. The differential equation
for a single intertwining operator and the trigonometric form of the Knizhnik-
Zamolodchikov equations were obtained in [FR]. We will review the basic prop-
properties of the intertwining operators, Knizhnik-Zamolodchikov equations, and their
solutions, following [KZ], [TK], [FR].
Knizhnik-Zamolodchikov equations present a fundamental example of consis-
consistent linear systems of differential equations with the additional factorization prop-
property. Systems of this kind are constructed from solutions of the classical Yang-
Baxter equation. For example, Knizhnik-Zamolodchikov equations are obtained
from the simplest rational solution of the classical Yang-Baxter equation. A. A. Be-
lavin and V. G. Drinfeld classified all nondegenerate solutions of the classical Yang-
Baxter equation meromorphic in a neighborhood of the origin [BD], We will state
their result at the end of this lecture.
3.1. Classification of intertwining operators.
Let Lx1,Lx0 be irreducible highest-weight representations of g, and let k be
a complex number. From now on, we assume that k is generic with respect to
each Aj (see B.30)), and thus the induced modules Vx,k — Ind?+ix are irreducible:
Vx^fc = L\^k. By definition, the space of vectors of degree zero in VAS. is identical
to Lx.
We are going to study g-intertwining operators $ : Vxt,k ~* V\0,k®V{z), where
z is a nonzero complex number. Here V\tk®V{z) denotes the completed tensor
product, which consists of all infinite expressions of the form JZSi wi ® vi>wi ?
V\,k,vi G V(z), such that Wi are homogeneous vectors and {degree(wi)} —» — oo.
By definition, the intertwining operator <& satisfies
C.1) <S>x[n] = (x[n] ® 1 + z" ¦ 1 ® rr)*.
The following theorem classifies such intertwining operators.

28 lecture 2. representations of lie algebras
Corollary 2.10.2.
[J±{z),J±(z)] = -J± (z).
x dz lx'^
We conclude this section with the generating function for the Virasoro algebra.
Denote
B.43)
n€Z
Then formula B.35) can be rewritten as follows:
B.44) ?(*) = - 1 ^ T"
Lecture 3. Knizhnik-Zamolodchikov Equations
In this lecture we -will begin to study a special class of intertwining operators
between a highest-weight module and a tensor product of another highest-weight
module and an evaluation representation for affine Lie algebras. These intertwin-
intertwining operators, also called vertex operators, or primary fields, were first studied
by V. G. Knizhnik and A. B. Zamolodchikov [KZ], who derived the celebrated
equations for vacuum expectation values (that is, some special matrix elements) of
products of such operators. A rigorous mathematical formulation of their work (for
g = sl2) was given by A. Tsuchiya and Y. Kanie [TK]. The differential equation
for a single intertwining operator and the trigonometric form of the Knizhnik-
Zamolodchikov equations were obtained in [FR]. We will review the basic prop-
properties of the intertwining operators, Knizhnik-Zamolodchikov equations, and their
solutions, following [KZ], [TK], [FR].
Knizhnik-Zamolodchikov equations present a fundamental example of consis-
consistent linear systems of differential equations with the additional factorization prop-
property. Systems of this kind are constructed from solutions of the classical Yang-
Baxter equation. For example, Knizhnik-Zamolodchikov equations are obtained
from the simplest rational solution of the classical Yang-Baxter equation. A. A. Be-
lavin and V. G. Drinfeld classified all nondegenerate solutions of the classical Yang-
Baxter equation meromorphic in a neighborhood of the origin [BD]. We will state
their result at the end of this lecture.
3.1. Classification of intertwining operators.
Let L\t, L\o be irreducible highest-weight representations of g, and let k be
a complex number. From now on, we assume that k is generic with respect to
each A; (see B.30)), and thus the induced modules V\^ = Ind?+?,\ are irreducible:
V\x,k — Lxuk- By definition, the space of vectors of degree zero in V\tk is identical
to Lx.
We are going to study g-intertwining operators $ : VXuk —> V\0]fc®V(z), where
z is a nonzero complex number. Here Vx^<S>V(z) denotes the completed tensor
product, which consists of all infinite expressions of the form YlZi Wi ® vuwi e
Vx,fci"t G V(z), such that Wj are homogeneous vectors and {degree(Wi)} —» — oo.
By definition, the intertwining operator $ satisfies
C.1) $x[n] = (x[n] ® 1 + zn ¦
The following theorem classifies such intertwining operators.

30 LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Theorem 3.1.1. Let g : LXl —> L\a <8> V be a Q-homomorphism. Then for
generic k there exists a unique Q-intertwining operator
such that for every vector w € VxLifc[0] = L\x the degree zero component of $g(z)w
is equal to gw.
PROOF. Assume for simplicity that the representation V is finite-dimensional.
Let us construct the operator <69 using the information we have. First of all, we
have to reconstruct $9(z)w, where w € Va^a^O] = Lx1 is a vector of degree zero.
We know two facts about the vector $9(z)w:
1) It is annihilated by the subalgebra g <S> tC[t];
2) Its zero degree component equals gw.
These conditions determine the vector uniquely. Indeed, the first condition
gives
But for generic k, the module V?o k is freely generated over g®iC[i] by (VaOia;[0])* =
L\o. Indeed, this statement is equivalent to saying that the contragredient module
V?o k is free over g <S> t~1C[t~1], which holds because for generic k, V?o k c^ VxOtk
(see end of Section 2.7). Thus, the restriction map
;0?S:, V(z)) -» Homc(il0, V) = LXo <8> V
is an isomorphism. Therefore, $9(z)u; is uniquely determined by its zero degree
component.
Now we can define the action of 4>9(z) on the whole module Vxi,/fc using induc-
induction on the degree and the identity
$9(z)x[n]w = (z[n]<g> 1 + zn ¦
This gives a well defined operator, because V\ltk is freely generated by its zero
component over g <S> t~1C[t~1].
Thus we have constructed an intertwining operator of the form C.2). Since no
choices could be made in the process of construction, such an operator is unique.
The proof for inifinite-dimensional V is completely parallel. D
3.2. Operator Knizhnik-Zamolodchikov equation.
The construction of the intertwining operator $9(z) given in the proof of The-
Theorem 3.1.1 shows that its matrix elements are Laurent polynomials in z. Therefore,
this operator can be expanded in a Laurent series
C.3) *9(z) = E$9[n]z~n-
ngZ
It is also seen from the construction that the Laurent coefficients are homogeneous
operators: 4>9[n] maps vectors of degree j to vectors of degree j + n. Thus, for
any A ? C the expression z~^$>9(z) can be regarded as a jj-intertwining operator
V\i,k —> V\0,k®z~AV[z, z], where <S> denotes the completed tensor product, and
z is considered as a formal variable. However, it is not an intertwining operator for
the bigger algebra {j unless we make a special choice of A.
3.2. OPERATOR KZ EQUATION
Proposition 3.2.1. The operator
z-^9(z) : VXlik -» V^k®z-^V[z,z-1]
is a g-homomorphism if and only if
C.4) A = A(Ai) - A(A0),
where A(A) is defined by B.25).
If A is defined by C.4) then we will denote z^"A*9(z) by
C.5)
n}z-"-*, A = A(Ai) - A(Ao).
ngZ
Proof. By definition, z~A*(z) is a g-intertwiner iff it satisfies
z*(z) = A ® z^)z*(z).
dz
It follows from C.5) and the fact that *9[n] has degree n that it suffices to
check this identity for level zero vectors, in which case it becomes
?
As before, let V* be the restricted dual to V. Obviously, as jj-modules, V*(z) ^
(V(z))'. Let u ? V, and let *?(z)w = u{$9{z)w) for w ? V\uk. We will regard
*?(z) as an operator V\lik —> V\Otk. The intertwining relation C.1) in terms of
$J(z) takes the following form:
C.6) [x[n],*»(z)] = zni
Let us now write the intertwining property
of all, the action of currents on V*(z) is given by
in terms of currents. First
TOO
n<0 n<0 ¦
Therefore, the intertwining property can be written in the form
C.8)
[J?@, *»(*)] =
.$9
These relations will also hold if $ is replaced by 4.
Note that the identity corresponding to the "plus" (respectively, "minus") sign
makes sense only in the region |C| < \z\ (respectively, \z\ < \C,\).
Now we are in a position to deduce the operator version of the Knizhnik-
Zamolodchikov equations. Let V = VM be a lowest-weight g-module with lowest
weight — p. Note that the Casimir element C acts in V^ by multiplication by

32 LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Consider the operators
C.9) $9(z) = z-AMi9(z) = E 4SW«""'4, A = A(Ai)
n€Z
where, as before, A(A) is given by B.25).
For o6g, let : Ja(z)<I>g(z) : be the normally ordered product:
C.10) : Ja(z)&u(z) :=J+(z)&u(z) - H{z)J~{z).
The following result, which we will call the operator Knizhnik-Zamolodchikov equa-
equation, has been obtained in [PR].
Theorem 3.2.2. Let B be an orthonormal basis of g. Then the operators
ig(z) satisfy the following system of differential equations:
C.11)
(* + Av)^;«SW = E:'
PROOF. The d-invariance property of <!9(z) can be written in the form
C.12) zTz*i{z) =
which is equivalent to
d -
Substituting into this equality Sugawara expression B.37) for d = — ?o, we
obtain
$
E[«[-»]
Using the intertwining relations C.8), we can rewrite it as follows:
3.4. KZ EQUATIONS FOR CORRELATION FUNCTIONS 33
Since ^2aeB $9a2u{z) = $cu(z) = (n,n + 2p)^(z), the last two terms cancel, and
we obtain
Multiplying both sides by
-, we get C.11).
?
Note that the equation C.11) can be understood as an equality of both power
series and analytic functions.
We also note that we never used the assumption that V^ is a lowest-weight
module; the proof above works for any module with weight decomposition such
that the Casimir element acts in it by a constant, in particular, for V = iM or
3.3. Gauge invariance of the intertwining operators.
In the previous section we deduced the commutation relations between the
intertwining operators $ and the element Lo of the Virasoro algebra. In this section,
we calculate the action of the entire Virasoro algebra on these operators. These
results are not used anywhere in this book.
Proposition 3.3.1.
C.13)
«*SW1 =
Idea of Proof. Substitute the Sugawara expression B.34) for Lm, then use
the intertwining relations as we did in the proof of Theorem 3.2.2. ?
Let j,j 6 C. Consider the space zpV^[z, z~1](dz)q spanned by formal "differ-
"differentials" v ¦ zp+n(dz)q, ueZ, with the following action of Vir:
Lm ¦ vzp+"{dz)9 = -v{p + n + q(m + l))z"+7l+m{dz)q.
This makes zpVli[z, z~1]{dz)q into a (Vir IX g)-module.
The following statement clarifies the representation-theoretical meaning of for-
formula C.13).
Corollary 3.3.2. The operator $3(z) is an intertwining operator Vxltk —>
V\0,k ® z~c^Vi\z, z~1](dz)A<-t^> for the entire gauge algebra Vir IX 0.
3.4. Knizhnik-Zamolodchikov equations for correlation functions.
Let L\,, 0 < i < N, be irreducible highest-weight g-modules, and let V^s,
1 < i < AT, be lowest-weight modules with lowest weights — fj,t.
Let us fix a level k 6 C generic with respect to each of A; (see B.30)) and
g-homomorphisms gi : L\t —» Lxt-1 ® Vw. Let
be the corresponding g intertwiners as in C.9). Consider the product
C 14)

LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Figure 3.1
This formula is visualized by Figure 3.1, where every line represents a g-module,
and every node represents an intertwiner, with the bottom line as the input and
two top lines as the outputs.
Since the operator i>9i+1 (zi+i) takes values in the completion V\,jfc®V^j+1 while
the operator i>9i(zi) is defined only on VXilk, it is not obvious why the product C.14)
makes any sense at all. At this stage, the best we can say about this product is that
it is defined as a formal power series in zx,..., zN. It is easy to show that this series
belongs to zrAl • ¦ • ^A"C[[^, -, ^]], where A{ = A(A<) - A^) + A(W).
Let u0 € (VXo,k[0])* = (LXoy,uN+1 e VXNtk[0] = LXn. Define the correlation
function
C.15) (uo,^(zi,...,z
Denote by V the following space:
C.16) V = Vtll®...®VtlN®LlN.
Note that this space has finite-dimensional weight subspaces, and the weights are
"bounded from below".
For fixed «o define a V-valued function ip(zi, ¦.., zn) by
C.17) V(*i, ¦ • ¦, zjv) = («o, *(zi, • • •, «jv)-)-
Equivalently, we can choose «i e V*^,..., un € V*N and write
C.18) ^i,...,««+1 (zi, • •., zw) = («o, *Ji (*i) ¦ ¦ • *S«(zjv W+i) e C.
Let us introduce some notation. For x € g, denote by (x)i,i = 1,..., N + 1,
the action of x on the i-th component of the tensor product V^ <8>... <8> V^.lV ® LXn .
Similarly, for A = x<S> y e g® g we denote A^ = (a;)i(y)j.
Our main result is the following theorem, originally derived in [KZ].
Theorem 3.4.1. The correlation function y>(zi,..., zjv) defined by C.17) sai-
is/ies i/ie following system of differential equations:
¦j=ijA
¦i,N+l
Zi
3.4. KZ EQUATIONS FOR CORRELATION FUNCTIONS 35
At the moment, we consider these equations as equalities of formal power series
in zi+1/zit with the fractions in the right hand side considered as formal power
series by l/(Zi - zj) = z^H^ai^/^)" for 3 > «• Later we wiU show that the
correlation functions are analytic for \z\\ > \z2\ > • • • > 0 and equations C.19) can
be interpreted as identities of analytic functions (see Section 3.6)
Equations C.19) are called the Knizhnik-Zamolodchikov equations. Usually,
they are written in a different form, namely:
C.20)
i= l,...,N + 1.
These two forms are obviously equivalent: if ip(zi,..., zN+1) is a solution of
C.20) then ij){z\,..., zN, 0) satisfies C.19). Conversely, if ip(zi,..., zn) is a solution
of C.19) then ij>{zu ..., zN+i) = i>{zx - zN+1, ...,zN- zN+1) satisfies C.20).
The rest of this lecture is devoted to the proof of this theorem.
Proof. The basic idea of the proof is to use the operator KZ equation (The-
(Theorem 3.2.2) and then use the commutation relation of 4 with currents.
Let ¦>f>u1,...,uN+1(zi, ¦ ¦ ¦ ,zn) be the scalar-valued power series defined by C.18).
Due to Theorem 3.2.2, we have
C.21)
—-ipUlt,.,tUN+1{zlt..., zN)
= (mo,
(...(fc+n^-*.
uw(zN)uN+1)
Now pull the currents J all the way to the right, and the currents J+ all the way
to the left:
- $aUi(zi)Ja {zi))...$UN(zN)uN+i)
J?(zi)\ ¦ ¦ ¦ iaUi (z,) ¦•¦*«„(zn)un+i)
Since uq and m^+i are zero degree vectors, we have Ja {z)un+i — —au^+i/z,
and (un,J+(z)w) = ~(J+(z)u,hw) = 0 for any w e V\Otk- Therefore, using the
commutation relations C.8) between currents and intertwiners, we can rewrite the

LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
previous formula in the form
)—ipuu...tUN+l{zu...,
a?BZi
a
As a rule, we will use the KZ equations in the form C.20). Note that C.20)
implies that any solution is translation-invariant: if 4>{z\,..., zN+i) is a solution
then ip(zi + c,..., zN+1 + c) = ip(zi,..., zN+1). We can also note that if \N = 0,
i.e. L\N is the trivial representation, then equation C.19) in N variables becomes
C.20) also in N variables, not in N + 1 which happens for nontrivial L\N.
Note that if uQ e (Vxo,fc[0])* is a lowest-weight vector with respect to g, then
the correlation function ip defined by C.17) takes values not in the whole space
V = V^ ® ... ® VMN ® L\n but in the subspace Vn ; similarly, if Ao = 0, so that
«o is g-invariant, then i\> takes values in Vs.
In this approach, we derived the KZ equations using the commutation relations
of *(z) with Lo. In fact, we could get the same equations using any Li (see
Proposition 3.3.1); for example, in [TK] the KZ equations are derived using the
commutation relations of *(z) with L-\. We have chosen Lo because this approach
can be generalized to the quantum case.
It is easy to see that all the arguments above work if we allow k to be non-generic
and replace Vx,,/fc by any highest-weight module with the same highest weight (for
example, L\uk), provided that the intertwiners <69i exist.
3.5. Consistency and g-invariance of
the Knizhnik-Zamolodchikov equations.
Theorem 3.5.1. The KZ system C.20) is consistent, i.e. for every p,q =
1,..., N + 1 we have
r, N+l
N+l
0
C.22) (*+fcV)-?__ T. -^-,(*+n#-- E -^- =°-
Sketch of proof. The proof is straightforward. Formula C.22) follows di-
directly from the relations Qpq = Qqp and [Qpq, Qpr + Qqr] = 0. The first relation is
trivial. Let us check the second relation. As usual, for a G g we denote by ap the
action of a on the i-th factor of the product V =
... <8> VllN ® L*X
Then
[np
a,b?B
3.6. ANALYTICITY OP THE CORRELATION FUNCTIONS
37
The last expression is zero because Q is invariant under the adjoint action of g: for
any b e g, we have [Q, b ® 1 + 1 ® b] = 0. Q
This shows that the KZ equations define a flat connection on a trivial vector
bundle with fiber V over the base XN+1 = {(zlt..., zN+1) e Cw+1|zi ^ Zj}; in
another language, these equations define a local system (locally constant sheaf)
over XN+i with fiber V. We discuss the notion of local system and related topics
later, in Lecture 7. We will denote this local system by VKZ, and for every open
V C XN+1 we denote by T/(:E>, VKZ) the space of solutions of KZ equations in V,
i.e. the space of flat sections of VKZ over V. It is a well-known fact that if V is
simply connected then Tf(V,VKZ) is isomorphic to V: for every point p € D, the
map Tf(D, Vkz) —* V : i> i-> ip(p) is an isomorphism.
Note that though V may be infinite-dimensional, its weight subspaces are finite-
dimensional. Since the KZ equations preserve weight subspaces, we can consider
the infinite-dimensional local system Vkz as a direct sum of finite-dimensional local
systems; thus, all the standard results on local systems apply to Vkz-
Theorem 3.5.2. The KZ system C.20) is Q-inva.ria.nt, i.e. for every x e g
and a solution ip of this system the function xip is also a solution.
Proof. It suffices to prove that
N+l
E
a,
,x\ =0,
which again follows from the g-invariance of fi. ?
This shows that the space of solutions of the KZ equations is naturally a
module over g. Moreover, for a simply-connected domain V, the isomorphism
Tf(T>, VKz) —> V : i> h-> V(f) discussed above is in fact an isomorphism of g-
modules. Theorem 3.5.2 also implies that the KZ equations can be considered in
the space of g-invariants V3 or in the space of singular vectors Vn . Note that
if V is a direct sum of lowest-weight modules, then any V-valued solution can be
obtained by applying W(n+) to V -valued solutions.
3.6. Analyticity of the correlation functions
Note that up to now our considerations (including the KZ equations) have been
purely formal. So far we have not learned how to multiply the operators *Uj(z0
with Zi being complex numbers: we know nothing about the convergence of the
corresponding power series. Remarkably enough, the KZ equations give us a key
to resolve this difficulty.
Let us regard C.19) as a system of differential equations for an analytic, V-
valued function i>{zi,... ,zpt). Let us introduce new variables Q =
l,...,iV-l,0v =
the following form:
C.23) i
Then it is easy to see that C.19) can be rewritten in
where the A; are holomorpbic functions in the region {|?j| < 1, 1 < j <
Moreover, because of the consistency property we have [A;@), Aj@)] = 0.

LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Now we can make use of the analytic theory of differential equations with
regular singularities.
Proposition 3.6.1. Let A;(Ci,.. ¦ ,Cm), I <i <m, be (n x n)-matrix valued
functions holomorphic in the region {\Q\ < 1, 1 < i < m}, such that the differ-
differential operators Cigfr ~ ^-i commute with each other. Assume also that for every
i, the eigenvalues of A;@) do not differ by a nonzero integer. Then the system of
differential equations
C.24)
1 < i < m,
has a unique (n x n)-matrix valued solution F [the fundamental solution) of the
form
C.25)
— i'ol.Cl) •••iCm
•••Sm
where Fa is an (n x n)-matrix valued function holomorphic in the region {\C,j\ <
1, 1 < j < m} and such that F0@) = Id.
For n = 1 this proposition is well-known in the theory of ordinary differential
equations (see, for example, [CL, Chapter 4]). For arbitrary n this statement can
be easily deduced from the results of [De, Section II.5].
Corollary 3.6.2. In the notation of the previous theorem, let f be a vector-
valued power series solution of system C.24):
C.26) /(Cl, . . . ,Cm) = Cf1 • • • dm/o(Cl, • • • ,Cm),
where f0 G C"[[Ci, ...,Cm]] and 8i e C. Then there exists a constant vector »eC"
such that /(Ci, ¦••) Cm) = F(?i,..., Cm)«, where F is the fundamental solution C.25),
and thus f converges to an analytic solution in the region {0 < \Q\ < 1}.
Proof. Let «(Ci, • - - > C™) = F~rf. Differentiating the relation / = Fv, we
obtain -S-v = 0,1 < i < m, which implies that v is a constant vector. Cl
In fact, the last statement of this corollary - namely, that any formal solution
of the form C.24) is analytic for 0 < \Q\ < 1 - can be easily proved without use of
Proposition 3.6.1.
Theorem 3.6.3. The correlation function C.17) is absolutely convergent in the
region \zi\ > |^| > • • • > \zn\ > 0 and defines a regular multivalued holomorphic
function in this region.
Proof. As we proved (see Section 3.4), this correlation function is a power
series solution of the KZ equations C.19) which can be written as V(Ci. • • • > Ov) =
Cil ¦¦¦CSnnMCu---,Cn-i), where ip0 e V[[Ci,... ,0v-i]]- Since the KZ equations
are consistent, and V can be represented as a direct sum of finite-dimensional weight
subspaces preserved by these equations, we can apply Corollary 3.6.2, which gives
us the statement of the theorem. D
Corollary 3.6.4. For |zi| > \z2\ > ••¦ > |zjv| > 0 the product of opera-
operators $Ul(zi)...$UN(zjv) is a well defined operator V\Nik —> V\Otk (i.e all matrix
coefficients of this product are convergent series).
3.7. CORRELATION FUNCTIONS SPAN THE SPACE OP SOLUTIONS 39
PROOF. We should show that the power series obtained by expanding the
product is convergent in the region \zx\ > |z2| > ¦ • • > |zjv| > 0. Because of the
intertwining property, it is enough to prove that if w is a vector of degree zero in
VxN,k, then the series Xw{zi,-,zN) = *ui(«i) ••• <&UN{zN)w is convergent. Xw can
be regarded as a power series with coefficients in Hom(VA*o k, VM1 ®.. .®V^W). Since
for generic k the module V?o k is generated over g by its zero-degree component
(compare with the proof of Theorem 3.1.1), it suffices to check the convergence of
this series when applied to vectors of degree zero in V^ k. This is equivalent to
checking the convergence of expressions (v, *Ul(zi)... $UN(zN)w), where v is of
degree zero. These expressions are nothing else but the correlation functions whose
convergence we have already proved. ?
Corollary 3.6.5. In the limit |zt| » ¦•• » \zN\ the correlation function
•0(zi,..., Zfi) defined by C.17) has the following asymptotics:
C.27)
where A* = A(A;)
C.28)
and
v = (uQ,gi...gN-)eV.
More precisely, this means that ip can be written in the form
C.29) ^ = Zl-Al...z-^Vn,
where ipQ is a function of Ci = z2/zi,... , Ov-i = ^W^w-i which is an analytic
single-valued function in a neighborhood of the point Q — 0 with values in V and
such that -ipo(O) = v.
PROOF. If we consider ip as a power series rather than an analytic function,
then C.29) easily follows from the explicit form of $ (see C.9)) and was already
used before. Since we know that this series is convergent, C.29) also holds as an
identity of analytic functions. ?
3.7. Correlation functions span the space of
solutions of the Knizhnik-Zamolodchikov equations.
In this section we show that under certain assumptions on \i,iii,k the cor-
correlation functions defined in Section 3.4 span the space of solutions of KZ equa-
equations C.19).
Let us recall the setup of Section 3.4. Let us fix Ajv,jU;. Denote by ? the
following collection of data:
C.30)
L\o.
The set of all possible ? spans the vector space
N
C.31)

40 LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Here the direct sum is taken over all Ao,..., \n-i € h*¦ However, it follows
from weight considerations that the only nonzero terms are those satisfying Aj +
Hi — A,_i 6 Q.
The vector space E has a natural structure of a g-module: we consider the
spaces of homomorphisms as trivial g-modules.
For each (eHwe defined in Section 3.4 the correlation function ijA(zi,..., zN)
(see C.17)). This function takes values in the space V = VM1 ® ... <8> V^N ® L\N
and satisfies the KZ equations C.19) in the domain |zi| > • ¦ ¦ > \zN\ > 0. Note
(see Remark 3.4.2) that if ? ? Hn~ (respectively, E») then ^ takes values in Vn
(respectively, in V8).
Let
C.32)
v =
cN\\Zl\ >¦¦¦> \zN\
Then V is simply connected, and thus the space V{V, Vkz) of V-valued solutions
of the KZ equations C.19) in V is isomorphic to V as a g-module (see Section 3.5).
Our goal is to show that "generically" the spaces E and T(V, Vkz) are isomor-
isomorphic. Let us recall that we have denned the notion of generic weight A ? t)* (see
B.8)); also, we have defined what it means for the level k to be generic with respect
to a weight A (see Theorem 2.8.2). In particular, if k is generic with respect to A
and A' e A + Q, then A(A) ^ A(A') (here, as before, A(A) = (A, \ + 2p)/2(k + hv)).
Theorem 3.7.1. Fix \N,fii,i = l,...,N, and the modules V^ as in Sec-
Section 3.4 such that they satisfy one of the following conditions:
1. "Generic case." The weights Ajv, Ajv+Mtv>- •• ,Xn+hn-\ Yix\ are generic
and k is generic with respect to each of them.
2. "Finite-dimensional case." XN, m ? P+, k(?Q, and the V^ are irreducible
finite-dimensional modules with lowest weight — ^.
Then correlation functions of the form C.17) span the space of V-valued solu-
solutions of KZ equations in T>. More precisely, the map
C.33)
where S is defined by C.31), is an isomorphism of g-modules.
Note that it follows from the weight considerations that in the generic case all
weights A, in C.31) are generic, and in the finite-dimensional case, all weights are
from P+.
Before proving this theorem, let us formulate and prove the following "complete
reducibility lemma":
Lemma 3.7.2. Under the assumptions of Theorem 3.7.1, the map
C.34) ~ "* / '
is an isomorphism of g-modules.
PROOF. First, note that
3.7. CORRELATION FUNCTIONS SPAN THE SPACE OF SOLUTIONS 41
In view of this, simple induction arguments show that to prove the lemma it
suffices to prove that the natural map
C.35) ^^Homg(i*, V^ ® Lx) ®?* —> V^ ® Lx
is an isomorphism if either both A and A + /j, are generic or both are from P+.
If A, A + [i e P+, then V^, L\ are finite-dimensional and it is well known that
C.35) is an isomorphism. Let us assume that A, A + /j, are generic. Since LI is
irreducible, the map C.35) is injective. On the other hand, since v must also be
generic, we have i* ~ M*,LX c± Mx. Using the same methods we used in the
proof of Theorem 3.1.1, we can show that in this case
a Q?\ urt~. / r * t/ o r*\ t/A— v
\o.o\i) noninlL,,, Vu. Qy L/X) — v
and therefore, the dimensions of the weight subspaces on both sides of C.35) coin-
coincide, which proves that it is an isomorphism. ?
Proof of Theorem 3.7.1. It follows from the consideration of asymptotics
" in the limit \z\ » • • • > zN (Corollary 3.6.5) and the previous lemma that
the map ? h-> ipt is injective. On the other hand, since both S and T(V, Vkz) are
isomorphic to V as g-modules (see Section 3.5 and the previous lemma), their weight
subspaces have equal dimensions, and thus the map ? t-+ ipt is an isomorphism. D
For computation of the solutions, it is more convenient to use a slightly dif-
different version of Theorem 3.7.1. Namely, let us assume that \n = 0. Then the
corresponding correlation functions ip^(zi,..., Zfj) take values in the space
and satisfy the KZ equations
C.37) (k + hv)-^-ip =
N
E
Simple modification of the proof above allows to prove the following result.
Theorem 3.7.3. In the setup of Section 3.4, assume additionally that \N = 0
and ^ii, k, Vm satisfy one of the following two conditions:
1. "Generic case." The weights hn,hn + Miv-i; • • • >fiv + • • • + Mi are generic
and k is generic with respect to each of them.
2. 'Finite-dimensional case." in 6 P+,k ^ Q, and the V^ are irreducible
finite-dimensional modules with lowest weight —jj,i.
Then correlation functions of the form C.17) span the space of V-valued solu-
solutions of the KZ equations in T>. More precisely, the map
C.38)
where E is defined by C.31), is an isomorphism of ^-modules.

42 LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Corollary 3.7.4. Under the assumptions of Theorem 3.7.3, correlation func-
functions of the form C.17) with «o being a lowest-weight (respectively, g-invariant)
vector in L\ span the space of solutions of the KZ equations with values in V
(respectively, V8).
Remark 3.7.5. Theorem 3.7.1 gives us a new way to identify the space of
solutions V;(T>, Vkz) with V. So far, we have only had a naive identification:
choose a point p € T> and map ip >-> ip(p). This depends on the choice of p. Now
we also have a "clever" identification: compose the isomorphisms
Tf(p,VKZ)~E~V,
constructed in Theorem 3.7.1 and Lemma 3.7.2, respectively.
This isomorphism can be described without the use of correlation functions and
the space E: namely, it assigns to a solution t\> a suitably renormalized asymptotic
of i/) in the limit |zi| ;§> ••¦ 2> | Z/v 11 which is the correct analogue of the nonexistent
"value of ^ at an infinite point". We will return to this in Lecture 8.
3.8. Trigonometric form of the Knizhnik-Zamolodchikov equations.
Let us assume that «0 is the lowest-weight vector in L\a and up/+i is the
highest-weight vector in LxN,k- Consider the correlation function
C.39) 4>{z\, ¦ ¦ ¦ izn) = (uq, *(zi, ..., zjv)ujv+i) ? V^ ® ... ® VllN
(see C.15) for notation). This function can be considered as a "part" of the correla-
correlation function ip(zi,..., zp/) defined by C.17). It follows from weight considerations
that in fact (f> takes values in the (finite-dimensional) subspace of vectors of weight
C.40) 4> e (Vm ® .
Let us derive a differential equation for
equations C.19) for i\>, which gives
C.41)
We start by rewriting the KZ
i,.. .,zN)aluN+i
where aj, a' are dual bases in g, and as before, (a)i stands for the action of a ? g
in the 2-th component of the tensor product.
Now, let us choose some special basis in g. Namely, for a e R+ let ea e
ga, fa 6 3~a be such that (ea, fa) = 1, and let xp,p = 1,..., r, be an orthonormal
basis in f). Then the elements Q,C defined in B.9) can be written as
C.42)
c=
3.8. TRIGONOMETRIC FORM OF THE KZ EQUATIONS
Therefore, we can rewrite the last term in C.41) as follows:
o6fi+
Here we have used the intertwining property of >& and n~-invariance of u0.
It is convenient to rewrite the last expression in a slightly different form. Note
that since 4> has weight Atv-Ao, we have Y,p T,f=i{xp)i(xp)j<l> = (hXN_Xo)i<t>. Thus,
C.43)
N
TV
E
where we have used the notation
C.44)
It is easy to check that [ea, fa] = ha and thus E^a/a + | Ep «P = \C + hp.
Therefore, C.43) takes the form
N
J=l
Substituting this in C.41), we get after simplification
C.45)
i±
]=1
3?i
Thus, we have the following theorem.

LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
Theorem 3.8.1. Let (f> be the correlation function C.39), and let the variables
Xi,i = 1,... ,N, be defined by Zj = eXi. Then 4> satisfies the following system of
differential equations:
C.46)
d
i
where
C.47) r(x) = e*QeX +In+
and fi* is defined by C.44).
Equivalently, we could rewrite C.46) as follows. Let
N
Then
C.48)
N
D
l,.
Equations C.46) or the equivalent equations C.48) are usually called the trigo-
trigonometric KZ equations, due to the appearance of the function ex in the definition of
r(x). In the spirit of this terminology, it is natural to call the original KZ equations
"rational".
As we have seen, rational and trigonometric KZ equations are closely related:
above we have shown how trigonometric KZ equations can be obtained from rational
ones. It is also possible to obtain the rational KZ equations from the trigonometric
ones.
3.9. Consistent systems of differential equations
and the classical Yang-Baxter equation.
Consistent linear systems of differential equations
C.49) -?-</>(z1,...,zN) = Ai(z1,...,zN)<j>(zu...,zN), <fr ? V, At ? End(V),
where V is a vector space, are of great interest in geometry and analysis. If the Ai
are holomorphic functions in a certain region V C CN, then the equations C.49)
represent a flat connection, V* = d, - Ait in the trivial vector bundle over V.
The flatness condition is represented by the standard commutativity relation, well
known to geometers and physicists:
C.50)
(compare with Lecture 7).
- djAi + [Ait Aj] = 0
3.9. CLASSICAL YANG-BAXTER EQUATION
So far we have seen two examples of such systems: rational and trigonometric
KZ equations. In a certain sense these examples are unique of the kind. In this
section we will make this statement precise.
One of the most remarkable classes of consistent systems of equations is asso-
associated with an arbitrary simple Lie algebra 0.
Let us say that a system C.49) is factorizable if
1. V = V\ ® V2 ® ¦ • • <8> Vjv, where V* are 0-modules;
2. There exists a function r(z) € 0 ® 0 of one complex variable, meromorphic
in a neighborhood of the origin, and a complex number k € C*, such that
C.51)
Ai(zu...,zN) = -
where s ? 0, and [s ® 1 + 1 ® s, r(z)] — 0.
Here ry is the usual notation: r^ = ~52p(av)i{bv)j when r = Y.Pap ® V
The notion of factorizable system was first introduced by Cherednik ([Ch2]).
Combining C.49) and C.51), we come to the following proposition.
Proposition 3.9.1. A factorizable system is consistent if and only if r(z)
satisfies the equation
C.52)
[rij(zi -Zj),rjk(Zj -zk)] + [rik(zi -zk),rjk(zj -zk)] + [ry(zt-Zj),rik(zi-zk)] = 0.
It is natural to take the modules Vi to be isomorphic to ?/@) with 0 acting
by left multiplication. Having found a factorizable system for Vi = U(g), we can
specialize it to any Vi.
If Vi = U(q), then equation C.52) reduces to a relation in the algebra (W@))®3.
This relation is the same for any triple i, j, k:
C.53)
))] [ (z1-Z2),rl3(zi-Z3)] = 0.
It is called the classical Yang-Baxter equation. Solutions of C.53) are called classical
r-matrices.
Observe that the classical Yang-Baxter equation is not differential but algebraic:
for factorizable systems, first derivatives in the consistency relations C.50) cancel.
This equation is a nonlinear matrix equation and is therefore fairly complicated.
Nevertheless, in 1982 A. Belavin and V. Drinfeld [BD] managed to classify all
solutions r(z) of the classical Yang-Baxter equation meromorphic in a neighborhood
U of the origin and possessing an additional nondegeneracy property: for at least
one point z 6 ?/', r(z) considered as a bilinear form on g* is nondegenerate.
Definition 3.9.2. Classical r-matrices r(z) and f(z) are called equivalent if
there exists an Aut g-valued function <f>(x) of one complex variable, meromorphic
near the origin, such that
- 22) =
- z2)).
Theorem 3.9.3. Let r(z) be a nondegenerate solution of the classical Yang-
Baxter equation C.53). Then r(z) extends to a meromorphic function on the entire

LECTURE 3. KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
complex plane and satisfies the unitarity condition ri2(—z) = —r2\(z). Moreover,
r{z) belongs to one of the following three classes:
1. rational r-matrices, i.e. those equivalent to an r-matrix which is a rational
function of z;
2. trigonometric r-matrices, i.e. those equivalent to an r-matrix of the form
f(eaz), where f is a $®Q-valued rational function;
3. elliptic r-matrices, i.e. r-matrices whose matrix elements are elliptic func-
functions of z {they exist only for 0 = sin).
Rational solutions have the form
C.54)
r{z) = — + P{z), P?0®0®C[z], c?C*.
Trigonometric solutions have the form
_ P(eaz)
C.55) -"¦'--
r(z) =
e-anzQ(eaz),
ea(n+\)z _
P,Q ?0®0®C[z],degP<n, deg(Q) < 2n, n ? N, PA) = tiCl, c? Cx
and satisfy the equation r(z + ^p) = r(z).
Elliptic solutions exist only for g = sin, and can be explicitly written as follows.
Let A = Ad A, B = Ad B ? Autg be the inner automorphisms of sin with A,B&
GLn given by
A(xi,X2,.- .,Xn) = (x2,- ..,Xn,Xi),
B(xi,x2,...,xn) = (x1,ex2,...,eTl~lxn),
Define the components fim>p e 0 ® g by fi = Sm7^=o nm,P, and (A ® l)nm,p =
em, (B ® l)nmjP = ep. Then the elliptic solutions in a suitable basis are given by
C.56)
r»,p(*)>
where 9rn:P(z) is the meromorphic function on C wit/i the following properties: it
has first order poles at z ? Zuji + Zo>2 <wmJ no other singularities, Reso6mp = 1,
and 9m,p(z + Wl) = ?m6m,p(z), ^(^ + ^2) = epem,p(z).
Elliptic solutions satisfy the equations r(z + nui1) =
first order poles at z ? Z Z
have
The proof of this theorem was obtained by combining the methods of complex
algebraic geometry and the structural theory of simple Lie algebras. The proof is
rather technical, and we do not discuss it here.
In describing rational and trigonometric solutions, we did not specify what
the polynomials P and Q may look like. For trigonometric solutions all possible
polynomials P and Q were completely classified in [BD]; we do not consider this
classification here. Rational solutions were completely classified by Stolin ([Stl],
[St2]), who, in particular, proved that P is a polynomial of degree < 1.
We are already familiar with two examples of nondegenerate classical r-matrices
arising from the KZ equations: r(z) = Ci/z and r(z) = n "'—P • N°w we can see
that the first of these solutions is rational and the second one is trigonometric.
Moreover, each of the two is the simplest representative of its class. This explains
3.9. CLASSICAL YANG-BAXTER EQUATION 47
to a certain degree the unique nature of the KZ equations and their special place
among consistent linear systems.
One may also consider "elliptic KZ equations", i.e. equations C.49) with Ai
given by C.51), and r being the elliptic r-matrix C.57). These equations, as well as
their rational and trigonometric counterparts, are also related to the representation
theory of affine Lie algebras. This relationship and other properties of the elliptic
KZ equations are discussed in [E].

Lecture 4. Solutions of the
Knizhnik-Zamolodchikov Equations
In this lecture we give explicit constructions of solutions of the KZ equations
in terms of certain integrals of hypergeometric type. Here we do not use the rep-
representation theory of affine Lie algebras. The relation of this approach with the
correlation functions will be discussed in the following lectures.
The first examples of explicit solutions of the KZ equations were obtained in the
physical literature [KZ], [FZ], [CF]. The general formula for 0 = sfe was written
down by V. V. Schechtman and A. V. Varchenko [SV1] and by E. Date, M. Jimbo,
A. Matsuo, and T. Miwa [DJMM]. In full generality the solutions were obtained
by Schechtman and Varchenko in [SV2]; see also [Matl] for the case 0 = sin and
[Chi] for another proof and further generalization.
4.1. The simplest solution of the KZ equations for 0 = sl2.
Prom now on, we use the following form of the KZ equations:
D.1) ,JU =
where iji takes values in the space
D.2) V=Vtil®...®VflN,
and V^ is the lowest-weight Verma module over g with lowest weight — im.
The relation of D.1), D.2) with the notation of Lecture 3 is given by \N = 0
and
D.3) K = k + hv.
In this lecture we will only consider solutions with values in the finite-dimen-
finite-dimensional spaces
N
D.4) W=(Vn~)\ A = -X> + /^ ^Q+
If n = ]nniCti, then the number \fi\ = ]T)n' will be called the level of the
corresponding solution (this has nothing to do with the level k of the modules over
Note that if we are in either the generic or the finite-dimensional case as denned
in Theorem 3.7.1 (this, in particular, implies that re ? Q), then it follows from
complete reducibility (Lemma 3.7.2) that any solution can be obtained by applying

50
LECTURE 4. SOLUTIONS OF KZ EQUATIONS
U(n+) to solutions with values in the subspace of singular vectors Vn , and thus
every solution can be easily reconstructed from solutions with values in the spaces
W above.
Let us consider the case 0 = s\2. Recall that sl2 is spanned by the elements
0 1
0 0
/ =
h =
1 0
0 -1
so that [h, e] = 2e, [h,f] = —2/, [e, /] = h. The invariant inner product in sl2 is
given by (x, y) = Ti(xy). The tensor fi for s\2 has the form
D.5)
¦ i/»®/».
The Cartan subalgebra of sl2 is one-dimensional: t) = C • h, and thus can be
identified with C so that /it-+l. This also gives the identification FT —> C : a i—> 2,
where a is the only positive root. Thus, we consider weights as numbers; under
this identification, (/?, 7) = 0"f/2 and p = 1.
The dual Coxeter number for sl2 is equal to 2, and thus k = k + 2.
It is also easy to calculate the dimension of the space W. Let ?>/v(m) denote
the number of ordered partitions of m into N nonnegative integer summands:
Simple combinatorial arguments show that
D.7) Piv(m) = ( m ).
\ m /
Since the elements emi^i ® ... <8> emw «/v, where the w; are lowest-weight vectors
in V^j, form a basis in V, it follows that dimVA = p/v(m) if A = — X)Mj + 2m.
Combining this with the complete reducibility (Lemma 3.7.2), we get the following
result.
Proposition 4.1.1. Let W be the space of singular vectors of weight A =
,i + 2m in V {see D.4)). Then for generic m (see B.8))
N + m - 2'
m
Let us construct the solution corresponding to the simplest case m = 0 (level
zero solution). In this case W is one-dimensional: W = C ¦ v,v — •
where Vi is the lowest-weight vector in V^,. Thus,
D.8)
... © vjv =
Introduce the multivalued scalar function
D.9) M*u--->*n) = \
and consider the W-valued function
D.10)
4.2. LEVEL ONE SOLUTIONS AND HYPERGEOMETRIC FUNCTION
Then it follows from D.8) that *0 satisfies the KZ equations D.1). This solution is
quite simple. To get more interesting solutions, we must understand what happens
when rra > 0.
4.2. Simplest level one solution and Gauss hypergeometric function
Let us consider the case N = 3, i.e. solutions in a tensor product of 3 spaces.
First of all, in this case the KZ equations can be rewritten as an ordinary differential
equation.
Proposition 4.2.1. Any solution il>(zi, z2, z3) of KZ equations D.1) can be
written in the form
D.11) ^(z1,z2,z3) = (zi - 23)(««+«i3+nM)/K/
where f(z) satisfies the following differential equation:
D.12) ^/W
X —
z\ - z2
z\ - z3
Proof. Introduce the variables
¦ , y = zi - z3, t = .
- 23
Then in these variables the KZ equations take the form
D.13)
fa12 q23
ax \ x x — l
dy
)
Since Q12 + Hi3 + Cl23 commutes with fiy, this shows that the function
depends only on x and satisfies equation D.12).
?
Let us now consider the case 0 = sl2,/i = 2, i.e. level one solutions. Assume
that we are in the generic case as defined in Section 3.7. Then the dimension of the
space W (see D.4)) is equal to two. If we additionally assume that fit ^ 0, then
the following vectors form a basis in the space W:
D.14)
<g> v2 ® v3 -
v2 ® ev3.

52
LECTURE 4. SOLUTIONS OF KZ EQUATIONS
Theorem 4.2.2. Any solution f(z) of the equation D.12) can be written as
follows:
where F(z) is a solution of the Gauss hypergeometric equation
D.16) z(l- z)—-^ + [c- (a + b+l)z]~ abF =
with
D.17) a = h3/k, b - -fii/K, c = 1 - (/
Proof. Explicit calculation shows that the action of fi12, fi23 in the basis
W2 defined in D.14) is given by
D.18)
Let us define
- Z) ^ f{z).
Then 5 satisfies the differential equation K-^g(z) = ( ^ + ^ \g{z), where
0,
Therefore, if we write g{z) = ^1(
system of differential equations:
4F - F
dz
Id :
+ F2(z)w2, then Fit F2 satisfy the following
^±MF2,
z-1
Denoting differentiation for brevity by a prime, from the first equation we get
F2 = ~zF[, which reduces the second equation to
Simplifying this, we get the hypergeometric equation D.16).
?
4.3. INTEGRAL FORMULAS FOR LEVEL ONE SOLUTIONS. 53
In particular, we can take the function F to be the Gauss hypergeometric func-
function 2Fi(a,b,c;z), which by definition is the only solution of the hypergeometric
equation D.16) satisfying F@) = 1 (see, for example, [WW|). In the disk \z\ < 1
this function can be represented by the power series
D.19)
where (a)n = a(a+l)... (a+n—1). It follows from equation D.16) that 2.F1 (a, b, c; z)
can be analytically continued to a multivalued analytic function in C \ {0,1}.
Proposition 4.2.3. For generic values of parameters, the following functions
form a basis of solutions of the hypergeometric equation D.16) in the neighborhood
of the origin:
FW=2F1(a,b,c;z),
FB) = z^^F^a - c + 1,6 - c + 1,2 - c; z).
The proof of this proposition is straightforward.
4.3. Integral formulas for level one solutions.
In this section we consider level one solutions for 0 = 5B (that is, /j, = 2 and N
arbitrary).
Let us introduce the convenient notation z = (zj,..., z^) and define the mul-
multivalued function
D.20)
3=1
Let zi,...,zjr?Cbe distinct points on the complex plane. Let C be a closed
contour in the plane not containing any of these points and such that the function
ij}\, regarded as a function of t, has a continuous branch along C. At self-intersection
points of C this branch is allowed to take more than one value, depending upon the
part of the contour along which the self-intersection point has been approached.
For generic weights fa this is equivalent to the condition that the index of C with
respect to each of the points Z\,..., zn be equal to zero. An example of a contour
satisfying this requirement and still homotopically nontrivial in C \ {21,... ,zn}
is shown in Figure 4.1 (this contour is known as the Pochhammer loop). Such
contours and their multidimensional generalizations will play a central role in the
construction of solutions of the KZ equations, and we will later classify them, but
now we need just one contour of this kind.
Figure 4.1. The Pochhammer loop

54 LECTURE 4. SOLUTIONS OF KZ EQUATIONS
As before, let v = vi ® ... ® v^, where Vi is the lowest-weight vector in Vm.
Let
i
D.21) *c(z)=Vo(z))
where tpo and t/ii are defined by D.9), D.20) respectively and as before, er denotes
the action of e € sb on the r-th component of the tensor product. We assume that
a continuous branch of ipi on C has been chosen so that the integral D.21) is well
defined. The function \f c, UP to a factor, does not depend on this choice.
Proposition 4.3.1. \fc is a W-valued solution of the KZ equations D.1).
Proof. This theorem is proved by direct calculation, which we organize in
such a way that it will be useful in the future (compare with [R]).
Introduce the following multivalued operator functions:
z,t).
Z — Zi
D.22)
Then one has the following lemma, the proof of which is straightforward.
Lemma 4.3.2.
D.23) K^Y-
We can rewrite the integral D.21) in the following form:
*c(z)=^o(z) f Y(x,t)-vdt.
Jc
Let us first prove that \fc(z) is a solution of the KZ equations. Since
solution, it suffices to prove that
is a
Due to Lemma 4.3.2 and the fact that (hi + njv = 0, this reduces to
f d
which is obvious. Thus, \fc is a solution of the KZ equations.
To complete the proof, we also need to verify that \&c ? W, that is, that
/(*c) = 0. Since fv = 0 and
4.3. INTEGRAL FORMULAS FOR LEVEL ONE SOLUTIONS.
we have
Jc
Jc
Z)V f K—iP1(z,t)dt = O.
Jc at
zi
a
Example 4.3.3: Integral formulas for the hypergeometric func-
function. Let N = 3 and put zx = 0, z2 = z, z3 = 1. Then the integral formula D.21)
gives
D.24)
v1
t — z
t — 1
On the other hand, Proposition 4.2.1 and Theorem 4.2.2 imply that \fc@,z, 1)
can be written in terms of solutions of the hypergeometric equation D.16). Com-
Comparing D.24) with D.15), we see that the function
Fc(z) =
f
Jc
f K-(t - z)-/"(l - i)-«/« —
Jc *
is a solution of the hypergeometric equation D.16). Making the change of variables
t = xz, we see that up to a constant (which arises from the ambiguity in the choice
of a continuous branch), the previous formula becomes
Fc(z) = f x-1-
Jc
dx.
Let us assume that z~l ^ [0,1] and choose C to be the Pochhammer loop going
around the points 0 and 1 (see Figure 4.1). Then it can be deformed to a 4-fold
cover of the interval [0,1], as shown on Figure 4.2.
Figure 4.2. Deforming the Pochhammer loop into an interval
Comparing the values of a single-valued branch of i~1~Ml/K(l — x)~ll'llK on
these four intervals, we see that for Re(/ij/«) < 0,Re(//2/re) < 1 we have
f
Jc
/K dx
/•I
) I x-1
Jo
-^!*- dx

56
LECTURE 4. SOLUTIONS OF KZ EQUATIONS
Thus, for Re(/ii/re) < 0,Re(/i2/«) < 1 and nt/is $ Z we see that the function
F{z)= [ z-1-"l/":(l-z)-'"/'t(l- zxY^'^dx
Jo
is a solution of the hypergeometric equation. It is easy to see that the restriction
Hi/is ? Z can be eliminated.
This solution is regular at z = 0, and therefore must be a multiple of the Gauss
hypergeometric function. Calculating F@), we finally recover the following classical
result, due to Euler.
Proposition 4.3.4. For Re c> Re 6 > 0,
4.4. Solutions of the Knizhnik-Zamolodchikov
equations for s^: arbitrary level.
Now we will study the structure of solutions of the KZ equations for sb for
arbitrary /j,. As before, we call the number m = ^ ? Z+ the level of the solution.
We will use the shortened notation z = (zi,... ,Ziv),t = (t\,... ,tm), and dt =
dti A--- Adtm.
For a fixed z, define the following operator-valued differential form in YI<m —
D.25)
^z,m = \[(tP- tn)"Y(z,t!). ¦ ¦ Y(x,tm)dt,
where Y is defined by D.22) (cf. [R.]). This is a regular multivalued differential
form in Yx<m.
Let us assume that we know how to integrate such forms. Namely, let us assume
that we have a certain "integral" Jc, which is defined on all multivalued forms of
the form
/K'-(rational function of t with no poles in YXjm)dt
P<n
pj
and such that the rule of integration by parts holds. For the case m = 1, an
example of such an integral is given by the integral over the Pochhammer loop (see
the previous section). We will discuss how to construct such integrals in general in
Lecture 7.
We can now generalize Proposition 4.3.1 as follows. Define the V-valued func-
function
D.26)
Jc
Theorem 4.4.1. The function \fc is a W-valued solution of the KZ equations
D.1).
Before proving Theorem 4.4.1, let us present formula D.26) in its original form
in which it appears in [DJMM], [SV1].
4.4. SOLUTIONS OF THE KZ EQUATIONS FOR sl2: ARBITRARY LEVEL 57
Define the function
D.27) if>m(z,t) = JJ(Zi — Zj) '* TT(*p — zj)~~" TT(*P ~ O*'
•<J . PJ P<"
Let m = (mi,... ,77i;v) be a vector of nonnegative integers such that J2mj —
rra. Define the rational function
m
D.28) pm(z,t)= Y Ylitn-z,^)-1,
s6Em n=l
where Sm is the set of all maps s : {1,..., m} —» {1,..., N} such that the number
of elements in s^) equals ttJj for all 1 < j < N.
Then
D.29)
It is easy to see that the right hand sides of D.26) and D.29) are identical.
However, the operator version D.26) is somewhat easier to prove.
The rest of this section is devoted to the proof of the theorem.
Proof of Theorem 4.4.1. Let us introduce the shortened notation
D-30) X(t)=II ('?-'»)"
p<n
and dt — dti A ¦ ¦ ¦ A dtm. We have the identity
dy ,-- 2
D.31)
Since \f0 = i>ov is a solution of the KZ equations, it is enough to show that
/ [
Jc
where
D.32)
dZi ^
Using D.23), we obtain
D.33)
r
/ [Vi,wa!,m]t;=
Jc $[
m .
= ?/ x(t)Y(z,h)...
,.\(ri
xY(z,tm)vdt,

58 LECTURE 4. SOLUTIONS OF KZ EQUATIONS
where the superscript p signifies that the term marked by this superscript stands
instead of Y(z, tp) in the product Y(z, ti) ...(** *)(p>... Y(z, tm) (the number of
factors in this product is always m).
The expression D.33) naturally breaks in a sum of three terms, corresponding
to the three summands in the parentheses. Let us simplify each of them.
Integrating by parts and using the obvious fact that all TJ and Y commute, we
can reduce the first summand to
p=l
...Y(z,tm)vdt
)M...Y{z,tm)vdt
771 n c\
¦r » / OY
~f^Jc dtp '
D.34) = Y, J ~~Y(z,h)...Ti(z,tp)^...Y(z,trn)vdt
= 2 E j?*(t)(ll Yi(*<ti))M*,tp^>i(*>U)
(tp - Zi)(tp - tn) *-? tn - Zj
To simplify the second and third summands, we use the following relations to
move the operators H(z,tp) (respectively, hi + m) to the right, where they kill the
vector v:
D.35)
This gives
D.36)
-2 ^
,p:n>p
% -vdt,
tp-Zi^itn-ZjHtp-Zj
D.37)
I
=2
4.4. SOLUTIONS OF THE KZ EQUATIONS FOR sl2: ARBITRARY LEVEL 59
Adding up D.34), D.36), D.37) we see that it suffices to prove the following
identity:
D.38)
.5L(*p-*)(<p-*»)
-^ (*P-
7 '
which easily follows from the identity
1 1
D.39)
(a-6)F-c) (a-6)(a-c) (a-c)F-c)
= 0.
It remains to check that \fc G W, i.e. that /*c = 0. Since the dependence on
z is now unimportant, we will drop the argument z.
We start with the obvious identities
D.40)
dt
Using these identities, we deduce that
D.41)
]« = - X(t)
(p)
... Y{tm)vdt
p=ln>p
^))'"' • ¦ • Y{tm)vdt
p=i
Integrating D.41) over C and using the rule of integration by parts, we get
p=l
p=l
Substituting in this formula the expression D.31) for derivatives of \, we find
that all terms cancel, which shows that *cef. ?

60 LECTURE 4. SOLUTIONS OF KZ EQUATIONS
4.5. Solutions of the Knizhnik-Zamolodchikov
equations for a general simple Lie algebra
Now let us describe (without proof) the structure of solutions of the KZ equa-
equations for a general simple Lie algebra 0 of rank r.
As before, we are looking for solutions with values in the space
= (Vn~)\ \ = -
We will call the vector n = Yl kai the multilevel and the number m = \/i\ =
U the level of the solution. Also, let
and define v : {1,..., m} —» {1,..., r} by
D.42) i/(p) = sifps_i <p<p3-
Define the function
D,43)
Let us now describe the proper analogue of pm(z, t). As for 5I2, let m =
(mi,... ,m,ff) be a vector of nonnegative integers. Let Mi = 5ZJ=iTOj- Let
L : {1,..., m} —» {1,..., r} be an arbitrary map such that the set L^^s) has lB ele-
elements for 1 < s < r. Let St be the set of all bijections 6 : {1,..., m} —> {1,..., m}
such that v(b(p)) = i(p), 1 < p < m.
Define the rational function /9m,i(z,t) by
D.44)
As in Section 4.4, assume that we have an integral Jc defined on the multival-
multivalued differential forms oj = Vv(z> t) (rational form) and such that the integral of an
exact form is equal to zero; again, we will show how to construct such integrals in
Lecture 7.
Let ei,...,er be the Chevalley generators of n+. For every m,ias above, let
(
Introduce the vector-valued function
D.45)
The following result was proved by Schechtman and Varchenko in [SV2].
SOLUTIONS FOR A GENERAL SIMPLE LIE ALGEBRA 61
Theorem 4.5.1. The function 4c»a W-valued solution of the KZ equations
D.1). Moreover, for generic A* and k, any solution can be written in the form
* = *c for some choice of the cycle of integration C.
We would like to remark that Knizhnik-Zamolodchikov equations are defined
not only for simple Lie algebras but also for all Lie algebras with an invariant
symmetric form, in particular, for Kac-Moody algebras. Formula D.45) directly
generalizes to the case where 0 is an arbitrary Kac-Moody algebra.
If 0 = sl2 then formula D.45) has to transform into D.26). To see how this
happens, it is enough to prove the following combinatorial identity with rational
functions:
D-46) ^/9m,L(z,t) = /9m(z,t).
L
To deduce this identity directly is an instructive exercise.

Lecture 5. Free Field Realization.
The solutions of the KZ equations that we have obtained in Lecture 4 have the
form of integrals of certain relatively simple functions, namely, products of powers
and rational functions. In the next few sections we will represent these functions
as matrix elements of products of vertex operators and generating functions for the
infinite-dimensional Heisenberg algebra ("bosonic fields"), respectively, acting in
Fock spaces. The vertex operators were first introduced in dual resonance models
(see [Man] for a review) and were used in a construction of representations of
affine Lie algebras [FK], [Segl]. Computation of matrix elements of products of
free bosonic fields is given by Wick's theorem [Wic], widely used in quantum field
theory.
Since we know that the solutions of the KZ equations are given by the ma-
matrix elements of the products of intertwining operators, defined in Lecture 3, this
naturally suggests a relation between vertex operators in Fock modules and in-
intertwining operators for affine Lie algebra. We will show that such a relation
does exist, by explicitly constructing for sb a so-called "free field realization", in
which highest-weight modules over 5B are identified with Fock modules, and in-
intertwining operators are written in terms of vertex operators and some auxiliary
operators (screening operators). This construction was first discovered by M. Waki-
moto [Wak] and was generalized to the case of an arbitrary simple Lie algebra by
B. L. Feigin and E. V. Frenkel [FF1], [FF2] (see also [GMMOS], [BF], [Ku]). For
0 = sin. there exist relatively simple explicit formulas for the free field realization,
but for other Lie algebras the formulas are quite complicated, which makes explicit
calculations very difficult. We refer the reader to the paper [ATY] for the deriva-
derivation of general formulas of Schechtman and Varchenko for the solutions of the KZ
equations for an arbitrary simple Lie algebra g from the free field realization.
5.1. Fock modules and vertex operators.
Let t) be a Cartan subalgebra of a finite-dimensional simple Lie algebra g.
Consider the Heisenberg algebra ft = t) <8> C^,*"] ffi Cc C g. Let A ? f)*, and let
H\ be the Fock module over ft. By definition, H\ is generated by a vector v\ such
that
E.1) h[n]v\ = 0, n > 0, h[0]vx = \{h)vx, cvx = vx,
where for h ? F) we denote h[n] = h®tn, and the universal enveloping algebra of
ft- = F)®(~1C[(~1] acts freely in H\. Clearly, the representation H\ is irreducible.
Since c acts in H\ by the identity, the commutation relation B.15) yields
E.2) [h[m

LECTURE 5. FREE FIELD REALIZATION.
The operators h(n) are called creation operators if n < 0, and annihilation
operators if n > 0. The vector v\ is called the vacuum vector.
If we regard H\ as a module over Sj^o = h ® t^Cft] © f) ® tC[t] © Cc, its
structure will no longer depend on A. For p, e f)*, denote by e*1 the isomorphism of
ij^o-modules H\ —» ifA+^ which maps vx to va+m. It is obvious that exe^ = ex+li.
It turns out that although the Heisenberg algebra is almost commutative, one
can produce operators with very nontrivial commutation relations by merely com-
combining creation and annihilation operators. One of the most important examples
of this sort of construction is the vertex operator
E.3)
X(n, z) = exp
acting from H\ to the completion H\+I of the module H\+I, This operator is well
defined once a continuous branch of the function zh" has been chosen.
Observe that the situation with multiplication of vertex operators is the same
as it was with intertwiners: it is not clear how to multiply two of them, since the
second operator is not defined on the image of the first one. The best thing we can
do formally is define the normal ordered product : X(ni,zi)X(fj,2,Z2) :, which is
defined as follows.
Definition 5.1.1.
: hi[n]h2[m] :
hi\n]h2[m], m > 0,
/»2[wi]fti[n], m < 0;
= : e"h[0] : = e"h[0].
This definition can be extended to monomials in h[n\ and z'*'0' in the obvious
way, by reordering the factors so that the creation operators stand to the left of
the annihilation operators, and terms e^ stand to the left of zh^. In particular, we
can define normal ordered products of vertex operators. One easily sees that such
normal ordered products are well-defined as operators Hx —> H^.
It turns out that the usual product also makes sense for certain values of z\, z%-
Indeed, let us formally expand the product X{p,\, zi)X(fj,2, z2) and l°°k at its ho-
homogeneous components. These components will be series in z\,z2 with operator
coefficients. If we find that for B1,-22) belonging to some region in C2 these se-
series converge to analytic functions, we can make sense of X(fii,zi)X(fj,2, z2) as an
operator Hx -> Hx+I11+I12.
Proposition 5.1.2.
well defined operator H\
If \zi\
Z2I, then the product X(m, zi)X((i2, zi) is a
l2, and it is given by
E.4)
Proof. To pass from X(pi, zi)X(/i2, z^) to : X([j,i,zi)X([j,2,Z2) :, we need to
permute the terms exp(M1t') and exp(M2~), where
E.5)
n>0
5.1. FOCK MODULES AND VERTEX OPERATORS.
65
We have
m,n>0
n>0
n>0
Since Mf and M2 commute with [Mf,M2 ], it follows that
E.7) eMi+eM*~ = eMi'eM"'e[M""'M»"l = eM^eMi+ f 1 - —) ""
We also need to permute the operators zih^ [0] and e^2. We have
E.8) Zj eM2 = eti2z1 x^'111'.
Summarizing our calculations, we find that normal ordering causes the product
X([i1,z1)X(n2,Z2) to multiply by the scalar factor A - |i)</«./«>z<'"''1'>
= B1 - 22)<M1'M2>, which is equivalent to E.4). ?
Corollary 5.1.3. // |zi| > ¦ ¦ • > \zN\ > 0, then
E.9) X(Mi, zi). ..X(hn,zn) = H(zi - zj)^'-^
, zN) : .
The following properties of the vertex operators, which are proved by direct
calculation similar to the one above, will also be useful later.
Proposition 5.1.4.
1.
E.10)
2.
E.11)
={\,(i)znX(n,z).
z-n~K Then
hx(zi)X(fi,z2) =: hx(Zl)X(fi,z2)
, z2),
The important role of vertex operators in representation theory is demonstrated
by the vertex operator construction of the basic representation of an affine Lie
algebra, obtained in [FK] and [Segl]. The basic representation is, by definition,
the irreducible representation iu 1 of g. This representation is integrable. Let us
assume that g simply-laced. Using the Weyl-Kac character formula, it is easy to
show that as an ^-module, this representation is naturally isomorphic to the infinite
direct sum of Fock spaces: io,i = ®peQ ^B (where Q is the root lattice of g), so
that Vfj 6 ?0,1 [— 2 ]• Therefore, it is natural to expect that the action of g on
this space can be expressed in terms of the operators h[n] and e'3,/? e Q. It can
be proved that this is indeed so, and that such a representation is given by vertex
operators.

LECTURE 5. FREE FIELD REALIZATION.
Theorem 5.1.5. Let g be a simply-laced Lie algebra: for every root a 6 R we
have (a, a) = 2. Let us choose a bimultiplicative function ?:(JxQ-t {±1}, where
Q is the root lattice of g, such that ?(a,P)e{P,a) = (-l)<a>/3> for a,/? € Q. Then
one can identify Lo,i = ©^gg Hp so that the action of g on ?0,i is expressed by
the following formulas:
hef),
E.12)
where Jx(z) =
nSZ
Jea(z)=X(a,z)Ca,
Jfa(z)=X(-a,z)C-a,
and
= e(a,/3).
This theorem will not be used in this book. In our setting vertex operators
will appear in the explicit formulas for the intertwining operators C.2) in a certain
realization of the highest-weight modules.
5.2. Matrix elements of products of vertex operators.
As before, let us assume that we have fixed a number k 6 Cx. For convenience
introduce the notation
E.13)
Let «*, be the linear function on
is a homogeneous vector of a nonzero degree.
Proposition 5.2.1.
E.14)
where fj, =
such that (*>*,, ?;M/) = 1 and (v^,,v) = 0 if v
and the function ipo is defined by D.9).
PROOF. It is easy to check that
E.15) : X(m'i,*i) ¦ ..X(n'N,zN) : v0 = v.
This formula combined with E.9) yields E.14) ?
Corollary 5.2.2. The function ip^(z,t) defined by D.43) has the following
presentation in terms of vertex operators:
E.16)
where A = —
+ (i.
5-3- Pm,L(*> *¦) IN TERMS OP 0, 7 OPERATORS 67
5.3. Interpretation of the rational part of solutions of the
KZ equations in terms of creation and annihilation operators.
We have interpreted the function ^asa matrix element of a product of vertex
operators constructed from the generators of the Heisenberg algebra. We want to
obtain a similar realization of the rational function pm,L.
We will only consider the case g = s^. In principle, similar constructions exist
for arbitrary g, but they are more complicated.
Consider the new Heisenberg algebra B with the basis /3n,jn,c, n e Z, and
relations
E.17)
[/?n,7m] = 6n,—mC,
,, An] = [jn,7m] =0, [c, /?„] = [C, 7n] = Q.
In physical literature, this Heisenberg algebra is sometimes called the Pj systen
We introduce the currents
E.18) p(z)=y^0nz-n-\ -Yf^ = V.v-z-"
nSZ
nSZ
Define the normal ordered product
E.19)
a a / /?n7m, n < 0,
I 7m/?u, n > 0.
If two elements commute with each other, their normal ordered product equals
their usual product. The normal ordered product of several factors ft and jj is
defined similarly: the factors must be regrouped in such a way that the factors with
positive subscripts stand to the right of the factors with negative subscripts, and
/?o to the right of 70 Clearly, there are many reorderings satisfying this condition,
but all of them give the same result.
We define the Fock module H(B) to be the Z-graded module over B generated
by a vacuum vector v (of degree 0) and with the relations
E.20)
cv = v, -ynv = 0, n > 0, /3nv = 0, n > 0.
For zGCx, the currents f}{z) and 7B) are well defined operators H(B) —>
H(B), where H(B) is the completion of H{B) with respect to the grading. One
can show that if \z\\ > \z2\, then the products 0(zi)y(z2) and j(zi)p(z2) axe well
defined operators H(B) —> H{B). Also, the normal ordered product of currents
: /?(ziO(z2) : is defined for any values of z\ and z2.
We will need the following fact about the normal ordered product.
Proposition 5.3.1. lf\z^\ > \z2\, then
E.21)
Proof. The difference between the two products is only in terms containing
l-n- Therefore,
u>0
a

LECTURE 5. FREE FIELD REALIZATION.
We will also need to recall the Wick theorem for the Heisenberg algebra.
Let Hbe a. vector space and w be an antisymmetric bilinear form on H. Endow
the vector space H = H © Cc with the structure of a Lie algebra:
E.22)
[hi + \\c,h2
€ H.
This Lie algebra is called a Heisenberg algebra; this generalizes the Lie algebra S)
considered before.
Let H = H+®H°®H~<$Cc, where H* are isotropic subspaces in H and H° is
the kernel of w. Such a decomposition is called a polarization. Once a polarization
is fixed, we can construct an if-module T generated by a vector u (which is called
the vacuum vector) and defined by the relations
E.23)
h+eH+, h°eH°.
This module is called the Fock module.. Obviously, T = Cu © H+T.
Denote by u* the linear function on T such that (u*,u) = 1 and u* vanishes
on H+T.
Let P(N) be the collection of all partitions of the set {1, -~,N} into N/2 un-
unordered pairs of numbers (for odd N, we let P(N) = 0). Every such partition p
can be expressed (not uniquely) by two sequences pi(n) and P2(n), so that pi(n) is
the smaller number in the n-th pair and P2(n) is the greater one.
Theorem 5.3.2. (Wick) Let hi,...,hN e H. Then
E.24)
N/2 N/2
{u*,hih2...hNu) = ^2 n<"*'/lPi(">/lp2(">") = 5T II
l p€P(N)n=l
In particular, if N is odd, («*, hih2... hNu) = 0.
Now we can express the rational function pm(z,t) defined by D.28) in terms
of the currents /?(z) and 7B).
Proposition 5.3.3.
E.25)
mi!.
Proof. By Wick's theorem,
E.26)
(«*,/3(tn)y(zs{n})u).
E.27)
We use Proposition 5.3.1 to compute the factors:
1 , .
7
U Z
1
Substituting E.27) into E.26), we obtain E.25).
D
5.5. FREE FIELD REALIZATION OF VERMA MODULES OVER s[2. 69
5.4. Factorization of solutions of the
Knizhnik-Zamolodchikov equations.
Now we can combine the presentations of the functions ^ and pmL deduced
in the two previous sections and obtain a factorization of integrands of solutions of
the KZ euations.
Define the operators
E.28) U(t)=X(-a',t)®l3{i), Zm{\,z) = X{\',z) ®-y(z)m,
where a is the positive root of sfe, and as before, X = \/^/k. These operators
act from the space H^ ® H(B) to the completion -ff^-a' ® H(B) (respectively, to
Then formula D.29) can be rewritten in the form
E.29) *c(z) = Y] [ Ko ® u',U{h)... U{tm)
¦ Zmi(in, 21)... ZTnN(nN,zN)v0 ® u)dt
where Ao = X] Mr ~ 2m, and the sum is taken over all m = (mi,..., wi/v) 6 Z;Jf
such that Yl mi — m-
In the next sections we will see how this factorization naturally arises from the
free field realization of Verma modules and intertwining operators for s^.
5.5. Free field realization of Verma modules over s^.
Comparing formula E.29) with the results of Section 3, where we proved that
the correlation functions obtained as matrix elements of intertwining operators for
B are solutions of the KZ equations, it is natural to ask if it is possible to write not
only the correlation functions but the intertwining operators themselves in terms
of vertex operators in Fock modules. In the next sections, we will show how this
can be done for g = slj-
We will use two Heisenberg algebras, A and B. The algebra A is spanned by
elements an, n 6 Z, and a central element c, with the relations
E.30)
[an,am] = 2n<5n,_mc, [c,an] = 0.
This is nothing but a special case of the Heisenberg algebra Sj defined in Sec-
Section 5.1 for g = SI2: an = ha[n], where a is the positive root of st2. As usual for sl2,
we identify I)* ~ C : a >-> 2. Thus, for n 6 C we will write X(n,z) = X(na/2,z).
The algebra B was defined in Section 5.3.
We will work with the Fock spaces Hx(A) and H(B). The module H\(A) is
generated over A by the vacuum vector v satisfying the relations
E.31)
cu = v, anv = 0, n > 0,
= Av, A 6 C.
The module H(B) over B was defined in Section 5.3.
It is clear that the space fix = H\{A) ® H(B) is naturally a representation of
A © B. It is easy to see that this representation is irreducible.

70 LECTURE 5. FREE FIELD REALIZATION.
Recall the definition E.18) of currents f3{z),i{z) and introduce the current
In on\ / N \ ^ —n—1
n€Z
Consider the following operators HX(A) ® H{B) -> H\(A) ® H{B):
E.33)
Je(z) =/3(z),
Jh(«) = - 2 :
where k = k + 2 and the normal ordered products are defined as in Definition 5.1.1
and E.19).
The following construction, called the free field realization, was found by Waki-
moto ([Wak]) and later studied by many authors. It is also known under the names
"Wakimoto realization" and "bosonization".
THEOREM 5.5.1. Formulas E.33) define an action ofsl2 on the spaceH\{A)®
H(B). If \,k are such that the Verma module M^^k is irreducible, then this
representation is isomorphic to M^^k.
Remark 5.5.2. This realization of sh resembles the well known realization of
by differential operators in one variable:
E.34)
e = j3, h= -27/3 + j, f = -72/3 + 3i,
where 8 = —, 7 = z.
dz
These operators act on the space C[z] and make this space into a s^-module. This
module is a contragredient Verma module, i.e. it is the graded dual to the lowest-
weight Verma module with lowest weight —j. For generic values of 3, this module
is isomorphic to the Verma module with highest weight j, though in general this is
not so.
In fact, this is more than an analogy: there is a general construction, using
the action of a Lie algebra on the large cell of the flag variety, which gives formula
E.34) when applied to s^. It was shown in [FF1, FF2] that this construction can
also be applied (with suitable changes) to affine Lie algebras, and for 5^ it gives
precisely formula E.33).
Remark 5.5.3. For nongeneric ^/k\ the module H\(A)®H{B), which is called
the Wakimoto module, is not necessarily a Verma module. Rather, it is something
"in between" a Verma module and a contragredient Verma module. For example,
if A = 0 then it is easy to deduce from E.33) that fvo = 0, where «o is the
highest-weight vector in Ho(A) ® H(B).
The rest of this section is devoted to the proof of Theorem 5.5.1. The proof is
more or less straightforward, though it requires some technique.
Proof of Theorem 5.5.1. Our goal is to find commutation relations between
the new currents Je,Jh,Jf. The first step is to find their commutation relations
5.5. FREE FIELD REALIZATION OF VERMA MODULES OVER S[2.
with a, /?, 7, which is straightforward:
[Je(z),an] = 0,
[JreW,/3B]=0,
[Jh(z),an] =
E.35)
[Jf(z),f3n] = 2zn :
Let J" be the coefficient of z~n~l in the power series Jx(z), for x = e,h,f.
Then relations E.35) can be rewritten in the form
E.36)
[J],E{z)\ = 2*n :
: -JHzna(z) + knzn~\
The next step is to calculate the commutation relations of J" with a product
of two of the currents a, C,7 and their normally ordered products. Relations E.36)
imply
E.37)
kntn~li(z).
To deal with the normally ordered products, we need the following identities,
which are proved in precisely the same way as E.21):
E.38)
>
7(^2)
t — Z2 t — Z\ '

72 LECTURE 5. FREE FIELD REALIZATION.
These identities imply
E.39)
: = : 7(zh(t)C(t) :
t-z
Taking the limit as t —> z in E.37) and using E.38) and E.39), we obtain
E.40)
(k + 2)nzn-1-y(z) + 2zn-
dz
Now we could repeat the same arguments and find commutation relations of J™
with products of the form : 72(z)/?(z) :, and therefore, with Jx(z). However, it is
easier to do the following trick.
Introduce the operators
/mn r jm rn]
ee ~We i Je i'
E.41)
Tmn r rm rnl i o rm-f-n
1th —We >Jhl + ZJe
jmn r Jm Tn] jm+n
Jef —We >Jf\ — Jh '
/mn I jm rn] , o jm+n
hf =Wh , Jf\ + lJf
To prove that E.33) is a representation of st2, it suffices to prove that all of the
operators E.41) vanish except J^y~n which should be equal to kn{x,y).
Lemma 5.5.4. The operators E.41) commute with the currents a(z),p(z),j(z).
Proof. The lemma is proved directly, using relations E.36) and E.40). For
example (this is one of the most difficult cases),
(*)], J}\ + [Jf, [J?,P(z)]]
=[2zm : 7(z)/3(z) : —s/iizma(z), J"] - (same thing with m «-> n)
=2zm(-zn : J2(z)/3(z) :
— (same thing with m
2Kzm+n-17(.z)(m - n)
0.
re)
(n - m)
D
5.6. INTERTWINERS IN FREE FIELD REALIZATION: LEVEL 0 73
Corollary 5.5.5. The operators E.41) act by scalars inH\.
Thus, the assignment x[n] >-> J?, x = e,h, /, defines a projective action of Ls[2
in 1i.\. Since it is known that the only nontrivial central extension of L5I2 is st2,
this shows that
/™n = Km(x, y)<5m,_n, x, y = e, h, f,
where if is a constant (the central charge). Let us calculate K.
We have
[Jh, M*)\ = [Jh, -2 : 7(*)/»(*) 0 + W, yfa*(*)\
= - Anz"'1 + 2{y/KJnzn-1 = 2knzn~l.
Therefore,
By virtue of Schur's lemma, we have the following result.
which shows that K = k.
It remains to show that H.\ is isomorphic to M^x,k as an s^-module for generic
V^A, k. This is demonstrated by the following properties of H\:
1. ua ® u is a singular vector, i.e. J"(u ® u) = 0 if re > 0 and J°v\ ® u = 0.
2. J°(«a ® w) = y/H\v\ ® w.
3. The dimensions of the homogeneous subspaces of fix and M^xk are the
same in all degrees.
These properties show that the natural homomorphism M^x k —> H\ has to
be an isomorphism when Mr^xk is irreducible, which happens for generic ^/kA, k.
The theorem is proved. ?
The proof could be significantly simplified if we used the technique of operator
product expansion (OPE); however, it would take us more time to explain this
technique.
5.6. Intertwining operators in the free field realization: level zero.
The purpose of this section and the next section is to construct the intertwining
operators explicitly using the free field realization.
Recall that in the case of g = sfe and generic A, A + /x, k for every integer m > 0
there exists a unique (up to a scalar) intertwining operator
where, as before, V^ is a lowest-weight Verma module with lowest weight —ft. This
operator will be called the intertwiner of level m.
If we use the free field realization of Verma modules given by Theorem 5.5.1,
then the intertwiner <f>m(z) becomes a map fix' —> /Hx'+)x'-2m' ® ^(z)- (Recall
that A' = \/t/k.) Thus, it is natural to expect that it admits an expression in
terms of the currents a(z), fi(z), 7(z). This turns out to be the case if we also add
the vertex operators X(fj,, z) defined in Section 5.1.
We start with the operator of level zero.
Abusing notation, we will often write X(n, z) instead of X(fi, z) ® 1, and simi-
similarly for a, /?, 7.

LECTURE 5. FREE FIELD REALIZATION.
Lemma 5.6.1.
E.43) #°(z)to = X(fi',z) exp(-7(z) ® e)(w ® «„),
where w € fix1 and v^ is the lowest-weight vector ofV^z).
Proof. Denote for brevity
E.44) E(z) = exp(-7(z) ® e).
The intertwining property of E.43) is equivalent to
E.45) [X(ii',z)E(z), J?®l]w®vll = zn(l ® x)X(fi', z)E(z)w ® v^
for all x es[2,TO eHx>-
It is seen from formulas E.36) that [J",-y(z)] commutes with -jj, and thus
E.46)
,y(z)\ ®e)E(z).
Let us consider the cases x = e,x = f.
For x = e, we have [J",7(z)] = z" an^ lx(v',z), J"] = °; tnus
[AV, z)E{z),J?] = (z» ® e)X(M', «)?(*),
which implies E.45).
For x = f, note first that it follows from Lemma 5.1.4 that
A short calculation shows that this implies
E.47) [JJ,X(n',z)]=n.
Recalling E.36) and E.46), we can now calculate the left hand side of formula
E.45):
E.48)
To calculate the right hand side of E.45), note that [1®/, --y{z)®e\ = -y(z)®h,
and it follows from the Campbell-Hausdorff formula that
= (A(//, z){-zn12(z) ® e) - iiznX(v', z)-y(z)) E{z)
= -zn~f(z)(j(z) ® e + n)X(fi', z)E(z)w ® v^.
Thus,
E.49)
/, E{z)\ =
', z)E(z)w ® uM = z"X(/i',
z) ® h - 72(z) ® e).
-/i7(z) - 72(z) ® e)w ® vM
z) ® e)X(n', z)E(z)w ® vM.
Comparing E.48) and E.49), we get the intertwining property E.45).
Thus, we have proved that the operator E.43) is an intertwining operator for
J", Xi. Since these operators generate s^, this shows that it is an sfc intertwining
operator. ?
5.7. INTERTWINERS IN FREE FIELD REALIZATION: POSITIVE LEVEL 75
5.7. Intertwining operators in the free field realization: positive level.
Now we will find <&m(.z) for m > 0. Recall the operators
E.50) U(t) = X(-a',t)p(t)
(as before, a' = q/^/k). They are called the screening operators.
Lemma 5.7.1.
Proof. The first identity is obvious. Let us prove the second one. The fastest
way to do it is to use the following trick.
Let us calculate the commutation relation of J" with P(t)X(a', z). Using rela-
relations E.36), E.47), we get
= Btn : -y(t)P(t) : -^tna(t) + kntn-l)X{-a',
Using Propositions 5.1.4, 5.3.1 we can express this in terms of normal ordered
products:
{-a', z))
[JJ,p(t)X(-a',z)} = Btn :
- v^t" (: a(t)X(-a', z) :
)
L — 2
= Bt" : /3(tO(i) : -2zn : 0(t)-y{z))X(-a',z) + kntn~lX{-a',:
- tn : a'{t)X(-a',z) : -t-2^" ~ ^X{-a!,z).
Letting z -»t and again using Lemma 5.1.4, we get
[J?,/3(z)X(-a',z)}= -zn :a'(z)X(-a',z) : +(k + 2)A(-a', z)^-zu
' dz
dz dz
dz
Remark 5.7.2. There is a seeming contradiction: this lemma shows that in-
integral Jc U(t) dt is an intertwining operator fix1 —> *H\i-a', though we know that
for generic A, k there are no such operators. The explanation is that there are no
cycles C along which one could integrate U(t).
Consider the operator-valued function
H : Tix- -> Hx'+s-2m' ® Vf,

76
defined by
E.51) S(z, h,..., tm)w = X(ji', z
LECTURE 5. FREE FIELD REALIZATION.
e)U(h)... U(tm)(w ® «„).
As we know, this product is well defined if |z| > |ti| > ¦ ¦ ¦ > \tm\ > 0. However,
it can be analytically continued into a larger region.
PROPOSITION 5.7.3. The operators can be represented in the form
m
E.52) S(z,t1,...,tm) = H(z-tj)-1-^KtJx/l'l[(ti-tjJ^Eo{z,tu...,tm),
J = l i<j
where So is a holomorphic operator-valued function in (C*)m+1.
Proof. Analogously to the normal ordering Proposition 5.3.1, we have
exp(-7(z) ® e)/3(t) = - j—^ exp(—y(z) ® e)+ : exp(-7(z) ® e)/3(t) :
E.53)
This identity implies
E.54)
m
)"
0(tm) -
tm-z
This along with Proposition 5.1.2 on the normal ordered product of vertex
operators means that the operator function H can be represented in the form E.52)
with Ho being a certain normal ordered product of vertex operators and currents
/?, 7. It is easy to see that the normal ordered product of this form is a well-defined
function for all values of z, U e C*. ?
As before, assume that we have an appropriate cycle C such that we can define
JC3' dt for any function E' of the form E.52). Let us also assume that for any S',
we have Jc -g-E' dt - 0, and that
E.55)
/ S(z, ti,..., tm)dti A • ¦ ¦ A dtm ? 0
Jc
(in Lecture 7 we will prove that such a cycle exists and is essentially unique up to
a factor). Then the intertwiner $m(z) is given by the following theorem.
THEOREM 5.7 A. Let E be the operator valued function E.51) and let the cycle
C be as above. Then
E.56)
4>m{z) = [ E(z, tu ..., tm) dh A • • • A dtm.
Jc
5.8. CALCULATION OF THE CORRELATION FUNCTIONS. 77
Proof. We need to show that E.56) satisfies the intertwining property. Again,
it suffices to check commutation relations with e[n] (trivial) and f[n). Using the
intertwining property E.45), which we used in the proof of the formula for level
zero intertwiners, and Lemma 5.7.1, we get
E.57)
l,2(z, tu ...,t
zn(l®fM(z,tu...,tm))wdt
... — (t^X(-ar, tv)) ... U(tm)(w ® Vy) dt.
It is obvious that the differential form on the right hand side of E.57) is exact.
Therefore, the integral of E.57) over C is zero, i.e.
( [J?® 1, / Sdt] + z"(l®/) / Hdt )to = 0,
\ Jc Jc J
which is the intertwining property for f[n\. The intertwining relation for h follows
from those for e and /. Thus we have proved that E.56) defines an intertwining
operator for sfe. D
5.8. Calculation of the correlation functions.
Now we are in a position to calculate the correlation functions. As in Lecture 3,
let A^ = 0 and fix Ao, •. •, Ajv-ijMii •. • ,/xjv such that A*_i = Ai + fii — 2mi,m.i 6
Z+,m,N = 0. Denote m = Y^,mi\ then Ao = Ylt^i — 2m. Let VMi be the lowest-
weight Verma modules with lowest weights — Hi. Recall also the notation V\tk for
the induced module (see B.28)).
Let us assume that fii,k satisfy the genericity conditions of Theorem 3.7.3.
Then for each i = 1,...,N there is a unique (up to a constant) intertwiner
E.58) Qite) : VXi,k - VXi_itk ® VMi,
(compare with E.42)).
Since Ao,..., Ajv-i are generic, we know that Va,,* = L\uk = M\itk, and thus,
due to the free field realization, V\uk — Hy (A) ® H(B). For Ajv = 0 this is not so;
however, as we already mentioned (see Remark 5.5.3), in this case Vo.fc = ?o,fc is a
submodule in H0(A) ® H(B).
Now, using the results of the previous section, we can write the intertwiner
$i(zi) m terms of the free field realization. This gives
E.59)
**(*)= f 3<(z1TJ,...,7i,i)dT{A-.-A<iTl
JC,
Note that #jv is a level zero intertwiner (mjv = 0) and thus involves no integration
(Cat = point).
Let C = Ci x •¦• x CW_i.
Consider the correlation function
E.60)
*m(z) = Ko,

LECTURE 5. FREE FIELD REALIZATION.
Using E.59) and renaming the variables t* as follows: r? = tM^i+j, where
Mi = m\ + ¦ ¦ ¦ + m,i (thus, we get variables tp,p = 1,..., m), we can write this
function in the form
E.61)
'N, zN) exp(—y{zN) ® eN)U(tMN_1+i) ¦ ¦ ¦ U{tm)v0)v dt,
where, as before, v — v^ ® -.. ® v^N is the product of lowest-weight vectors in V^.
Applying formulas E.9) and E.54), we can reduce E.61) to the normal ordered
form, after which the calculation becomes trivial:
E.62)
n (
l<p<Ti<m
where Q is the normal ordered product:
E.63)
Q=:X{v'1,z1)...X{lilN,zN)X{-a',tl)...X(-a',tm)
It follows from the definition of normal ordering that
and thus we get
E.64)
(tP-tn)iY(tuz)...Y(tm,z)vdt.
Note that the only dependence of the right hand side of E.64) on the partition
m is in the choice of the contour C = C\ x ¦ • • x Cjv-i-
Thus, we have obtained the same expression for the correlation function, and
thus, for a solution of KZ equations as in Section 4.4 (formula D.24)), using the
free field realization.
Lecture 6. Quantum Groups.
In this lecture we give a brief review of the theory of quantum groups. Quantum
groups first appeared as certain auxiliary algebraic structures in the "inverse scat-
scattering method" developed by L. D. Faddeev and his collaborators (see [FRTl] and
[FRT2] for historical remarks). Departing from the physical models, V. G. Drinfeld
[Drl] and M. Jimbo [Jl] independently discovered a g-deformation as a Hopf al-
algebra of the universal enveloping algebra of an arbitrary Kac-Moody algebra, thus
providing a vast class of examples of quantum groups.
The theory of Hopf algebras used to be a traditional branch of abstract al-
algebra. Examples of Hopf algebras coming from topology and group theory have
been known to mathematicians for a long time. However, most of these Hopf alge-
algebras were either commutative or cocommutative. The discovery of quantum groups
by Drinfeld and Jimbo following earlier works of Faddeev and collaborators pro-
provided fundamental examples of Hopf algebras which are neither commutative nor
cocommutative, bringing the theory of Hopf algebras to the forefront of modern
mathematics.
Drinfeld has also shown in [Drl] that quantum groups admit in addition a
so-called quasitriangular structure, or the universal i?-matrix, which satisfies the
quantum Yang-Baxter equation and thus establishes a relation with the represen-
representations of braid groups. Explicit expressions for the universal i?-matrix were first
obtained in [Drl], [Ro2], [LS], [KR2]. In a subsequent paper [Dr2], Drinfeld
studied further the structure of quantum groups, in particular the Casimir element.
These results imply that representations of quantum groups have a structure of a
braided tensor category [RT]. The same category will appear later in connection
with the monodromies of the Knizhnik-Zamolodchikov equations.
In spite of the fact that the theory of quantum groups is a relatively new
subject, only about a dozen of years old, there are a number of good books about
it (see, for example, [Kas], [CP2], [L2]). The general theory of Hopf algebras is
presented in the earlier monographs [Ab], [Sw]. We refer the reader to these works
for further details.
6.1. Hopf algebras and their representations
Let us recall the axioms of a Hopf algebra.
A Hopf algebra over C is a complex vector space A equipped with five additional
structures:
1. multiplication — a linear map m:4®A->i;
2. comultiplication - a linear map A : A —* A <g> A;
3. unit - a linear map i:C^i;
4. counit — a linear map e : A —> C;

78
LECTURE 5. FREE FIELD REALIZATION.
Using E.59) and renaming the variables t'j as follows: rj = tMi_l+j, where
Mi = mi + ¦ ¦ ¦ + nu. (thus, we get variables tp,p = 1,..., m), we can write this
function in the form
E.61)
Jc
¦¦¦ x X(n'N, zN) exp(—y(zN) ® eN)U(tMN_1+i)... U(tm)v0)v dt,
where, as before, v = v^ ®... St^ is the product of lowest-weight vectors in VMi.
Applying formulas E.9) and E.54), we can reduce E.61) to the normal ordered
form, after which the calculation becomes trivial:
E.62) JC iKj iiP l<p<n<m
x(v*Xo,Qvo)vdt,
where Q is the normal ordered product:
). ..X(-a',tm)
E.63)
Q = : X(fi\,Zl).. .
xexP(|:-,(,
It follows from the definition of normal ordering that
and thus we get
E.64)
(tP-tn)iY(t1,*)...Y(tm,x)vdt.
Note that the only dependence of the right hand side of E.64) on the partition
m is in the choice of the contour C = C\ x • • ¦ x Cjv-i-
Thus, we have obtained the same expression for the correlation function, and
thus, for a solution of KZ equations as in Section 4.4 (formula D.24)), using the
free field realization.
Lecture 6. Quantum Groups.
In this lecture we give a brief review of the theory of quantum groups. Quantum
groups first appeared as certain auxiliary algebraic structures in the "inverse scat-
scattering method" developed by L. D. Faddeev and his collaborators (see [FRTl] and
[FRT2] for historical remarks). Departing from the physical models, V. G. Drinfeld
[Drl] and M. Jimbo [Jl] independently discovered a g-deformation as a Hopf al-
algebra of the universal enveloping algebra of an arbitrary Kac-Moody algebra, thus
providing a vast class of examples of quantum groups.
The theory of Hopf algebras used to be a traditional branch of abstract al-
algebra. Examples of Hopf algebras coming from topology and group theory have
been known to mathematicians for a long time. However, most of these Hopf alge-
algebras were either commutative or cocommutative. The discovery of quantum groups
by Drinfeld and Jimbo following earlier works of Faddeev and collaborators pro-
provided fundamental examples of Hopf algebras which are neither commutative nor
cocommutative, bringing the theory of Hopf algebras to the forefront of modern
mathematics.
Drinfeld has also shown in [Drl] that quantum groups admit in addition a
so-called quasitriangular structure, or the universal .R-matrix, which satisfies the
quantum Yang-Baxter equation and thus establishes a relation with the represen-
representations of braid groups. Explicit expressions for the universal i?-matrix were first
obtained in [Drl], [Ro2], [LS], [KR2]. In a subsequent paper [Dr2], Drinfeld
studied further the structure of quantum groups, in particular the Casimir element.
These results imply that representations of quantum groups have a structure of a
braided tensor category [RT]. The same category will appear later in connection
with the monodromies of the Knizhnik-Zamolodchikov equations.
In spite of the fact that the theory of quantum groups is a relatively new
subject, only about a dozen of years old, there are a number of good books about
it (see, for example, [Kas], [CP2], [L2]). The general theory of Hopf algebras is
presented in the earlier monographs [Ab], [Sw]. We refer the reader to these works
for further details.
6.1. Hopf algebras and their representations
Let us recall the axioms of a Hopf algebra.
A Hopf algebra over C is a complex vector space A equipped with five additional
structures:
1. multiplication - a linear map m : A® A^ A;
2. comultiplication - a linear map A : A —> A ® A;
3. unit - a linear map i : C —* A;
4. counit - a linear map e : A —> C;

LECTURE 6. QUANTUM GROUPS.
5. antipode - an invertible linear map 7 : A —> A.
These five maps must satisfy the following relations for x,y,z G A:
(i) associativity of multiplication: m(x ® m(y ® z)) = m(m(x ® y) ® 2);
(i*) associativity of comultiplication: (A ® IdA)(A(a;)) = (Ma ®A)(A(a;));
(ii) axiom of unit: m(t(l) ® 2) = mB ® t(l)) = a: (i.e. t(l) is the unit in A);
(ii*) axiom of counit: (e ® Id,i)(AB)) = (Id^ ®e)(A(a:)) = 2;
(iii) axiom of antipode: m((Idx ®7)(A(x))) = m(G® IdA)(A(x))) = t(e(x));
(iv) consistency of m and A (=A is an algebra homomorphism): A(m(x®y)) =
m ® m(P23(A(x) ® A(y))), where P23 is the map Am -> A®* given by
Pz3(Zl ® %2 ® X3 ® 24) = 2i ® 23 ® 22 ® 24;
(v) consistency of e and t: e(t(l)) = 1;
(vi) consistency of m and e: e(mB ® y)) = e(x)e(y);
(vi*) consistency of A and t: A(t(l)) = t(l) ® t(l);
(vii) consistency of 7 and m: 7G71B ® y)) = mopGB) ® 7B/)), where m°p is the
notation for multiplication in the opposite order: mop(x ® y) = m(y ® 2);
(vii*) consistency of 7 and A: AGB)) = G ® 7)(Aop(a;)), where Aop is the
notation for comultiplication in the opposite order: AopB) = Pi2(AB)),
P12 : A ® A —> A ® A being the permutation map, P\2{x ® y) = y ® 2;
(viii) consistency of 7 and 1: 7AA)) = t(l);
(viii*) consistency of 7 and e: e(<y(x)) — e(x).
Thus, any Hopf algebra is an associative algebra with a unit and an associative
coalgebra with a counit.
The axioms of a Hopf algebra imply that in order to define the structure of
a Hopf algebra on an associative algebra with unit it is enough to describe this
associative algebra by generators and relations and then define the action of the
comultiplication, counit, and antipode on the generators. We will later apply this
method to the construction of the quantum groups. In fact, it can be shown that
the unit, counit, and antipode of a Hopf algebra are uniquely determined by its
multiplication and comultiplication.
From this definition it immediately follows that if A is a finite dimensional Hopf
algebra then the dual space A" also carries a natural structure of a Hopf algebra:
the maps (l)-E) for A* are given by m' = A*, A' = m*, J — e", e' = (,*, 7' = 7*.
This Hopf algebra is called the Hopf algebra dual to A.
Replacement of A by Aop defines another Hopf algebra A°p. However, if A is
commutative then Aop is isomorphic to A as a Hopf algebra, through the isomor-
isomorphism 7 (in general, 7 is an antiautomorphism).
It is obvious that tensor multiplication respects the axioms of a Hopf algebra.
Hence, if A and B are Hopf algebras then A ® B also has a natural structure of a
Hopf algebra: m = m.A'SimB, A = Aa®Aj, 1 = ia®i-b, e = ?a®?b, 7 = 7a®7b-
Example 6.1.1.
(i) Let G be a group, and let A = C[G] be its group algebra. Set for g,h e G
F.1) m(g®h)=gh, A(g) = g® g, j(g) = g~\ e(g) = 1,
= e,
where e is the identity element in G. These formulas define the structure of a Hopf
algebra on C[G]. It is easy to see that this structure encodes the group structure of
G: if H is another group then any morphism of Hopf algebras C[G] —> C[H] comes
from a group homomorphism G —> H, and vice versa.
6.2. DEFINITION OP QUANTUM GROUPS.
If G is finite then the algebra A* dual to A is the algebra T{G) of complex-
valued functions on G:
F.2)
, h) = f(gh),
(ii) Let G be a linear algebraic group, and let A = V(G) denote the space
of polynomial functions on G. Then formulas F.2) define the structure of a Hopf
algebra on A.
(iii) Let g be a Lie algebra, and let U(g) be the universal enveloping algebra of
g. This is an associative algebra with a unit. Define the comultiplication, counit,
and antipode on g by
F.3)
AB) =
+ l®o;, e(x) = 0, 7B) = -2, i?g,
and extend these maps to U(g) using the axioms of a Hopf algebra. This can be
done in a unique way, and the result is a Hopf algebra structure on U(g). This
algebra is the dual algebra to T'(G) in an appropriate sense.
It is said that a Hopf algebra is commutative if m = mop and cocommutative if
A = Aop. It is obvious that a finite-dimensional Hopf algebra A is commutative if
and only if A* is cocommutative. Any group algebra is cocommutative, while the
algebra of functions on a group is commutative. The universal enveloping algebra
of a Lie algebra is cocommutative.
A representation of a Hopf algebra A is a left module over A as an associative
algebra. In particular, we can endow C with a structure of representation of A by
2A = eB)A, i?i, A 6 C. This representation is called trivial.
If V is a finite-dimensional representation of A then one can define two dual
modules to V: V* and *V. As a vector space, each of them is the space dual to
V. The action of A in V* is: af(v) = f(-y(a)v), a € A, v €V, f eV*. The action
of A in *V is: af[v) = f(-j~1(a)v), a e A, v e V, f 6 *V. In general, *V is not
isomorphic to V*. It is easy to show that "V* is canonically isomorphic to V for
any V (through the transformation i>(f) = f(v)), but V* and "V are in general
different from V. Note that it follows from the axioms of a Hopf algebra that the
canonical maps of vector spaces V* <8> V —* C, C —> V ® V are A-homomorphisms.
An important property of Hopf algebras that general associative algebras do not
possess is the possibility to tensor representations. If pi : A —> End(Vi), i = 1,2,
are two representations of A, then their tensor product p : A —* End(Vi ® V2)
is defined by p(x) = p\ ® p2(A(x)). Axiom (iv) insures that this map is also a
representation of A. Axiom (i*) guarantees associativity of the tensor product:
V\ ® (V2 ® V3) = (Vi ® V2) ® Vi holds true for any three representations V\, V2, V3
of A. If A is cocommutative (e.g. C[G], U(g)), then we also have commutativity of
the tensor product: the map v\®V2 >-» V2®v\ defines an isomorphism of A-modules
Vi <8> V2 ~ V2 ® V\. However, if A is not cocommutative then, in general, there is
no reason to expect any relationship between V\ ® V2 and V2 ® Vi.
6.2. Definition of quantum groups.
Now we are ready to introduce Drinfeld-Jimbo quantum groups associated to
an arbitrary Kac-Moody algebra. Let A be a generalized Cartan matrix, which we
for simplicity assume to be irreducible, and g (respectively, ge) the corresponding
Kac-Moody (respectively, extended Kac-Moody) Lie algebra (see Definition 2.6.1,

LECTURE 8. QUANTUM GROUPS.
formula B.23)). We assume that the Cartan matrix is symmetrizable, i.e. there
exist numbers di such that rfiOy = djdji. We fix the di by the condition that
they are positive integers whose greatest common divisor is 1. In this case, as was
discussed in Lecture 2, ge can be endowed with a nondegenerate bilinear form {{, ))
such that atj = 2«?i'*i$ ¦ We normalize this form so that dt = ({ai,ai))/2, or,
equivalently, ({a, a)) = 2 for short roots.
Warning: this normalization differs from the one that we used for finite-
dimensional and affine Lie algebras before, where we required that {a,a) = 2 for
long roots.
Recall that this form gives us a way to identify he ~ f)*, and under this iden-
identification hi = 2ai/{{ai,ai)). In this lecture we will use this identification, which is
different from the one used in the previous lectures.
Prom now on q = e* will denote a fixed nonzero complex number (( ? C), and
qx will be a notation for etx, where x is a number or an operator. Throughout the
text we assume that q is not a root of unity.
We define, following Drinfeld and Jimbo, a deformation Uq(g) of the Hopf alge-
algebra U(g). As an associative algebra with unit, Uq{g) is generated by the elements
e;, fit 1 < i < r> Qh, h ? b, satisfying the following defining relations:
q° = l, qa+b = qaqb, a,6ef),
qheq~h = qa^eu qh hq~h = q~
_ q~dihi
?*-?-*
1-atj
F.4)
(-1)"
l-ay-n
= 0,
(-1)"
where [n]i =
The comultiplication in Wg(g) is given by
F.5)
F.6)
The counit and antipode are given by
e(eO = e(/0 = 0, e(qh) = 1,
The key result in [Drl] and [Jl] is the following theorem.
Theorem 6.2.1. Formulas F.4) - F.6) define a Hopf algebra structure on
The proof of this statement is tedious but straightforward.
6.2. DEFINITION OP QUANTUM GROUPS. 83
The Hopf algebra Uq{g) is called a quantum group. Replacing in the definition
the Cartan subalgebra f) by the extended Cartan subalgebra f)e, we get a definition
of the extended quantum group W9(ge).
Remark. This definition differs from the original definitions of Drinfeld and
Jimbo. To get their definition, we must take a subalgebra of Uq{g) generated by
eitfi and Ki = qhi. Another version, considered by Lusztig, is to allow qh for h
running through some lattice in f), containing the coroot lattice. We use the largest
possible set of generators to avoid some technical difficulties; all the results are
essentially the same for all of the above versions.
Example 6.2.2. Let g = s\2- In this case, the corresponding quantum group
is generated by the elements e, /, qxh,x € C, with the relations
q-xh = q2xe
qxhfq~xh = q~2xf,
Aqxh = qxh ® qxh
A/ = /® 1 + q~h®
xh ® qxh,
= -qhf, j(qh) =
Let Uq{g+) denote the subalgebra of Uq(g) generated by qh,h e f), and e^;
similarly, let Uq{g~) be the subalgebra generated by qh,h e (), and ft. One easily
checks that they are Hopf subalgebras. One can also define the extended versions
of these subalgebras, replacing t) with f)e.
Representation theory of quantum groups is to a certain degree parallel to
the classical case. We only consider representations with weight decomposition:
V = 01" such that for every v e V-,qhv = qi+Wv^ e^^v) (they are also
called type I representations). Here the weights /j, run through the space f)* for
Uq(g) and through the space f)e for Uq(ge).
We can define highest weight representations of quantum groups. To every
weight A there corresponds the Verma module M| = Ind^'|°+,Xx, where X^ is a
one-dimensional space in which all ei act by zero and qh acts by multiplication by
qW>. Verma modules admit weight decomposition.
Similarly to the classical case, we define the submodule I\ in M' to be the sum
of all proper submodules in M'. Then the quotient Lqx = Mi/I\ is an irreducible
representation of the quantum group Uq(g). More generally, we will say that V is
a highest-weight module if it is a quotient of a Verma module.
The following theorem, proved in [LI], [Rol], describes the structure of finite-
dimensional representations of Uq(g).
Theorem 6.2.4. Let g be a simple finite-dimensional Lie algebra, and let q e C
not be a root of unity. Then:
1. The module L\ is finite-dimensional if and only if A e P+ ¦ For A e P+,
dim L\ = dim L\, and the dimensions of the weight subspaces of L\ and L\
are equal.
2. The category C(g,q) of finite-dimensional Uq(g)-modules with weight decom-
decomposition is semisimple, with simple objects Lqx, A e P+.

84
LECTURE 8. QUANTUM GROUPS.
We can also define contragredient Verma modules. For a module M with weight
decomposition, define Mc to be the restricted dual to M (that is, Mc = 0(MA)*
as a vector space) with the action of Uq(g) given by
(gv*,v) = {v*,r(g)v), v* e Mc,v e M,g e W?(g),
where the anti-automorphism of algebras r : W?(g) —»W?(g) is defined by
r(/4) =
r(ab) = r(b)r(a).
Note that r is a coalgebra automorphism, and thus the usual isomorphism of vector
spaces (Mi <g> M2Y — Mf ® M% is a W9(fl)-morphism.
The module (M?)c is called the contragredient Verma module. It has the same
dimensions of weight subspaces as Mj[, but in general is not isomorphic to M^.
There is a canonical morphism of Wg(g)-modules M' —» (M')c, defined by v\ i-> aj,
and it can be shown that the image of this morphism is exactly the irreducible
module L\.
Finally, it is also easy to define lowest-weight Verma modules and contragredi-
contragredient lowest-weight Verma modules, similarly to the classical case.
Let p G t) be an element such that {(p,hi)) = 1, or, equivalently, «i(p) = rf, (if
the Cartan matrix is nonsingular, this defines p uniquely; otherwise, we can choose
any p satisfying these conditions). Then for every x € W9(g),
(R I7\ 2/ \ 2p_ —2p
lO-i 1 'y \~C) == Q XQ •
Indeed, it suffices to check this on generators, which is trivial.
Recall that for every finite-dimensional module V we have defined a dual mod-
module V. The module V* coincides with V as a vector space, but has a different
structure of W9(ge)-module.
Proposition 6.2.3. Let V be a finite-dimensional module overUq($e) with a
weight decomposition. Then the map
F.8)
Sv : V ->¦ V",
v i-> q2pv
is an isomorphism of modules.
Proof. This immediately follows from F.7).
D
6.3. Quasitriangular structure and braided tensor categories.
Quantum groups have an additional important structure which is not present
in a general Hopf algebra - the quasitriangular structure. This structure insures
that V\ ® V2 is canonically isomorphic to V2 ® V\ for every pair of representations
Vi, V2. However, the isomorphism between these two products is no longer the
trivial permutation of factors, as it used to be for W(g), but a certain nontrivial
linear map, individual for each pair of representations.
Let A be a finite-dimensional Hopf algebra.
6.3. QUASITRIANGULAR STRUCTURE AND BRAIDED TENSOR CATEGORIES. 85
Definition 6.3.1. A quasitriangular structure in A is an invertible element
R € A <g> A, R = J2i °« ® bi, «i. h € A, such that:
F.9)
where
RA(x) = Aop(x)R,
(A®ldA)(R) = R13R23,
(IdA®A)(R)=R13R12,
Rl2 =
bi <g> 1 = R <g> 1,
F.10)
R13 =
R23 =
are elements of A ® A ® A associated with R.
A Hopf algebra A with a quasitriangular structure is called a quasitriangular
Hopf algebra; the element R is called the universal R-matrix of A.
Let Abe a, quasitriangular Hopf algebra, R the universal .R-matrix of A, and
Vi, Vi two finite-dimensional representations of A. Then Vi <g> V2 and V2 ® Vi
are isomorphic as representations of A: the map Rv1y2 = ^"^ |vi®vb, where P :
Vi<g>V2 —> V^®Vi is the permutation of factors, establishes an isomorphism between
Vi ® V2 and V2 ® Vi (this property follows from the first equation in F.9)).
We list here some further properties of the .R-matrix:
Proposition 6.3.2. The universal R-matrix R satisfies the following identi-
identities:
F.11)
F.12)
F.13)
Also, the element R°p = P(R), where P(a
R-matrix for the Hopf algebra Aop.
Proof. We have
(IcU
G®Idx)(-ff) = (Id^®7-1)(i?) = .
= b <g> a, is a universal quantum
® ldA)(R))R = (e ® ldA ® ldA)(R13R23)
= {e®ldA®ldA)(A®ldA)(R)
= ii (by axiom (ii*) of a Hopf algebra).
Therefore, since R is invertible, we obtain (e <g> Id^)(.R) = 1^. The second relation
(ldA®e)(R) = 1A is obtained similarly.
To prove F.12), write
= {m®ldA)((~,®IdA®IdA)((A®ldA)R))
= (te <g> ldA)(R) (by axiom (iii) of a Hopf algebra)
= U®U (by F.11))

LECTURE 6. QUANTUM GROUPS.
which proves that G® 1<1a)(R) — -R; the second identity is proved analogously.
Applying 7 <g> 7 to F.12), we get
which implies F.13) because 7 <g> 7 is an antiautomorphism of A <g> A.
Finally, it is obvious from the definition that
R°pAop(x) = A(x)R°p,
(ldA®Aop)(R°p) =.
which implies that Rop is a quasitriangular structure for Aop.
The most natural way to explain the meaning of all the axioms and properties
of the universal .R-matrix is to use the language of braided tensor categories, which
we briefly recall here (see [Mac, MSI, Kas] for more details). By definition, a
braided tensor category is an additive category C with a distinguished object 1, a
bifunctor <g> :C <g> C —> C, and functorial isomorphisms
associativity isomorphism avi,v2,v3 : (Vi ® V2) ® V3 ~ Vi ® (V2 ® V3),
unit morphisms Ay : 1 <g> V ^ V, py : V" ® 1 — V,
commutativity isomorphism o-Vl^y2 : V\ <g> V2 — V2 ® Vi
, which Eire given by
which have to satisfy the following axioms:
(l)triangle axiom: two morphisms (X <g> 1) ® V —> X <
px <g> Id and (Id®Ay)ax,i,r, are equal
B)pentagon axiom: all the maps ((Vi <8> V2) ® V3) ® V4
which can be obtained by iterations of a,a~1, are equal
C) 2 hexagon axioms: the isomorphisms Vi ® (V2 ® V3) —» (V2 ® V3) ® Vi, given
by crv,,v2®V3 and by a~1(Idv2 ®opv1,v3)o(o'Vi,va 8Wv3)«"\ aie equal, and similarly
fora-1.
Theorem 6.3.3. Let A be a quasitriangular Hopf algebra. Then the category of
representations of A is a braided tensor category with 1 = C, trivial unit morphisms
and the associativity morphisms (i.e., the same as for the category of vector spaces),
and the commutativity morphism given by av,w = Rv,w-
PROOF. The fact that the unit morphisms and the associativity morphism
axe indeed ^4-morphisms follows from the axioms of Hopf algebra, and R is an di-
dimorphism due to the first equality in the definition of the .R-matrix. The triangle
and pentagon axioms are obvious; the hexagon axioms follow from the second and
third identity in F.9). ?
In a similar way, axioms involving the antipode can be summarized by saying
that the category of finite-dimensional representations of a quasitriangular Hopf
algebra is a rigid braided tensor category, which means that every object V has a
dual V along with the functorial morphisms 1—* V ®V* ,V* ®V ^> 1 satisfying
some natural axioms. However, we will not use this notion.
6.4. QUANTUM YANG-BAXTER EQUATION
6.4. Quantum Yang-Baxter equation
and representations of braid groups.
As before, let Ahea, quasitriangular Hopf algebra.
Proposition 6.4.1 (Quantum Yang-Baxter equation). The universal quan-
quantum R-matrix satisfies the following relation in A <g> A <g> A:
F.14) R12R13R23 — -R23-ffl3-ffl2-
Proof.
R12R13R23 = Ri2(A®ldA)(R) = (Aop ® UA)(R)RX2
D
This immediately gives us a connection with Artin's theory of braid groups.
We recall the definition.
Definition 6.4.2. The braid group in N strands is the group BN with the
generators 61,..., bN-i and relations
F.15)
bj =bjbit \i-j\ >
bih+ibi = bi+1bibi+1.
This group has a simple interpretation as a group of braids formed by N strands
with fixed top and bottom and the group operation being pasting one braid to an-
another. The braids are considered up to isotopy. Figure 6.1 shows the braids corre-
corresponding to the generators bi and the isotopy corresponding to the defining relation
F.15). It can be shown (see [Bir]) that this geometric definition is equivalent to
the algebraic definition above.
i i+i
Figure 6.1. Generators and relations in the braid group.
Note that there is a canonical homomorphism a : Bn —> Sjf given by bi i-> s<.
The kernel of this homomorphism is called the pure braid group and is denoted
PBN.
Theorem 6.4.3. Let A be a quasitriangular Hopf algebra, V\,..., V/v - repre-
representations of A. Let Ri : V\ ®... <g> Vtt —» V\ ® ... <g> Vi+1 ® K ®... ® Vat be given by
Ri = Pi,i+\RviV,+i, where Piii+1 is permutation of factors i,i + l. Then we have the
following idenity of operators Vi <g>... <g> Vjv —> Vi ®... <g> Vi+2 ® Vi+1 ® K ®... ® Vn :

LECTURE 6. QUANTUM GROUPS.
Proof. This immediately follows from the quantum Yang-Baxter equation
F.14) ?
Corollary 6.4.4. Let W be a representation of A. Then the map
F.16) bi<-*Ri, i-l,...,N -1,
defines a representation of the braid group Bn in the space W®N.
The same results hold in an arbitrary braided tensor category.
Theorem 6.4.5. LetC be a braided tensor category. Define functorial isomor-
isomorphisms $%YZ :X®{Y®Z)^Y®{X®Z) by
{we leave it to the reader to put appropriate indices for a).
Then:
1. Denote
P12 = 0x.Y.z<au ¦¦X®{Y®(Z®U))-*Y®{X
Pfs = ld®f3YZU : X ® (Y ® {Z® V)) -> X ® (Z <g> (Y ® ?/))¦
Then we have the following identity of operators Vi ® (V2 ® (V3 ® f))
F.18) 012023/5
2.
F-19) P%,Yli
The proof of this theorem is straightforward and is left to the reader.
6.5. Quantum double construction.
The classical examples of Hopf algebras considered in Section 6.1 either do not
admit a quasitriangular structure at all or admit a trivial quasitriangular structure:
R = \A ® 1A. In this section we will learn to produce nontrivial examples of
quasitriangular Hopf algebras. The following construction, due to Drinfeld, creates
a quasitriangular Hopf algebra from an arbitrary finite-dimensional Hopf algebra.
Let Abe a, finite-dimensional Hopf algebra. Let Oi be a basis of A, and let a%
be the dual basis of A*. Consider the vector space D(A) = A ® ^4*op. We want
to introduce a Hopf algebra structure in D(A) so that the natural embeddings
A <-+ A <g> A*°p, A*op ^ A <g> A*op will be Hopf algebra homomorphisms. One
way to do it is to take a tensor product of Hopf algebras, but this is not very
interesting. Instead, let m'fci and y^kl be the structure tensors of the multiplication
map A®A®A-* A and comultiplication map A —> A® A® A, respectively (in the
basis Oi), and denote by a{ the matrix of the inverse to the antipode in A. Define
multiplication in D(A) by
F.20) o3o( = ^"m^^o'o,,
8.6. QUANTUM DOUBLE CONSTRUCTION FOR W,(ge)
where summation is implied over repeated indices. It can be shown that formula
F.20) is consistent with the axioms of a Hopf algebra and uniquely defines the
structure of a Hopf algebra on D(A). This algebra, which was first constructed by
Drinfeld in [Drl], is called the quantum double (or Drinfeld's double) of A.
Theorem 6.5.1. The Hopf algebra D(A) is quasitriangular, with
F.21) JR
Example 6.5.2. Let A = C[G] be the group algebra of a noncommutative
finite group G (see Example 6.1.1). Then D{A) = C[G] n F{G), where F{G) is
the algebra of functions on G. Therefore, D(A) is a quasitriangular Hopf algebra
which is neither commutative nor cocommutative. This Hopf algebra is called the
quantum double of the finite group G, and its representations are G-equivariant
vector bundles on G. It is probably the simplest nontrivial example of a quantum
group.
6.6. Quantum double construction for Uq(ge)
In this section, we show how one can define a quasitriangular structure on the
quantum group W9(ge). It is based on the fact that Wg(ge) is almost the quantum
double of W,(g+), which allows us to use the results of the previous section; how-
however, since Uq(ge) is infinite-dimensional, one must make necessary changes. The
construction of this section is due to Drinfeld; detailed exposition can be found in
[T]. As before, q is either a formal variable or a complex number which is not a
root of unity (in fact, the universal .R-matrix can also be defined for q a root of
unity, see [L2], but we will not use this).
Recall that we have defined the subalgebras Uq{gf) C Uq(ge) as subalgebras
generated by qh,et (respectively, f{). As vector spaces, they are isomorphic to
C[t)e] (giW^n*), where C[\)e] is the group algebra of be, with basis qh,h e t)e, and
Uqixi^) are the subalgebras generated by the elements e* (for plus) and /; (for
minus).
We define the restricted dual space W^g*)* to W9(g±) to be the vector space
C[f)*] <g>Uqin*)*, where W./n*)* is the restricted dual (by degree) to W^n*). The
pairing between C[f)e] and C[(j*] is given by the formula (qh, qx) = qxl>h\ The Hopf
algebra structure in Ug(g±) is introduced in a standard way: the multiplication,
comultiplication, unit, counit, and antipode in W^g*)* are dual to the comulti-
comultiplication, multiplication, counit, unit, and antipode in Uq(g~), respectively. The
following theorem, which has no classical analogue, was proved in [Drl].
Theorem 6.6.1. There exists an isomorphism of Hopf algebras 9 : Uq(sl) —*
W9(g+)*op such that for every h e t)e, 9(qh) = 1h, where we identify f)e and \fe by
means of the form ((,)).
This identification gives rise to a nondegenerate bilinear form (Drinfeld pairing)
MqCe) ®M<i(9e) ~* C satisfying certain invariance properties.
Note that this theorem does not hold for q = 1: in this case one of the algebras
is commutative and the other is not.
This shows that we can identify the restricted quantum double of Uq(g+) with
the tensor product

LECTURE 8. QUANTUM GROUPS.
This algebra contains a two-sided ideal H generated by the elements qh ® 1 -
1 <g> qh. It is easy to check that this ideal is respected by all the structures of the
Hopf algebra. Therefore, the quotient by this ideal will also be a Hopf algebra.
Theorem 6.6.2. The map
f®li-. q",
l®qh^qh
can be uniquely extended to a Hopf algebra isomorphism
F.22) 4> : Wg(gj~) ®Uq(g~)/H ol Uq(ge).
Let us now construct the universal .R-matrix for the algebra D(Uq(gt)) and
push it forward to the algebraUq{ge), which should give us a quasitriangular struc-
structure for Uq(ge).
Let O; be a basis of Uq(n+) which consists of homogeneous elements, and let
ai be the dual basis of Wg(n"~) with respect to the Drinfeld pairing defined by
Theorem 6.6.1. Let xp be an orthonormal basis of hc with respect to the inner
product ((,)) on he- Define
F.23)
R =
(recall that q = e*). This expression is infinite and does not belong to the tensor
square of the algebra. However, it makes sense as an operator in the product U\ ® U2
of any two highest-weight modules Ui,U2 overW9(ge). The following result was first
obtained by Drinfeld in [Drl].
Proposition 6.6.3. The expressionR defined by F.23) satisfies the R-matrix
axioms F.9), and thus satisfies Yang-Baxter equation F.14), regarded as a relation
between operators in tensor products of highest-weight modules.
Abusing the language, we will say that R defines a quasitriangular structure
onUq(9e).
Sketch of proof. Let ahi = qha,i. Then {ahi} is a basis of Uq(g+). To prove
our proposition, we would like to make sense of the equality
F.24) R = y^Qfti ®Ohi-
To make sense of the elements a*hv we extend the algebra Uq{xT) by adding
new elements I*, A e be- introduced by Lusztig (see [L2]). These elements satisfy
the defining relations
1\ =
= 0, A
¦ = > I*.
F.25)
-U/i = /jI
6.6. QUANTUM DOUBLE CONSTRUCTION FOR M,(ge) 91
Denote the algebra thus obtained by Uq{$~).
The action of Uq(n~) in any highest-weight W9(ge)-module (or tensor product
of highest weight modules) naturally extends to Ug(g~): the elements lx act by the
identity on vectors of weight A and by zero on vectors of any weight // ^ A.
Note that strictly speaking, the new algebra Uq (gj) is not a Hopf algebra since
our definitions of the unit and comultiplication are illegal (the sums are infinite).
However, this does not create any problems as long as we are working with highest-
weight modules.
The Drinfeld pairing {,), defined in Theorem 6.6.1, can be extended to a pairing
between Uq(g+) and Uq(g~) as follows:
F.26)
), aeUq(n+),
(as before, we identify t) with h*).
This definition is suggested by the relation (qh, qX) = qrtx) and the equality
Qh = Ylx 9A(ft)l> satisfied in every highest-weight module. One can check that the
extended pairing satisfies the invariance relations:
F.27)
(xy,z) = (x®y,A°e(z)), (x, yz) = (A(x), y® z),
<7(aO.7(iO> = (x,y), <!,*> = e(x), (y, 1) = e(y).
With respect to this pairing, we have o^ = l/>a\ Therefore, we can define an
expression
F.28)
R =
which makes sense as an operator in a tensor product of highest-weight mod-
modules. Drinfeld's quantum double theorem 6.5.1 (suitably modified for the infinite-
dimensional case) implies that this expression satisfies relations F.9). It remains
to observe that in a product of highest-weight modules
which immediately yields F.23). ?
From now on, when we talk about the universal .R-matrix for Uq(ge), we will
have in mind the one given by F.23).
For finite-dimensional and affine Lie algebras one can explicitly compute the
universal ^-matrix. These formulas can be found in [LS, KT1, KT2, KR2, Ro2],
Here we give the formula for g = sl2 (see [Drl]).
Example 6.6.4. The universal R-matrix ofUq(sl2) is
F.29) R =
It can be checked directly that F.29) satisfies relations F.9).
Combining the results of this section with Theorem 6.3.3, we get the following
result.

92
LECTURE 6. QUANTUM GROUPS.
Theorem 6.6.5. LetC{ge,q) be the category of finite-dimensional representa-
representations with weight decomposition of the quantum group Uq(ge) {we assume that q is
not a root of unity). Then this category has a structure of a braided tensor category,
with the associativity and unit morphisms the same as for the category of vector
spaces, and with the commutativity morphism given by ovw = Rvw = PRvw,
where R is the universal R-matrix F.23) forUq(ge).
It is useful to note that, although for finite-dimensional g the category C(g, q)
is semisimple, with the same simple objects L\, A € P+, and multiplicities N*v =
dimHomU|j(B)(i^,iJ ® L%) as the category of representations of g, they are not
equivalent as braided tensor categories, which is obvious because R2 ^ Id. In fact,
they are not equivalent even as tensor categories, which is less obvious.
6.7. Quantum Casimir element.
Let A be a quasitriangular Hopf algebra. Define the element
F.30)
This element lies in A if A is finite-dimensional; for A = Uq($e) it is n°t so, but
this expression is well defined as an operator in any highest-weight module. The
element u was defined by Drinfeld [Dr2], who also proved the following identities
involving this element.
Proposition 6.7.1.
72 (x) = uxu l for every x G A
u 1 = m((f x ®IdA)((R°p) 1)]
«) = (u®«)(i?opJR) 1 = (RopR) Ji
F.31)
F.32)
F.33)
where R°p was defined in Proposition 6.7.1.
We refer the reader to [Dr2] for the proof.
Note that F.31) implies 72(u) = u.
Let us recall that we have also defined an element q2p e Uq(ge) such that
J2{x) = q2pxq 2p (see F.7)).
Theorem 6.7.2. Let A = Uq(ge). Define the quantum Casimir element C by
C = q2pu :.
Then C is well defined in every highest-weight representation ofUq(ge), is cen-
central (that is, it commutes with the action ofUq(ge)), and satisfies
If V is a highest-weight module with highest weight A, then
F.35) C\v = >
8.8. INTERTWINING OPERATORS AND THEIR COMMUTATION RELATIONS.
93
Proof. The fact that C is central and F.34) follow immediately from the
previous proposition and F.7). To prove F.35), it suffices to check this property
on the highest-weight vector v of Ml, where we have
C^v = q~2puv = q~2pmo(-y®\)(R°p)v
= q-2"m o G <g> l)(gs>> x-®x» ^ oi ® at)v
= q~2pm o
a
This result immediately implies the following corollary. Recall that for any two
representations V, W we have defined Rv,w = PR\v®w ¦ V <g> W -* W <g> V. Let
us for brevity write R2 for RwyRv,w '¦ V <g> W —> V ® W.
Corollary 6.7.3. Let Fa, V^, VL- be highest-weight modules overUq(ge) with
highest weights \,n,v respectively. Then for any embedding of'Uq(gB) -modules V\ C
Vu ® Vv we have
where
Proof. It follows from the previous theorem that R2 = A(C)(C-1 <g> C),
which along with F.35) gives the statement of the corollary. ?
6.8. Intertwining operators and their commutation relations.
Let g be a simple finite-dimensional Lie algebra. Consider the category C(g, q)
of finite-dimensional representations of Uq(g) with weight decomposition. Since this
category is semisimple, with simple objects LA, A € P+, all morphisms in this cat-
category can be described in terms of the spaces of homomorphisms Hom^(B)(L', V).
In particular, we will show that the commutativity and associativity morphisms,
defining the structure of a braided tensor category on C(g, q), can be interpreted as
commutation relations between intertwining operators. This material will be useful
in the next lectures. The statements of this section can be specialized to the usual
enveloping algebra U(g) by setting q = \.
Let V be a finite-dimensional representation of Uq(g). Denote
F.37) Hy^ = HomWg(g)(L^, V <g> L?),
where, as before, L\ is the irreducible finite-dimensional module with highest weight
A. If we have two finite-dimensional representations V, W, then it follows from
complete reducibility that the composition of intertwiners $j <g> $2 |-> A ® $2) ° $i
gives an isomorphism
F.38) ffiffig^,,

94 LECTURE 8. QUANTUM GROUPS.
This decomposition can be illustrated by the following diagram:
Let R+ = PR, Rr = P{Rop)~l = [R+)'1- Multiplying an intertwiner $ :
L\ —» V <g> W ® L? by the matrix ^^ ® IdLj, we get an intertwining operator
Lqx —> W <g> V ® ZJ, which we will denote by B**. Using F.38), we can consider
B* as an isomorphism
(o.oaj tiwv{A,v) : ^pri;,
The matrices B* are called the exchange matrices, and are nothing but rewrit-
ings of the commutativity isomorphism (or more exactly, the isomorphisms p±,
defined by F.17)) in terms of the spaces of intertwiners.
Proposition 6.8.1. The exchange matrix satisfies the following identities
(compare with F.18), F.19)):
1. Braid relation: ifV,U,W are finite-dimensional representations, then
F.40) B12B23B12 = B23B12B23,
where both left-hand side and right-hand side are isomorphisms
U ® W ® V <g> L«),
^, V ®
or, more explicitly,
and, as usual, B^2 = B± <S> Id, B^
2. Unitarity relation: 6*6* = Id.
Proof. The first relation is just the rewriting of the braid relation for R, or
the Yang-Baxter equation for R, in terms of the spaces of intertwiners; the second
is immediate from the definition. ?
The relation F.40) is called in statistical mechanics the star-triangle relation
and plays an important role in statistical mechanics; see [DJMO], [P].
In fact, the exchange matrix is also closely related with the associativity iso-
isomorphism. Indeed, let us rewrite the associativity isomorphism a : (T^<g>L?)<g> W ~
V <g> (L% <g> W) in terms of the spaces of intertwiners as follows:
F.41)
, v) :
6.8. INTERTWINING OPERATORS AND THEIR COMMUTATION RELATIONS.
95
In particular, if g = sfe, and V, W are irreducible finite-dimensional representa-
representations with highest weights i,j, then the spaces of intertwiners axe zero- or one-
dimensional, and if we fix a normalization of the intertwiner i^ : Vi —> V, <g> 14,
then a becomes a matrix, depending on 6 parameters i,j,X,u,n,jx'. Elements of
this matrix (which are quite nontrivial even for q = 1) are called the (quantum) 6j-
symbols ; in physical literature they are also known as Racah coefficients. Explicit
formulas for quantum 6j-symbols can be found in [KRl].
On the other hand, it follows from the hexagon axiom that the following dia-
diagram is commutative:
Thus, we can write
F.42)
W ®
B+w(\, v) = R2aylw(X, v)
: ©M, H?w
where ^ : ©M, H?w » ©„
with the i?-matrix. Since both
©„#&„,
: ®,,H*W
©n' ^wv are compositions
i, R2 have a very simple form - for example, their
squares are diagonal matrices, see F.36) - we see that essentially the exchange
matrix B is the same as the 6j-symbol. In particular, for s^ one can write B =
D2a~1 D^1, where D\,D2 are diagonal matrices.
We can also consider the case when all the highest weights A, /j., v,..., used
above, are generic (thus, L\ = M', etc.), while keeping the representations V, W,...
finite-dimensional, and define the spaces Hxv by the same formula F.37). These
spaces can be described explicitly: the linear map {,) : HomM?(g) (Mx, M? ® V) —>
Vx~v defined by ($) = (u* ® Idv)($v\), where v\ is a highest-weight vector of
Ml and v1^ is a lowest-weight vector of (Mfr)*, is an isomorphism (compare with
C.36)):
F.43) Hx
In other words, for every generic A and v ? Vx M there exists a unique operator
*X : M\ —> M^ <g> V such that &{{v\) = v ® v^ + lower order terms. Similarly,
we define for homogeneous v € V,w ? W the intertwiner $^®w = (Id®*™) o $^ :
L\ —t V ®W ® LI, where v = A + weight(u) + weight(w). It follows from the
results of Section 3.7 that it gives an isomorphism
(V ® W)x~" ~
, V
Thus, we can again define the matrix B^w(\,v) : (V ®W)X-V ~ (W ®V)X~",
as in F.39). Taking the direct sum over all u1 and writing B±(A) = PB±(A), we
get the operator ByW(X) : V ® W —> V ® W, which can be explicitly defined by
F.44) (i?±
Note that the operators B±(A) preserve weight, but are not Wg(g)-homomorphisms.

98 LECTURE 6. QUANTUM GROUPS.
Proposition 6.8.2. The exchange operator B±(\), defined by F.44), satisfies
the following relations.
1. The modified (or dynamical) Yang-Barter equation: for any three finite-
dimensional representations V, W, U
1 '
where Btfw(\ - h^)*^ <g> w <g> u) := B^w(\ - urt(u))±(i) <g> w <g> «), v e
V, w e W, u e U ifu is homogeneous, and wt(u) denotes the weight ofu (the
other notation is defined similarly).
2. The unitarity condition:
F.46)
The proof of this proposition is parallel to the proof of Proposition 6.8.1.
Lecture 7. Local Systems
and Configuration Spaces.
Now we will look at the geometry hidden behind our construction of solutions
of the KZ equations; in particular, we will prove that the cycles of integration which
we needed in the previous lectures indeed exist and, for generic values of parameters,
give all solutions of the KZ equations. These cycles span certain homology groups
with coefficients in local systems, studied earlier by K. Aomoto [Al], and the
integral formulas for the solutions of the KZ equations can be viewed as a pairing
between the homology and cohomology groups. This approach was developed by
V. V. Schechtman and A. N. Varchenko, and is presented in detail in the monograph
[VI]. In dealing with cohomology with coefficients in local systems, the language
of sheaf theory and cohomology of sheaves is very useful. However, we have tried
to organize our exposition so that the main results do not refer to sheaf theory as
far as possible.
We refer the reader to textbooks on algebraic topology and algebraic geometry
(such as [GH], [GR], [Har], [Iv]) for detailed exposition and proofs of the results
briefly reviewed here. All the manifolds considered in this lecture are either C1
real manifolds or complex analytic manifolds considered with the usual topology.
Unless stated otherwise, the manifolds are assumed to be nonsingular.
7.1. Local systems.
Let us start by recalling some basic definitions from differential geometry. Let
ibeaC1 real manifold, and E a finite-dimensional vector bundle over X. Denote
by U{(X) the bundle of differential forms of degree i on X, and let U{(X, E) =
E®U,%(X). Recall that a connection V is a map of local sections of vector bundles
satisfying the following condition: for any local section s of E and a function / on
X we have
G.2)
V(/a)=/V(s)+s®d/.
Equivalently, this shows that for any vector field f we have a map V^ : E —>
E : s i-> (V(s),f) such that Vj(/s) = /V?(s) + (<%/)«. In particular, if we choose
a coordinate system x',i = 1,..., n, on X, then we can introduce V; = Vg/gxi. It
is easy to see that V; can be written in the form
/7 o\ V7 — __ i A {r\
u .01 Vj — q - ~t~ **iy^*),
where the Ai are operators acting on the fibers of E.

96 LECTURE 8. QUANTUM GROUPS.
Proposition 6.8.2. The exchange operator B±{\), defined by F.44), satisfies
the following relations.
1. The modified (or dynamical) Yang-Baxter equation: for any three finite-
dimensional representations V, W, U
F.45)
where B]^W{X — h^)^(v <8> w <8> u) := B^W(X — «/<(«))*(?; ®w® u), v €
V, w € W, u € U if u is homogeneous, and wt(u) denotes the weight ofu (the
other notation is defined similarly).
2. The unitarity condition:
F.46) B^yiX^BvwiXf = ldv®w .
The proof of this proposition is parallel to the proof of Proposition 6.8.1.
Lecture 7. Local Systems
and Configuration Spaces.
Now we will look at the geometry hidden behind our construction of solutions
of the KZ equations; in particular, we will prove that the cycles of integration which
we needed in the previous lectures indeed exist and, for generic values of parameters,
give all solutions of the KZ equations. These cycles span certain homology groups
with coefficients in local systems, studied earlier by K. Aomoto [Al], and the
integral formulas for the solutions of the KZ equations can be viewed as a pairing
between the homology and cohomology groups. This approach was developed by
V. V. Schechtman and A. N. Varchenko, and is presented in detail in the monograph
[VI]. In dealing with cohomology with coefficients in local systems, the language
of sheaf theory and cohomology of sheaves is very useful. However, we have tried
to organize our exposition so that the main results do not refer to sheaf theory as
far as possible.
We refer the reader to textbooks on algebraic topology and algebraic geometry
(such as [GH], [GR], [Har], [Iv]) for detailed exposition and proofs of the results
briefly reviewed here. All the manifolds considered in this lecture are either C1
real manifolds or complex analytic manifolds considered with the usual topology.
Unless stated otherwise, the manifolds are assumed to be nonsingular.
7.1. Local systems.
Let us start by recalling some basic definitions from differential geometry. Let
X be a C1 real manifold, and E a finite-dimensional vector bundle over X. Denote
by n{(X) the bundle of differential forms of degree i on X, and let SV(X, E) =
E ® f22(X). Recall that a connection V is a map of local sections of vector bundles
satisfying the following condition: for any local section s of E and a function / on
X we have
G.2)
V(/s) = /V(s)
Equivalently, this shows that for any vector field f we have a map V{ : E —>
E : s i-> (V(s),?) such that Vj(/s) = /V?(s) + (d^f)s. In particular, if we choose
a coordinate system xl,i = 1,... , n, on X, then we can introduce V* = ^7g/gX'- It
is easy to see that V, can be written in the form
G.3) V, = —-+Ai(x),
where the Ai are operators acting on the fibers of E.
97

LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
Definition 7.1.1. A connection V is called flat if we have [V$, V,,] = V^^j
for any two vector fields ?, JJ.
Definition 7.1.2. A local system over X is a vector bundle equipped with a
flat connection.
If we choose a local coordinate system on X, then the connection is flat iff
[Vi,V.,-] = 0.
A connection V gives rise to the "parallel transport" of elements of E. Namely,
let 7 : [0,1] —> X be a path. Then for every v e Sy(o> there is a unique section s of ?
over a neighborhood of the path 7 such that V-yS = 0. This gives an isomorphism
M7 : JEy(o) -> ?^A) : sG@)) >-> sG(l)), which is called the holonomy operator
along 7.
Theorem 7.1.3. //V is a.#at connection, then M,, depends only on the ho-
motopy class off.
Corollary 7.1.4. Let its fix a base point x0 ? X. Then 7 >-> M-, is a repre-
representation of the fundamental group irj (X, xo) in EXa.
In fact, it can be shown that Corollary 7.1.4 gives a bijection between local sys-
systems on X (up to isomorphism) and finite-dimensional representations of vy{X, xo).
Definition 7.1.5. Let U <Z X be open, and let s be a section of E over U.
We say that s is flat if V^s = 0 for every vector field f in V. The space of flat
sections of E over V will be denoted T f(U, E).
Lemma 7.1.6. Let U C X be open and simply connected. Then for every p e U
the map F/(f, E) —> Ep : s >-> s(p) is an isomorphism.
This lemma is the base of another approach to local systems. Namely, let us
consider the sheaf of flat sections of E. Then Lemma 7.1.6 implies that this sheaf
is locally constant; again, it can be shown that any locally constant sheaf of vector
spaces of finite rank is obtained from a local system. Thus, "local system" and
"locally constant sheaf" are Synonyms.
Also, for a local system we have the notions of dual local system and tensor
product of local systems. If E, E1 are local systems on X, then the tensor product
in the sense of vector bundles E <g> E' carries a natural flat connection defined by
V(s<g>s') = (Vs)<g>s'+s<g> Vs'. Similarly, if E* is the dual vector bundle, then we can
define a connection in E* by Vs*(s) = —s*(Vs) for any local sections s, s* of E, E*
respectively. This condition means that the natural pairing E ® E* —» C°°(X) is
compatible with the connection: (Vjs, s*) + (s, V^s*) = 9^{s, s").
If we have a smooth map of manifolds / : Y —> X and a local system E on
X, then it naturally defines a local system fE on Y such that (/*?% = ¦&/(„)•
For example, if / is an embedding then f*E is just the restriction of E to the
submanifold Y. Abusing the language, we will sometimes say "sections of E over
f(Yy even when / is not an embedding, meaning by this "section of f*(E) over
Y". In fact, for local systems one can define f*E not only for smooth but also for
continuous maps /. The simplest way to do this is to use the representation of local
systems as locally constant sheaves.
All of the above can be applied as well to complex manifolds with appropriate
changes; for example, we have to consider holomorphic functions and differential
forms rather than smooth ones.
7.2. COHOMOLOGY AND HOMOLOGY OP LOCAL SYSTEMS 99
7.2. Cohomology and homology with coefficients in local systems.
Let X be a C°° manifold of dimension n, and let E be a smooth vector bundle
with a connection on X. As before, let Sl'(X, E) = E <g> €l\X) be the bundle of
smooth E-valued i-forms on X. Then we can extend the connection V to maps
rfv : Uk(X, E) -» Uk+\X, E) by the rule dv{s ® w) = V(s) A w + s ¦ du. Thus, we
have the sequence of maps of vector bundles
G.4) Q-*E-»Sl\X,E)-> >€T(X,E)-*Q.
Passing to global sections, we get maps of vector spaces
G.5) 0 -> T(E) -> T(U\X,E)) - > r{U"(X,E)) -» 0,
where r(ft*) are the global sections of fi* over X.
Proposition 7.2.1. 7/V is aflat connection then G.5) is a complex: d% = 0.
We will denote this complex by W(X,E).
Thus, we have the cohomology of this complex H'DR(X,E), which is called
the (de Rham) cohomology of X with coefficients in the local system E, or simply
de Rham cohomology of the local system E.
If the local system E is trivial, i.e. E = C and V{ = <%, then for a C00 real
manifold we have H'{X,E) = H'(X,C), the usual cohomology of X. We will
discuss the complex case below.
We can also define the dual notion - singular homology with coefficients in local
systems.
Let Efc be a fc-dimensional simplex. Recall that a singular fc-simplex in X is a
continuous map a : Sfc —> X. A singular fc-chain with coefficients in E is a formal
sum ?\ Ejo-j , where the o~j are distinct singular fc-simplexes, and the ?j are flat
sections of the local system a\E over Sfc. Note that since T,k is simply connected,
every local system over it is trivial. The set of all fc-chains in X is obviously a
vector space (infinite-dimensional); we will denote it by Ck{X,E).
Now we need to define the boundary operator d : Ck(X,E) —» Ck~l(X,E).
Recall that the usual boundary operator d : Ck(X) -> Ck~1(X) is given by da =
J2j=o{-iysj, where Sj is the singular simplex defined by the restriction of a to
the j'-th face of Sfc (which is canonically identified with Sfc-i). Define the new
boundary operator by d(ea) = X)*=o(-1)'ejsj> where e, is the restriction of ? to
the j-th. face of S^.
It is easy to check that d2 = 0. Therefore, (C.(X,E), d) is a complex of vector
spaces. This complex is called the singular chain complex of X with coefficients in
E.
The homology of the complex C.(X,E) is called the singvlar homology of X
with coefficients in E, and is denoted by H,(X,E).
In classical differential geometry, differential forms can be integrated over sin-
singular chains. By virtue of the Stokes formula, this integration descends to a pairing
between homology and cohomology. De Rham's theorem claims that this pairing is
nondegenerate. It turns out that these results can be generalized to local systems.
Let\E be a local system over a manifold X with fiber V. Then differential k-
forms with coefficients in E can be integrated over singular fc-chains with coefficients
in the dual local system E* as follows. If u e Uk(X,E) and a : Sfc -> X is a
singular fc-simplex, then a*w is a differential fc-form on Sfc with coefficients in the

100 LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
local system a E: a u> € Clh(Eh,a E) = Clk(T,k) ® V (the last identity is possible
because S^ is simply connected). Thus, we can integrate this form over ?&. This
gives us an element of V, which can be paired with e 6 F/(Sfc, cr E ) ~ V to give
a number. This is by definition the integral of w over the cycle c = ea:
G.6) fw = (e, f a w).
Jc J^k
Proposition 7.2.2 (Stokes formula).
G.7) [du= [ w.
Jc Jdc
Proof. This follows directly from the usual Stokes formula. Q
Corollary 7.2.3. Integration of forms over chains defines a pairing
B:H (X,E )<$>HDR(X,E)^C.
De Rham Theorem. Let X be a C1 real manifold. Then the pairing B
is nondegenerate. Therefore, H (X,E ) is naturally isomorphic to HDR(X,E) ,
provided that at least one of them is finite-dimensional.
We can also easily define a version of the complex G.4) for a nonsingular
complex manifold, replacing the smooth forms by holomorphic ones. However, if we
define the de Rham cohomology as the cohomology of the complex G.5) of global
differential forms, in many cases we will get something trivial. For example, on
compact manifolds there are too few global forms. For this reason, the cohomology
of the complex G.5) for a complex manifold usually do not give much information
about the topology of X.
However, under some additional assumptions on X we still have an analogue
of the de Rham theorem. Recall that a complex manifold is called affine if it is
isomorphic to a closed subvariety in CN. In particular, any manifold of the form
X = C™ \ D, where D is a subvariety of codimension 1, is affine.
Complex de Rham Theorem. Let X be an affine complex manifold, and
let HDR{X, E) be the cohomology of the complex G.5) of global holomorphic forms.
Then the integration pairing B is nondegenerate. Therefore, H (X, E ) is naturally
isomorphic to HDR(X, E) , provided that at least one of them is finite-dimensional.
If X is a C1 real or affine complex manifold, we'll drop the subscript, denot-
denoting the cohomology of the complex G.5) simply by H (X,E), and calling it the
cohomology of the local system E. For nonaffine complex manifolds, this needs to
be changed (see below).
Corollary 7.2.4. For a complex affine manifold X of complex dimension n,
Hi(X, E) and Hl(X, E) may be nonzero only for i <n.
Note that as a topological manifold, X has dimension In, so it would be natural
to expect that Hi(X,E) may be nonzero for n < i < 2n.
For the reader familiar with sheaf theory, we mention that the simplest way to
prove the de Rham theorem both for a real C1 manifold and for an affine complex
manifold is to prove that the de Rham cohomology HDR(X, E) coincides with the
7.3. CONFIGURATION SPACES AND ORLIK-SOLOMON ALGEBRA. 101
cohomology of the locally constant sheaf ? of flat sections of E, after which we can
use standard algebraic topology results (see the proof for the C°° case in [GH]). To
prove the equality H'DR{X, E) = H'(X, ?) we note that by the Poincare lemma, the
complex of sheaves G.4) is in fact a resolution of the sheaf ?. Thus, we can calculate
H'(X, ?) as the hypercohomology of the complex of sheaves G.4). Finally, since the
sheaves Q*(X, E) are acyclic, this hypercohomology coincides with the cohomology
of the complex of vector spaces G.5). In the C°° case, the acyclicity follows from
the existence of a partition of unity for these sheaves (see [GH]). In the complex
case, it follows from the fact that every affine manifold is a Stein space and Cartan's
famous Theorem B (see [GR, VILA]).
This gives the most natural definition of cohomology of a local system E which
works for an arbitrary manifold X: it is just the (sheaf-theoretic) cohomology of
the locally constant sheaf ?:
H'(X,E)=H'(X,?).
In particular, if E is trivial then this cohomology coincides with the ordinary sin-
singular cohomology of X as topological space.
Many of the results above can be proved for algebraic varieties over C, and,
even more generally, for schemes over an algebraically closed field of characteristic
zero, as was shown in papers of Grothendieck and his students and collaborators.
However, discussion of this topic is definitely beyond the scope of our book.
7.3. Configuration spaces and Orlik-Solomon algebra.
In this section we show how one can explicitly calculate the cohomology of
certain local systems on the complements of a union of hyperplanes in Cn.
Let S be a hyperplane arrangement, i.e. a finite set of codimension 1 affine
hyperplanes in Cn, and let X be the complement of the union of these hyperplanes.
For any hyperplane H 6 S, specify a linear function iu in C™ such that H is the
set of solutions of the linear equation f#(z) = 0.
Let A = {\h}h€S be a collection of complex numbers. Define a multivalued
function ip on X by
G.8)
H€S
Definition 7.3.1. For every S, A as above we denote by C\ the one-dimen-
one-dimensional local system over X whose local sections are functions of the form ip(z)f(z),
where / is a holomorphic function in X, and the connection is given by Vj = <9<.
It is easy to see that the same local system can be described as a trivial one-
dimensional bundle with the connection given by
G.9)
V%f= dif +
In particular, the flat sections are given locally by / = ip l-
We will be interested in calculating the cohomology of this local system. This
cohomology has been studied by many authors; we only give brief statements of the
results, referrring the reader to the monographs [OT], [VI] for detailed exposition.
Define the Orlik-Solomon algebra OS'(X) as the graded exterior algebra gen-
generated over C by the differential 1-forms dlog?n for all H e S. This algebra is

102 LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
finite-dimensional, and it can be described by generators and relations, which was
done by Orlik and Solomon.
Every element / of the OS algebra can be considered as a differential form on
X with coefficients in the local system C\ by ui = tpf. We will call such forms
hypergeometric. It is obvious that hypergeometric forms form a subcomplex of
the complex n*(X,?A)- This subcomplex will be denoted by Q'hg(X,C\). The
following theorem was proved in [SV2j.
Theorem 7.3.2. For almost all A, including A = 0, the inclusion
is a quasi-isomorphism of complexes, i.e. it induces an isomorphism of the coho-
cohomology.
As usual, "for almost all" means "for all except for a countable union of subvari-
eties of codimension 1". For A = 0 this theorem was proved by Orlik and Solomon.
The full set of conditions on A was found in [ESV] and refined in [STV], where it
is also shown that for the local system appearing in the integral formulas for the
solutions of KZ equations these conditions are closely related to the Kac-Kazhdan
conditions B.29).
Theorem 7.3.2 implies that the cohomology of configuration spaces can be cal-
calculated with the help of hypergeometric forms. Since the space of hypergeometric
forms is finite-dimensional and we know all linear relations between them, this
essentially reduces the problem to combinatorics.
Example 7.3.3. Let X = C \ 0, ^(z) = zA. Then the only hypergeometric
1-form is zx^f, and the only 0-form is zx. Since d(zA) = \zx^f, we see that for
A = 0 we have H1 = C, and for almost all A 7^ 0, H1 = 0. In fact it is easy to check
that in this case H1 = C if A 6 Z and H1 = 0 if A ? Z; thus, Theorem 7.3.2 holds
for A^(Z\{0}).
Now, let us consider an even more special case of arrangements of hyperplanes
- so-called discriminantal arrangements, or configuration spaces.
Let X/v denote the space of all complex vectors z = (zi,..., zjv) ? C^ such
that z, 7^ Zj when i 7^ j. For m > 0, we have a natural map Xm+jv —> X/v which
forgets all components of the vector except the first N components. This map is a
fiber bundle. Denote the fiber over a point z 6 Xjv by Kz,m. Obviously,
G.11)
= {t = («,,... ,tm) 6 Cm\ti
zp}.
It will be important for us that we have a natural action of the symmetric
group 5jv on X/v; similarly, we have a natural action of Sm in every fiber KZitn.
Let us specify a set of complex numbers Q = {qpn,rPj, 1 < p < n < rn, 1 <
j < N}. As before, this set of exponents defines a one-dimensional local system
Cq on KZ)tn corresponding to the function ip given by
G.12)
p<n
The cohomology of this local system was calculated by Aomoto ([Al]) under
some non-integrality conditions, which generalize the condition A ^ (Z \ {0}) of
Example 7.3.3. As in all proofs of this type, he first proved that one can compute
7.4. COHOMOLOGY OF LOCAL SYSTEMS AND SOLUTIONS FOR fil2 103
cohomology using hypergeometric forms (which is nothing but Theorem 7.3.2 in this
special case) and then used combinatorial identities in the Orlik-Solomon algebra.
We will give his answer in the cases which are of interest for us in the next sections.
Note also that if the exponents qpn,rPj are symmetric under the action of Sm
(this is only possible if qpn = q,rpj = r,), then they also naturally define a local
system Cq on the quotient space Yz?m = Kz,m/5m. It is easy to show that
(.'•!<>) •" I" z,mt'-Q) = (ti (lz,mi Cq)) m.
This cohomology was also calculated by Aomoto.
7.4. Cohomology of configuration spaces with coefficients in local
systems associated with the Knizhnik-Zamolodchikov equations for sl2-
In this section we fulfill the promise given in Lecture 4. Recall that there we
wrote integral formulas for solutions of the KZ equations, which involved integrals
of multivalued differential forms. We promised to explain how to integrate such
forms later. That is what we will do in this section.
As before, let z = (zi,..., zjv), t = (ti,..., tm). Fix complex numbers (which
is the same as weights for s[2) fj,j,j = 1,...,N, and a parameter k e Cx. Now,
let us fix some z and consider the local system C = C((i,k) on Yz>m (see G.11))
associated in the sense of Definition 7.3.1 with the function
i — Zj) 2« J_J_(*p — ¦Zj) * J_J_('p -*n)"-
P,i P<n
G.14)
,frm(*. t) =
The product ]Ji< j B« ~~ zj) ^~ obviously is a constant on every Yx^m and thus does
not change the local system; we've included it to agree with previous notation and
for future use.
Recall that in Lecture 4 we have constructed solutions of the KZ equations for
0 = s\2 in the space V = M~L ® ... <g> M~N, where M~ is the lowest weight Verma
module with lowest weight — fi. These solutions had the form
G.15)
= Y] I
J
<MZ, t)ftn(z, t) dt ¦ Vn
where the sum is taken over all partitions m = (mi,...,mjv) such that mi >
0, "fT, "»i = m, um = emiVi ® ... Si emNvn and the pm are certain rational functions
of z,t regular on Y^^ (see D.28)). Thus, we see that the integrand is in fact an
element of the twisted space of holomorphic differential forms Q.m(YXtm,C)\ since
dimyzm = m, any such form is closed. Therefore, they can be integrated over the
cycles C 6 Cm(yZ|tn, ?*), and the integral in fact depends only on the class of C in
Hm(Yzttn, C). Thus, we need to know what the corresponding homology space is.
This can be deduced from a general result due to Aomoto ([Al]).
Let us start with the case m = 1, Kz>i = C \ {zi,..., zry}.
Theorem 7.4.1. Let m = 1. Then for almost all fj,, is:
1. H0(Yx,uC*) = 0.
2. dim.Hi(yZ|i,?*) = dimH1 (Yzd,C) = N - 1.
3. The forms
dt

LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
form a basis of if1^^,/^)
4. The Pochhammer loops (see Figure 4.1) d going around
basis in Hi(YZtUC*).
and zjv form a
As before, "for almost all" means "for all except a subvariety of positive codi-
mension". Later we will discuss when this theorem fails (see Section 7.7).
Corollary 7.4.2. For every z and almost all fiun, the map C >-> *o> where
*o is defined by G.15), is an isomorphism between H\{yx^,C*) and the space of
W-valued solutions of the KZ equations, where W is the space of singular vectors
of weight - ? m + 2 in V {see D.4)).
Now, let us consider an arbitrary level m. The following result was proved by
Aomoto in [Al].
Theorem 7.4.3. For any z and almost all fi,K we have:
1- Hi{YZ)m, ?*) = 0 for i < m.
2. dimHm(Kz,m>>C*) = dimHm{Y^m,C) = m!pjv-i(m), where pk(m) is the
number of partitions of m in an ordered sum of k nonnegative integers (see
D-6)).
However, we need a smaller space. Namely, it is easy to see that all the forms
ipmPmdt appearing in the integral formula G.15) are symmetric under the action of
Sm. Thus, the integral in G.15) depends only on the image of C in the symmetrized
homology space Hm(Y^m,C*)Sm = i?m(Kz,m,?*) (see the end of the previous
section). These homology spaces were also computed by Aomoto.
Theorem 7.4.4. For any z and almost all fj,,K we have:
1. Hi(Yz,m, ?*M~ = 0 for i < m.
2. As Sm -module, H^Y^,™, ?*) is isomorphic toppt-\(m) copies of the regular
representation, and thus
m, mN = 0,
Y^m,?.)Sm = pN-i(m).
3. The forms
G.16) ipm(z,t)pm(z,t)dt-vm, m= (jni,...,jnjv), ]P
form a basis in Hm(Y^rn,C)s-.
The basis in the homology space can also be described explicitly if we use
relative homology; we will do this in Section 7.6.
Corollary 7.4.5. For every z and almost all ^t, is, the map C i—> ^lc> where
^c is defined by G.15), is an isomorphism between Hm(Yx<m,C*)Sm and the space
W of singular vectors of weight — ^ fj,i + 2m in V (see D.4)).
In fact, this statement is true for arbitrary values of /ii (assuming that k is
generic), though this is much more subtle, since for some values of fn the dimension
of the space W can jump. We formulate here the corresponding statement, referring
the reader to [VI] for the proof.
Theorem 7.4.6. For every z,fj,i and almost all k, the map C >—> *c where
*c is defined by G.15), is an isomorphism between Hm{Y^^m, C)Sm and the space
W of singular vectors of weight — ]P Mi + 2wi in V isee D-4)).
7.5. GAUSS-MANIN CONNECTION.
This is closely related with the isomorphism between this homology space and
spaces of intertwining operators for quantum groups, which we will discuss later
(see Section 8.7).
7.5. Gauss-Manin connection.
To complete the analysis of the integral formulas for solutions of the KZ equa-
equations for 3B, we need one more step. So far we have developed cohomology and
homology theory for one local system. For the KZ equations, we need to consider
families of local systems. Indeed, we have a bundle
G.17)
i-JV+m ¦
¦XS
such that the fiber over a point z e Xjv is Ylrn (see Section 7.3). To construct
a solution, we must choose a cycle C in every fiber (at least, locally, i.e. in a
neighborhood of a point z), and in order for the proofs of Lecture 4 to be valid
these cycles must be chosen in some compatible way.
Here is the appropriate mathematical language. First of all, note that the
function xpm(z,t) defined by G.14) gives rise to a local system ?(fJ.,it) on XN+m,
and local systems on each Y^m are obtained by restriction of this one. Next, the
fibration G.17) gives rise to the vector bundle H (Xtf+m/Xf^,C ) whose fiber at
a point z is equal to H (Kz,m, C ).
It turns out that this vector bundle carries a natural flat connection, called the
Gauss-Manin connection. In our case it can be denned quite easily. Namely, let us
take some point z e X^. Let C 6 Hi(Yzrn,C ) be a cycle. Considering all V^m
as subsets in Cm, it is easy to see that in fact C defines a cycle in each Y^i m for
z° close enough to z (this also requires that we have a local system on X^+m, not
only on each fiber); thus, we have a local section of the bundle Ci(Xjv+m/Xjv, C ).
Let us define a connection in C (X^+m/X^,C ) by the condition that all local
sections obtained in the above described manner are flat. One easily checks that
this defines a flat connection in the homology bundle H (Xjv+m/X^,? ). Similarly,
we can pass to 5m-symmetric homology and define a flat connection in the bundle
H (XN+m/XN,C )s~.
Now we can summarize most of the results of the previous sections in the
following theorem.
Theorem 7.5.1. For any /m and almost all k, the map C
an isomorphism of the local systems
establishes
G.18)
Hm(XN+m/XN,C
where Hm(Xff+rn/Xfi,C )Sm is endowed with the Gauss-Manin connection and
Wkz is the trivial vector bundle over X^ with the fiber W = (V" )A, A = — X) Mi +
2m (see D.4)), endowed with the Knizhnik-Zamolodchikov connection.
In more elementary terms, this theorem can be formulated as follows.
Theorem 7.5.2. In the notation of the previous theorem, let z € Xjv and
let us identify the homology spaces Hm{Yzim) for z° in some neighborhood of z as
described above. Then for any fii and almost all k, the map Ch^c, defined by
G.15), is an isomorphism of Hm(Y^m)Sm with the space ofW-valued solutions of
KZ equations in the neighborhood of z.

LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
We finally note that Theorem 7.5.1 (or the equivalent Theorem 7.5.2) fails for
rational values of k. For example, it was shown in [FSV] that if k e Z+ and
the highest weights m < k, then the image of the map G.18) (in other words,
those solutions of the KZ equations that can be obtained from integral formulas)
coincides with the so-called bundle of conformal blocks, which is a subbundle of
Wkz- This is, of course, just one reflection of the deep structures of conformal field
theory which gives rise to the KZ equations. We will briefly outline this in the final
lecture.
7.6. Relative homology
So far, we have proved the general theorem which shows that we have suffi-
sufficiently many cycles to get all solutions of the KZ equations. However, we do not
have any construction of these cycles; even in the simplest case where the cycles can
be described in terms of the Pochhammer loops, it is still not clear how to integrate
over such cycles explicitly. It turns out, however, that we can replace these cycles
by usual simplexes.
Example 7.6.1. Let X = C \ {0,1} and let E be the one-dimensional local
system on X associated in the sense of Definition 7.3.1 with the function
G.19)
a,/3eC.
In this case for generic a, C we have Hi(X; ?"*) ^ C, and the generator of this
group is the Pochhammer loop C shown on Figure 4.1.
Let / be a meromorphic function on C, regular everywhere except possibly at
the points 0,1, and let Re a 2> 0,Re C » 0. Then, deforming the contour C as
shown in Figure 4.2, we see that we have the following result:
G.20)
f
Jc
z) dz = A - e27riQ)(l - e
2ni/3
f
Jo
/#(*) dz
(this was used in deriving the integral formulas for the hypergeometric function in
Section 4.2).
Now note that the identity G.20) shows that the integral /„' /(z)\I>(z) dz, which
is originally defined only for Re a 3> 0, Re C 2> 0, can be analytically continued for
all values of a, /3 ^ Z. This analytic continuation defines a functional on H1(X; E)
and thus an element of the homology group Hi(X;E*); we will denote this element
by [0, ljre9 and call it the regularization of the simplex [0,1].
Thus, we have seen in this example that the interval [0,1], which is not a cycle
in the sense of the definition in Section 7.2, still defines, for generic values of the
parameters, an element of the homology group by analytic continuation. This idea
goes back to Hadamard.
In the general case, we have a similar result. The corresponding notion here is
that of relative homology.
DEFINITION 7.6.2. Let X = X \ D, where X is a (complex or real) manifold
and D is a closed subset. Define the space of relative chains Ck(X,D;E) to be the
space of all finite linear combinations of the form ^elai, where cr* is a singular
fc-simplex in X (not X!) and ?j is a flat section of E over <Ji(Sfc) n X.
7.6. RELATIVE HOMOLOGY 107
Note in particular that all singular simplexes which are completely inside D
are equal to zero in Ck(X, D; E).
It is easy to check that the boundary operator d described in Section 7.2 is
correctly denned in C.(X,D;E) and d2 = 0. Thus, we get a complex, whose
cohomology is called the relative homology of X and denoted by H,(X,D;E). In
particular, if X = X then this coincides with the usual homology H,(X;E).
It obvious that we have an embedding Ck{X\ E) C Ck(X, D\ E) which is com-
compatible with d. Thus, we have a map of homology:
G.21) H.(X;E)-+H.(X,D;E).
In general, this map is neither injective nor surjective.
Example 7.6.3. In Example 7.6.1, we have [C] = (l-e27ria)(l-e27ri/3)[0,1] and
the map G.21) is surjective for a, C <? Z. In fact, it can be shown that for a, C ? Z,
this map is an isomorphism. Note that Hi(C, {0,1}; E*) = C • [0,1] independently
of a,/?, whereas the dimension of the usual homology ifj(C\{0,1};E*) does depend
on a, f3; for example, for a = C = 0, dim Hi (C \ {0,1}; E*) = 2.
Let us consider the hyperplane arrangements discussed in Section 7.3: X =
Cn \ union of hyperplanes, X = Cn and E = C\. Assume that all hyperplanes are
real, i.e. are specified by equations with real coefficients. Then we have (see [SV2])
Theorem 7.6.4. For generic values of the parameters \h the map G.21) is
an isomorphism.
We have already seen an example of such an isomorphism in Example 7.6.3.
This theorem can be refined: we can formulate some sufficient conditions for
this map to be an isomorphism. Let an edge be any subset L of the set of all
hyperplanes S such that flnei H ^ 0. Denote \L — ~52H€L \h- The following
theorem was proved in [SV2].
Theorem 7.6.5. If for any edge L we have \L <? Z, then the map G.21) is an
isomorphism.
In fact, it suffices to consider this condition only for so-called dense edges, but
this is too technical for our book.
Now we complement Theorem 7.4.3. Recall the notation Yx<m C Cm,C(fi, k).
Let us denote by Dzm the union of hyperplanes tt = tj,tt = zfc, so that Yz^m =
Cm \ -DZ]tn. It turns out that the corresponding relative homology is independent
of the local system and can be easily described, which was done by Aomoto ([Al]).
Theorem 7.6.6. For all ih,k we have:
1. i?i(Cm,?)Zim;?(/i,«;)) = 0 for i < m.
2. ie(z?R". Define
Let {Di} be the set of all bounded connected components ofY^m (each of
them is the interior of some convex polytope). Fix arbitrarily a section of
C(fi,k)over each Di. Then

LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
Combining this result with Theorem 7.6.4, which gives an isomorphism between
relative and usual homology, we see that for generic ft,itwe have
1- Hi(yim;?(/i,«)) = 0 fori < m.
2. tfm(Kz,m; C(fi, «)) = 0C • [Di\.
A typical simplex Di looks like this:
Zl < tjl < <ii < ¦ • • < 22 < tq < ¦¦¦ < Ztf.
There is an alternative basis in the homology space which can be used for
complex 2 as well. Let us consider all ways to represent the set {l,...,m} as
a disjoint union of N - 1 ordered sets. It can be described by a partition m =
(mi,...,mjv_i) such that m* > 0,?"i» = "i, and a bijective map 0 which assigns
to a pair a, 6, l<o<JV — 1,1<6< ma, an index i = <p(a, b) e 1,..., m.
Theorem 7.6.7. Lei zi,...,zjv 6e such that the intervals [z^zm], where
j = l,...,JV—1, do not intersect except at zn ¦ For every m,<f> as above let Dm(j>
denote the following subset ofYZtTn:
G.23)
= {h,...,tm\U =
0 <
Then for almost all (ii,K the simplexes L
mology, and thus also in the usual homology Hm(Yx,m,?(
These two types of bases are presented in Figure 7.1.
za- zN) ifi = <p(a,b),
form a basis in the relative ho-
hoz2
Figure 7.1. Two bases in the relative homology Hm(Yxrn,C).
Note that, similarly to the constructions of the previous section, we have a
natural action of the symmetric group Sm and a Gauss-Manin connection on the
relative homology groups.
Similarly to the ordinary homology, there is an appropriate dual object, which
can be described as a certain type of cohomology. This cohomology will not be
used anywhere else in the book, but it is interesting enough in its own right, so we
give a brief review. Let us first discuss the case of a real C°° manifold.
Definition 7.6.8. Let X be a real C°° manifold, and let E be a local system on
X. As before, assume that X = ~X\D for some X. Then the relative cohomology of
E (denoted H' (X, E)) is the cohomology of the following complex of vector spaces:
G.24)
7.6. RELATIVE HOMOLOGY
109
where ri(fi*) are the global sections of fi* over X which vanish in a neighborhood
oiD.
This is a special case of the more general notion of cohomology with a given
family of supports. For example, if X is compact then H, (X, E) coincides with the
cohomology with compact support.
Note that integration gives a natural pairing between the spaces of relative
chains C.(X, D; ?"*) and the forms r.(n*(X; E)). As_in Section 7.2, one can verify
that this pairing descends to a pairing between H.{X,D;E") and H*{X;E).
Theorem 7.6.9. Under the assumptions of the previous definition, the pairing
H.(X,D;E*)® HT{X;E)^C
is nondegenerate.
If X is a complex manifold, then the definition above does not make sense:
there are no holomorphic forms which vanish in a neighborhood of D. However, we
can rewrite the definition in a form which makes sense for complex X.
Recall (see the end of Section 7.2) that we have denoted by ? the locally
constant sheaf of flat sections of E, and the de Rham cohomology of E coincides
with the cohomology of the sheaf ?. Denote by j the (open) embedding j : X «-> X.
We quote the following well-known result (see, e.g., [Har]).
Lemma 7.6.10. For every sheaf F on X, there is a unique sheaf jiT on X
such that:
1. // U CJC is open and U C X, then j\F(U) = T{U).
2. IfxeX\X, then (j,f)x = 0.
For example, if T = W(X, E) is the sheaf of differential forms, then j\T is the
sheaf of differential forms which vanish in a neighborhood of D. Note that now this
definition works for complex manifolds as well.
Proposition 7.6.11. Let X be a C°° manifold. Then
H'(X,j,?)=Hf(X,E).
Sketch of proof. This follows from the acyclicity of j,U' and the exactness
of j\. ?
This proposition justifies the following definition, which now works in the com-
complex case as well.
Definition 7.6.12.
G 25^ fJ^(JC F1^ TT*C\ i f-\
Theorem 7.6.13. The pairing Hi(X,D;E)®H?(X,E) -> C is nondegenerate.
Note that there are no restrictions on the local system in this theorem.
It is tempting to write the usual homology Hl{X,?) also in terms of X, i.e.
find some extension j?? of the sheaf ? to X such that Hi(X,?) = H'(X,j-??).
Unfortunately this is impossible, but it becomes possible if we replace sheaves
by complexes of sheaves, and cohomology by hypercohomology. The appropriate
algebraic language is that of derived categories and perverse sheaves (see, e.g.,

110 LECTURE 7. LOCAL SYSTEMS AND CONFIGURATION SPACES.
[GM]). For readers familiar with this, we point out that we have the following
result:
IP{X,?) = IP(X,Rjt?).
7.7. The case of arbitrary 0.
In this section, we briefly describe what happens if we replace the Lie algebra
3B considered in the previous sections by an arbitrary simple Lie algebra g.
As before, let V = M~t <g> ... <g> M~N, where M~ is the lowest-weight Verma
module over q with lowest weight — fj,, and let W = (Vx)n be the subspace of
singular vectors of weight A = — Yl M» + M> where M = S h&i- Denote rn = \fj,\ =
E
Recall that we have constructed integral formulas for the solutions of the KZ
equations with values in W. These solutions have the form
= [ i>li{z,t)p(z,t)dt,
J
G.26)
where p(z,t) is a certain VA-valued rational function, C is an appropriate cycle
and ipp is given by D.43):
l<p<n<m
where v : {1,... ,rn} —> {1,... ,r} is a fixed map such that
Similarly to the discussion of the 5B case, it is clear that the function ip^ defines
a local system C on the same space Yim which we used before, and thus C must
be taken to be an element of the homology space Hm(YZtm, ?*). Again, in fact the
integral only depends on the symmetric part of C. However, the symmetry group is
now different. Namely, we must replace the group Sm used before by its subgroup
G.27)
Sfi = {se Sm\v{i) = i/(s(i)) for all i}.
Clearly, 5M is just a product of symmetric groups: 5M = S^ x ••¦ x Sir. The
important thing is that the local system C is invariant under the action of S^,
which follows from the symmetries of formula D.43), and thus we have a natural
action of 5M on the homology and cohomology of this local system.
Theorem 7.7.1. Let z 6 X^. As before, let us identify the homology spaces
Hm{Yx',m>?*) forz' in some neighborhood ofz as described above. Then for almost
all fj,i,« the map C 1—> \I>c> defined by G.35), is a surjective map of Hm(YZtm,C*)s>'
onto the space of W-valued solutions of the KZ equations in the neighborhood of z.
This shows that the integral formulas give all solutions of the KZ equations.
However, it is not true that this map is injective, even for generic values of the
parameters. It can be shown (see [SV2]) that the homology space Hm(Yz,m) <* for
generic n can be naturally identified with the space of singular vectors of weight
A in the tensor product of Verma modules over the the Lie algebra g', which by
definition is spanned by the same generators ei,fi,ht as g but without the Serre
7.7. THE CASE OF ARBITRARY 0. m
relations; thus, n'~ is just a free Lie algebra with generators /*. In order to get the
space of singular vectors in Verma modules over usual Lie algebra, we have to use
intersection homology, or, equivalently, perverse sheaves.

Lecture 8. Monodromy of the
Knizhnik- Zamolodchikov Equat ions.
In this lecture we study the monodromy of the KZ equations. We show that
the monodromy of the KZ equations gives rise to braided tensor categories and the
categories that originate from the KZ equations coincide with those arising in the
representation theory of quantum groups, reviewed in Lecture 6. The study of the
monodromy of the KZ equations was started in [TK]. It seems that the relation be-
between this representation and the one arising in the theory of quantum groups was
first noticed by physicists (see, for example, [AGS1], [AGS2]). The mathemat-
mathematical approach was developed by T. Kohno ([Kohl], [Koh2]) and V. G. Drinfeld
[Dr3],[Dr4]. Later it was refined by D. Kazhdan and G. Lusztig [KL1-4][L3],
who used extensively the representation theory of affine Lie algebras. This ap-
approach does not use Schechtman-Varchenko integral formulas for the solutions of
the KZ equations.
Another approach, based on a careful analysis of the geometry of cycles, was
started in [AGSl, AGS2], [FW1] for g = sl2 and was developed for arbitrary
simple Lie algebra by Varchenko ([VI],[V2]). The most general form of this result,
which uses the language of intersection homologies, can be found in [BFSj.
Throughout this lecture, g is an arbitrary simple Lie algebra and k ? Q.
8.1 Monodromy of Knizhnik-Zamolodchikov
equations and the braid group
In this lecture, we consider the KZ equations in the form D.1):
N
O I ^—i ^*Z7
k—ip = I y —
where ip is a function of Z\,..., z^j with values in the space V = V\ ® ... <g> Vjv,
where the Vi are representations of Q. In this section we always assume that the Vi
are finite-dimensional. Also, unless otherwise stated we assume that k ? Q.
As was discussed in Lecture 3, this system is consistent. Thus, it can be
interpreted as a flat connection in a trivial vector bundle with the fiber V over the
configuration space
X-N = {(Zl, ¦ ¦ ¦ ,zn) 6 C \Zi^Zj},
or equivalently, as a local system over Xn (see Lecture 7). In this language, solu-
solutions of the KZ equations are flat sections of the corresponding local system. We will
use the following notation: for any subset U C CN we will denote by Tf(U, Vkz)
the space of flat sections of the KZ local system over U.

114 LECTURE 8. MONODROMY OP KZ EQUATIONS
In this lecture we will discuss the monodromy of this local system. As in
Section 7.1, for any path 7 : [0, lj —> Xjv we denote by M7 the operator of holonomy
along 7. Since in this case all the fibers are identified with V, we consider M7 as an
operator in V. It is easy to see that this operator depends only on the homotopy
class of 7.
It follows from the analyticity of the KZ equations that if t/>Bi,... ,2jv) is a
solution in a neighborhood of a point z0, then the analytic continuation of ip is also
a solution. Therefore, the holonomy operator M7 can be described as the operator
of analytic continuation along 7.
Proposition 8.1.1. For any 7, M7 :V-^Visa $-homomorphism.
Proof. This follows from the g-invariance of Q. ?
Note that this proposition together with complete reducibility of V implies that
each M7 preserves the subspace of singular vectors in V and is uniquely defined by
its action on this subspace.
Let us fix a base point zo = B1,..., 2/v) 6 Xn- Consider the loops 7 such that
7@) = 7A) = zo; the corresponding operators M7 will be called the monodromy
operators. It is easy to see that in this case 7 >-> M7 is a representation of the
fundamental group tti(Xjv,zo) in V, which is called the monodromy representa-
representation. This definition depends on the choice of the base point; however, since Xjv
is connected, it is easy to see that if we choose another base point then all the
monodromy operators with respect to this new base point can be obtained from
the old ones by conjugation, and thus give the same representation of tt^Xjv).
The fundamental group of Xjv is well-known. Recall the braid group Bn and
the pure braid group Pi?jv(see Definition 6.4.2).
Theorem 8.1.2 (Artin).
1.
n1(XN)=PBN.
2. Let the symmetric group Sjv act on Xjv by permutation of variables, and let
be the corresponding quotient space. Then
TTi(Xff/Stf) = Bff.
We refer the reader to [Bir] for the proof, giving here only the construction
of the homomorphism Bff —> tti(Xn/Sn). Let us choose a base point zo =
(z\,... ,2;v) such that Zi 6 R, 21 > z% > • • • > 2/y. Then 6j corresponds to a
transposition of zt and zi+i such that Zi passes above Zi+1 (see Figure 8.1).
Z-,
¦ z.
Figure 8.1
In general, the KZ local system is not a local system over Xn/Sn, and thus
we do not have a representation of the braid group Bn in V. Instead, for every
a e SN let V = VCT-iA) ® ... ® Va-i(N), and let a : V -* Va be given by
8.2. ASYMPTOTICS OF SOLUTIONS OF THE KZ EQUATIONS 115
<j(i>i <g>... <g> i>jv) = i>c7-i(i) ® • • • ® vff-i^y Fix a base point z € D and let 7 be a
loop in Xjj/Sn (which we can consider as an element of Bn), Then we can lift it
to a path (not necessarily a closed loop) in Xjj. Define the operator
(8.1) M^ = aM^:V -^ V,
where a = GG) is the image of 7 under the canonical map Bjv —* Sn- In particular,
we define M^{z) = (M7j)±:l, where ji is the path shown on Figure 8.1.
Theorem 8.1.3.
f = Id,
(8.2)
Proof. The first identity is immediate from the definitions; the second follows
from the relation 7*7^+17^ = 7i+i7i7i+i in the fundamental group of Xfj/Sij. ?
Corollary 8.1.4. Assume that all representations Vi are isomorphic: V\ =
... = Vft = W, so that V = W®N. Fix a base point z e RN, zx > ¦ ¦ ¦ > zN. Then
the map bi 1—> M^(z) is a representation of the braid group Bn in V, which does
not depend on the choice ofz (up to isomorphism).
The main goal of this lecture is the study of this representation, in particular,
of the operators Mt. We will call these operators the "half-monodromy operators".
Note, however, that even in the SI2 case it is difficult to write explicit formulas for
these operators.
8.2. Asymptotics of solutions of the
Knizhnik-Zamolodchikov equations
As was discussed before, a change of the base point results in the conjugation
of the monodromy operators. It turns out that there is a preferred choice of the
base point, under which the monodromy operators take an especially simple form.
Namely, one has to take for z "infinite point at which 21 3> 22 3> .. - 3> zjv" • That
is, instead of considering the monodromy as an operator which relates values of
a solution and its analytic continuation at the base point z, we will consider the
operator which relates their asymptotics in the limit |zi| 3> \z2\ 3> ... 2> \zn\-
The rest of this lecture will be devoted to the study of these asymptotics and
corresponding "monodromy operators".
Prom now on, we fix the subset D C Xfi by
(8.3) D = {(z1,...,zN)eRN\zl>z2> ¦¦¦> zN}.
Note that D is simply-connected and thus the space Tf(D, Vkz) of V-valued
solutions of the KZ equations on D is isomorphic to V: if we choose a point zo € D,
then ip h—> V>(zo) is an isomorphism between the space of solutions and V.
To define the asymptotics, let us make the following change of variables, fol-
following [V2]:
(8.4)
u2 =
22 - .
2l - .
— Z\ — 22,
. , UJV-1 = -
2
Zi H hzjv.

116
LECTURE 8. MONODROMY OF KZ EQUATIONS
Then (zi,..., zjv) >—¦> («i, • • •, «jv) is one-to-one (moreover, the inverse mapping
is polynomial) and therefore we can consider any function / = /(z) on D as an
analytic function of ut defined on some subset Du C CN. Note that all the co-
coordinates «!,..., Uff-i are positive on D and that the closure of Du contains the
origin. This change of variables is chosen so that if we have a curve z(t) such that
z(t) -> 0 as t -> 0, then
¦Ui(t)
Now we can define limits and asymptotics.
Definition 8.2.1. Let / be a smooth vector-valued function on D. We say
that
lim ^ /(z) = v
for some vector v if, when rewritten as a function of u using (8.4), / satisfies
lim /(«) = v.
In a similar way, one defines asymptotics.
Definition 8.2.2. Let / be a smooth vector-valued function on D. We say
that / has the asymptotics <p{z)v as Z\ 3> • ¦ ¦ 2> zjv for some scalar function 4> and
vector v (notation: / ~ <f>(z)v) if one can write
f(z)=<P(z)(v + o(z))
for some vector-valued function o{z) which, considered as a function of u, can be
analytically continued to a regular function on some neighborhood of the origin in
CN, and o@) = 0.
Replacing in these definitions D by Do — {(zi,... , zjv)|zi > z2 • • ¦ > zjv =
0}, we get the definitions of limits and asymptotics as Z] > • ¦ ¦ > zjv-i 2> 0.
Note that if /(zl5... ,zn) is such that f(z1:... ,zn) = f{zi +c,...,zpj + c), then
lim2l3>...>3JV /(z) = limzi;s>...3>o /(¦*), and both limits exist simultaneously.
The study of asymptotics of solutions of the KZ equations is based on the
following lemma (see [V2]).
Lemma 8.2.3. In terms of the variables ui: the KZ equations take the following
form:
(8.5)
d
K- if) =
Q/V-1
Reg, t =
where Reg stands for some operator-valued rational function ofut regular atu = 0,
and
(8.6) n« = y;ny.
Proof. Explicit calculation (rather lengthy).
?
S.2. ASYMPTOTICS OF SOLUTIONS OF THE KZ EQUATIONS 117
Example 8.2.4. Consider the case N = 3. Then ux = zx - z2, u2 =
(z2 — z3)/(zi — z2), «3 = Z) + z2 + z3, and the KZ equations have the form
d
K-t W =
au
Q1
ft23
+ —
  +
(compare with D.13)).
This lemma, together with the consistency of the KZ equations, implies that
the operators (ii commute. Later we will construct a basis of common eigenvectors
for these operators.
Now we can formulate the key proposition of this section.
Proposition 8.2.5. Let us assume that k <? Q. Then:
1. For every vector v e V which is a common eigenvector ofQt with eigenvalues
Ui there exists a unique solution ipv of the KZ equations in D such that
3.7)
This solution is called the asymptotic solution corresponding to v.
Let va be a basis in V consisting of eigenvectors for Qi. Then the corre-
corresponding asymptotic solutions ipa form a basis in the space of solutions of
the KZ equations on D. Thus, we have the isomorphism of g-modules
and this isomorphism is independent of the choice of the basis va.
Proof. The proof of this theorem is based on Proposition 3.6.1. Indeed, if
we have a fundamental matrix solution F(ui,..., u^) of the KZ equations, written
in the form (8.5), then for every eigenvector v the function F(ui,..., u^)v is the
asymptotic solution corresponding to v, which is obvious from the definitions.
To check the assumptions of Proposition 3.6.1, we need to show that for each
k, the eigenvalues of the operator fit + • • ¦ + Ojv-i do not differ by a nonzero integer
multiple of k, which follows from irrationality of k (since all these eigenvalues are
rational — see the explicit formula for the eigenvalues in the next section). ?
Note that the condition k ? Q is essential: the map <j> has zeroes and poles for
certain rational values of k.
We will refer to the pair (D,<j> : Vf(D,VKz) — V) as an asymptotic zone.
One should imagine this asymptotic zone as an infinite point in a certain com-
pactification of D, and the isomorphism <f> plays the role of the usual isomorphism
4>z ¦ T/(-D, Vkz) — V defined for every finite point z & D by <t>zD>) = ip(z)-
The isomorphism <j> constructed above can be described more explicitly - at
least, for k large enough. Namely, we have the following theorem.

118
LECTURE 8. MONODROMY OF KZ EQUATIONS
Theorem 8.2.6. There exists a constant M, depending only on the represen-
representations Vi,..., Vfj, such that for |«| > M and for every solution rp{z) of the KZ
equations we have
(8.8) m = Ijm n1 «r<*+""K1-l)^W = ft|m>0 II ^"VW-
i=l i—\
Proof. It suffices to choose « such that all eigenvalues A of the sums fij +
¦ ¦ • + Qjv-i on V satisfy |A/«| < 1, and to check (8.8) for ip an asymptotic solution,
in which case it is obvious. ?
8.3. Asymptotics of the correlation functions.
We have mentioned that the operators fij, defined by (8.5), commute, which
can be easily deduced from the identity [ft 12 + ^13,023] = 0, already used in the
proof of consistency of the KZ equations. Let us construct the common eigenbasis
for these operators, assuming that each of the representations Vi is irreducible.
Choose some irreducible finite-dimensional representations Li,i = 0, ...,N — 2,
and nonzero intertwining operators ft : ?;_i —> Vi <g> Li, i = 1,..., N — 1, where
we let Ljv-i = Vjv- Consider the intertwining operator F = /jv-i • • • /1 : Lo —>
V\ ® ¦ ¦ . ® Vjv; graphically, it can be represented by the tree shown on Figure 8.2.
Figure 8.2
Then the vectors F(v), with v running through a basis in Lo, f°r all possible
choices of Li and ft form a basis in V\ <g>... <g> Vjv, which follows from complete
reducibility. Note that we have
(8.9)
> 1 ® a® A;
(a) (a in the zth place)
where a runs through some orthonormal basis in g and Afc : U($)
the comultiplication. Therefore,
QiF(v) =
i^) - C{Vt) - C(Li))F(v)
8.4. MONODROMY WITH RESPECT TO AN INFINITE BASE POINT.
119
where C(V) is the value of the Casimir element in V (see B.9)).
Thus, we have constructed the basis of eigenvectors for Clt in V. The next
proposition shows how we can construct asymptotic solutions using the represen-
representation theory of affine Lie algebras.
Theorem 8.3.1. Let v e Lo and ft be as above. Let /* : L* -* L\_x ® Vt be
the adjoint operator to ft. Assuming that Li ~ L\t, let
: VKjk
VXr_lik
Vi(zi), i = 1,..., N - 1,
be a Q-intertwiner, as in C.9), and let ip(zi,...,z^-i) = (v,&x ...<I>jv_r} ?
V\ <g> ... <g> Vn be the corresponding correlation function C.17). Then i>(z) is the
asymptotic solution of the KZ equations corresponding to the vector F(v).
Proof. This has already been proved. See Corollary 3.6.5. ?
8.4. Monodromy with respect to an infinite base point.
In Section 8.1, we have defined the notion of asymptotics of a solution of KZ
equations in the limit Zi > ¦ ¦ • > zN, which gave an isomorphism <fiZl»—^>zN :
Ts(D,VKz) ^ V (recall that Tf{D,VKz) denotes the space of solutions of KZ
equations with values in V on the domain D = {z\z\ > ¦¦¦ > z^}). One can
generalize this, defining other asymptotics. In particular, it is easy to define for
every permutation a e 5jv a notion of asymptotics as zff-i^ 3> • • • 3> zCT-i(jv),
which gives an isomorphism
(8.10) <pa : r(<r(D), VKz) ^ V
(we leave it to the reader to write a formal definition).
Let / be a V-valued solution of the KZ equations on D. Choose a point z E D.
Let 7* be a path in Xjv connecting z with cr(z) such that every point zt passes
above (for 7+) or below (for j~) the point Zj if j > i (this defines the path uniquely
up to isotopy). Then we can analytically continue / along this path to the point
<j(z). This gives us an isomorphism of spaces of solutions on D and cr(D).
Let M*(oo) : V —> V be defined as the following composition:
(8.11) M±(oo) : V ~ Tf(D,VKZ) ^ Tf(a(D),VKz) ^ V
where the isomorphisms are, respectively, taking asymptotics at 21 2> • • • 2> 2jv,
analytic continuation along the path 7* described above, and taking asymptotics
at 2CT-
A)
In particular, let
), and let
These operators are similar to the halfmonodromy operators Afj(zo), defined
by (8.1): if in (8.11) we replace the isomorphisms of taking asymptotics at 21 2>
• • • 2> 2jv, ^ct-i(i) 3> • • • 3> zCT-i(W) by the isomorphisms <^z, <t>a(x) of taking value at
z (respectively, <r(z)), then we get the definition of Afj(z). Sometimes the operators
Mj(oo) are also called the connection matrices.
Theorem 8.4.1.
t @0) M± @0)^@0).

LECTURE 8. MONODROMY OF KZ EQUATIONS
Proof. For |k| 2> 0, this theorem immediately follows from Theorem 8.1.3
and the following lemma.
Lemma 8.4.2. Let A(z) = YliLl1
Then for \k\ » 0,
^ (compare with Theorem 8.2.6).
(8.13)
= lim A(ai(z))Mi(z)A^1(z).
This lemma follows immediately from Theorem 8.2.6.
For arbitrary k € C \Q, the theorem follows from the statement for
analytic continuation.
0 by
Corollary 8.4.3. // all V = W,V = W®N, then bt .-» M*(oo) defines a
representation of the braid group Bn in V, which is isomorphic to the representation
given by monodromy with the base point z (see Corollary 8.1.4).
It will be shown in the next section that in a sense, the operators M;(oo) are
simpler than the operators Mi(z). However, they are still difficult to calculate
except in some very simple cases. The most important of these cases is described
by the following theorem.
Theorem 8.4.4. Let g = s\2, N = 3, and let Vi = V)ii,i = 1,2,3, be irreducible
finite-dimensional representations with highest weights in, such that the space of
singular vectors W = ((V^t <g> V^2 <8> V^,)n )A,A = —Mi — M2 ~~ M3 + 2, is two-
dimensional with basis W\,w2 given by D.14). Then the restriction of the operators
Mx (oo) to W in the basis Wi,u>2 is given by the following matrix:
^7riqr(i>-a)r(c) c^ila-c) rF-q)rB-,
(Q 1 A\ nVi ( \ V / " FF)r(c—a) FF—c+l)r(l —a) I v"-
where
(8.15)
Proof. The proof is based on the results of Section 4.2, where it was shown
that in the case considered here, the KZ equations essentially reduce to the hyper-
geometric equation in one variable x = (zi — z2)/(zi — Z3), and thus, its solutions
can be written in terms of the Gauss hypergeometric function. For convenience, we
replace the variable x by z = 1 — x = B2 — z3)/(zi ~ Z3)-
Let us denote by Wo, W^ the spaces of solutions of the hypergeometric equa-
equation D.16) with the parameters a, b, c given by (8.15), in a neighborhood of z = 0
(respectively, z = 00) with a cut along the ray z € R_. Each of these spaces is
two-dimensional. Denote by A^ the isomorphisms Wq —> W^, given by analytic
continuation in the upper (respectively, lower) haJfplane. These two analytic contin-
continuations are different, since the point z = 1 is a singular point of the hypergeometric
equation. Then it is easy to deduce from Proposition 4.2.1 and Theorem 4.2.2 the
following result.
/
12 (
1
-1
^c^7TiorF-a)r(c)
<J7Ti!.r(a-6)r(c)
1 \
Ml , ^-21
, c are given by
a =
M3 l Mi
—e~
= (
c -
piri(o-c)
1
M2 +
K
T(b-a)TB-c)
T(a-b)TB-c)
r(a-c+l)r(l-6)
M3
8.4. MONODROMY WITH RESPECT TO AN INFINITE BASE POINT. 121
Lemma 8.4.5. The operators Afj^oo) : W —> W can be written as follows:
(8.16) Mf^oo) = <t>aoA±D>Q)'1,
where the isomorphisms <p0 : Wo —> W, ^ : W^ —> W are obtained by taking the
asymptotics at 0 (respectively, oo) of the W-valued function
(8.17)
f(z) = F(z)Wl +(z- 1)—F'(z)w2,
M3
where F(z) € Wo (respectively, W^), and Wi,w2 are defined by D.14).
If we choose as basis of Wa the functions ui = 2Fi(a, b, c;z) and u2 =
z1~c2Fi(a-c+l,b-c+l,2-c;z), where 2-Fi is the Gauss hypergeometric function
(see D.19)), then it is easy to see, using the identities 2-F\@) = 1 and 2F{@) = ab/c,
that
(8.18)
w2,
. fJ.2 + ?13 - «
4>o(u2) = W2
(in fact, both ui and u2 are asymptotic solutions in the sense of Proposition 8.2.5).
Similarly, explicit calculation shows that as a basis in W^ one can take ui =
?-a2.Fi{a, 1 -c + a, l-ft + a^),^ = z^^^b, 1 - c + b, 1 - a + ft.z), and
(8.19)
3(tii) = wi -u>2,
(U2) = W \W
(again, ui and u2 are asymptotic solutions around oo).
The most difficult part is to relate ui,U2 with the analytic continuations of
ui, u2. This can be done using the following identity, valid for z $. R+ (see, e.g.,
[Bat]):
(8.20)
which gives
r(a - b)T(c)
T(a)r(c-b)
{-zy\F1(b, 1 - c + 6,1 - a + 6; z'
3.21)
rF-a)rB-c)
:
,i(»-c+i) r(a-b)TB-c)
+ e
Combining formulas (8.18)-(8.21), we get the statement of the theorem. ?

LECTURE 8. MONODROMY OF KZ EQUATIONS
8.5. Commutation relations for intertwining operators.
Let V be a finite-dimensional g-module, and let A, /x € P+ . Denote
(8.22) H*Y = Homg(IA, !„ ® V)
(compare with F.37)). As was discussed before, if k ? Q, then for every g € fl?vr
there exists a unique g-intertwiner 4>3 : V\k —> V^.fc ® V(z). Similarly, if V, W are
finite-dimensional g-modules and g\ € H?v,g2 € jH^vv> tnen define
We can extend this by linearity to define l>3 for every g € ©^ JT^ ® fl^ =
Homg(?A, LU®V ® W). This operator is well-defined as a function of 21,22 for
|zi| > 1221 as long as we have chosen a branch of Iog2i (see Corollary 3.6.4). In
particular, it is uniquely defined for z\ > z2 > 0.
Now, for z2> Zi > 0, define the operator A±&s(z1,z2) : Lx,k —* LVik®V(zi)®
W{z2) to be the analytic continuation of the operator ^9{zi,z2) along the path
when z\ goes above (respectively, below) z2. As usual, when talking about analytic
continuation of an operator, we mean the operator whose matrix coefficients are
analytic continuations of the corresponding matrix coefficients of <&3.
Theorem 8.5.1.
1. The operators A±<&g{z\,z2) are well-defined for z2 > Z\ > 0, and are g-
intertwiners.
2. For z2 > zi > 0,
(8.23)
where the exchange matrices B*1 = ByW(v", A*) is given by
Bvw(i/*,\*) : Homg(L*,y ®W® Lx) -> Homg(L*, W ® V ® L\),
g i—>A/i (oo) ° g*
Here Mf (oo) : V ®W ® L\ —> W ®V ® L*x is the half-monodromy operator
(8.12), and we identify
(8.24)
Homg(IA, Lu ®W®V)= Homg(i;, W ® '
Homn(LA, Lt, ® y ® VP) = Homo(L*, V^ ® W ® Lt).
Proof. The fact that the operator A±$9(zi,z2) is a g-intertwiner is obvious,
since conditions of commuting with the action of jj are given by a system of equations
whose coefficients are analytic functions of z. Next, let us fix an element u € L*v
and consider the correlation function 4>i,{zi, z2) = (u,<&9-) € V®W®LX. Since the
operator is uniquely defined by its correlation function (see Lecture 3), it suffices
to check that ]/)? can be analytically continued. But Vs satisfies the KZ equations
C.19) in N = 2 variables, and thus can be analytically continued. Similarly, to
prove B), it suffices to check that the correlation functions on both sides coincide,
which follows from the fact that Vu is the asymptotic solution in the region |2i | S>
\z2\ corresponding to the vector {u, </(¦)} € V ® W ® L\ (see Theorem 8.3.1), and
from the definition of the operator M. ?
8.5. COMMUTATION RELATIONS FOR INTERTWINING OPERATORS. 123
This theorem implies the following factorization of monodromy operators.
Corollary 8.5.2. Let Vi,..., VN be finite-dimensional representations of g.
Then the action of the operators Af±(oo), describing the half-monodromy of the KZ
equations with respect to an infinite base point (see (8.12)), on the space
Roms!(,LXo,V1®...®VN)= 0
is "local":
(8.25)
where
f-(oo) =
is given by (8.24).
Note that Corollary 8.5.2 can be proved without reference to correlation func-
functions, by analysis of behavior of the monodromy operators M7(z) as z -» oo, but
such a proof would be much more technical and boring.
Thus, we see that the structure of monodromy with respect to the infinite
base point is encoded by one operator fiy v, v = Af*(oo) : V\ ® V? ® Vz —>
Vi ® Vi ® V3, which relates the asymptotics of solutions of KZ equations in 3
variables 2^22,23 (when written in the form C.20); if we rewrite them in the form
C.19), they become equations in 2 variables) in the asymptotic zones z\ s> 22 3> 23
and 22 3> z\ 3> 23. As we have shown before, these equations can be reduced to
just one ODE in one variable (see Lemma 4.2.1). This is the main advantage of
the operators Ma(po), describing the relation between asymptotics of solutions in
different asymptotic zones, over the the monodromy operators Ma(z), describing
monodromy-with respect to a base point z: the operators Ma(z) do not have a
factorization like (8.25).
It follows from Theorem 8.4.1 that the operator /? satisfies the following iden-
identities:
(8.26)
= Id,
where we use the same conventions as in F.18).
Rewriting these identities in terms of the operator
: Homg(XA, V ® W ® ?„) -+ Homg(IA, W ® V ® Lv),
(see (8.24)), we get the following proposition.
Proposition 8.5.3.
(8_27) B± 5^=Id,

124 LECTURE 8. MONODROMY OF KZ EQUATIONS
where we use the same notation as in Proposition 6.8.1.
Finally, we note that the operators Bvw{\,i') can be defined for generic A
(provided that V, W are finite-dimensional). Repeating the constructions of Sec-
Section 6.8, we can also define the operator Bvw(^) and check that it satisfies the
dynamical Yang-Baxter equation F.45).
8.6. Equivalence of categories and Drinfeld-Kohno theorem
Identities (8.26), (8.27) for the operators /3,B are the same as the identities
F.18), F.40) for the commutativity morphism in the category of representations
of a quantum group. However, the operator j3 clearly is not the same as the usual
commutativity morphism in the category of finite-dimensional representations of g
(given by v ® w >—> w <g> v), since its definition uses the monodromy of Knizhnik-
Zamolodchikov equations. Therefore, one naturally asks if there is a structure of
a braided tensor category on Rep g which would give this /?. Such a category was
constructed in [Dr3, Dr4] and is called Drinfeld category.
To describe this category, let us again consider KZ equations in 3 variables
Zi,z2,z3. Recall (see Lemma 4.2.1) that these equations can be reduced to just one
equation
where x = (zi - z2)/(zi - z3). Then the asymptotics of the original KZ equations
as z\ » 22 » 23, 22 > 21 > Z3 are tne same as asymptotics of (8.20) as x —» 1
(respectively, x —> —oo).
However, equation (8.20) has one more singularity x = 0, which in terms of the
original variables 21,22,23 corresponds to 2! - 23 » 21 - z2 > 0. It is convenient
to assign to each asymptotics a tree as in the figure below.
V,
v2 M
Z2» Z, » Z3
Z,-Z3»Z1-Z2
Figure 8.3. Trees illustrating different asymptotic zones for the
KZ equations in 3 variables.
Thus, we can define, similarly to the definition in Proposition 8.2.5, the iso-
isomorphism
(8.29) 4>Zl_Z3>2l_Z2>0 : Tf(D, VKZ) = V,
where, as before, Tf(D, Vkz) is the space of solutions of KZ equations in 3 variables
on the domain D = {zi > 22 > 23}.
8.6. EQUIVALENCE OF CATEGORIES AND DRINFELD-KOHNO THEOREM
Define Drinfeld associator *vi,v2,v3 : (Vi ® V2) ® V3
(V2 ® V3) by
(8.30)
: V
, VKZ)
V.
Theorem 8.6.1. Let k ? Q, and let C(g) be the category of finite-dimensional
representations ofg. Then: aVlv2v3 = $VlV2v3,ffVjV2 = Penin^, and\,p the same
as in the category of vector spaces, define on C(fl) a structure of a braided tensor
category {see Section 6.3). In this category, the morphism A^VjVa : Vi®(V2®V3) ->
V2 ® (Vi ® V3), defined as in F.17), coincides with the half-monodromy operator
Mf(oo), defined by (8.12).
We will denote this braided tensor category by C(fl, k).
We refer the reader to the original papers of Drinfeld or to the expositions in
[CP2, Kas, SS] for proof of this theorem, which is not difficult. For example,
the pentagon axiom, which claims that two isomorphisms ((Vi ® V2) ® V3) ® V4 —>
Vi®(V2® (VJi®^)) are equal, follows from the fact that both isomorphisms describe
the relation between asymptotics of solutions of KZ equations in the asymptotic
zones 21 — z4 » zi - z3 » Zi — 22 and Z\ » Z2 > Z3 ~> 24. Since the region
Zi > Z2 > 23 > Z4 is simply-connected, the operator relating asymptotics is uniquely
defined. The hexagon axiom can be proved in a similar way. Finally, to check
that the half-monodromy operator Aff^oo) is given by *v2ViV3Pi2e±'rir!i:!/'c$^V v
(which is formula F.17) for the operator 0), we note that in terms of equation
(8.20), M* is the operator of (suitably renormalized) holonomy along the path
connecting x = 1 with x = -00 in the upper (for plus sign) or lower (for minus
sign) halfplanes. Each of these paths can be deformed to a composition of 3 paths
as shown in Figure 8.4, and the holonomies along these three paths are exactly
$i7rin//c d $
Figure 8.4. Illustration of Mf(oo) = $v-2vi
Remark 8.6.2. The operator pe^a/^ in fact appears as the operator relating
the asymptotics of KZ equations in 2 variables in the zones Zi > z2 and z2 > Zi.
Remark 8.6.3. It can be shown that different asymptotic zones for the KZ
equations in n variables can be identified_with the maximally degenerate points
in the Deligne-Mumford compactification Mo,n of the moduli space of genus zero
surfaces with n marked points and nonzero tangent vector at these points. These
points are labelled by trees with n leaves similar to those in Figure 8.3.
By now, we have two braided tensor categories, which both are deformations
of the category of finite-dimensional representations of g: the category C(g, q) of
representations of the quantum group Uq(g), defined in Lecture 6, and Drinfeld's
category C(g, k), which coincides with the category of representations of g as an
abelian category, but has associativity and commutativity morphisms defined in
terms of asymptotics of KZ equations with the parameter k <? Q.
The following fundamental theorem, which was proved by Drinfeld [Dr4] over
the ring of formal power series in ?, and by Kazhdan and Lusztig [KL1-4, L3)

126 LECTURE 8. MONODROMY OF KZ EQUATIONS
for numerical values of k, establishes equivalence of these categories (for an exact
definition of what it means for two tensor categories to be equivalent, see, for
example, [Kas]).
Let
(8.31) m=p23l,
where cti and as are, respectively, the long and the short roots in R. Thus, m e
{1,2,3}, and m = 1 for simply-laced root systems.
THEOREM 8.6.4. Let the ¦parameter k of the KZ equations be irrational, and let
Uq(o) be the quantum group with the parameter q = emlmK, where m 6 {1,2,3} is
defined by (8.31). Then the categories C(g, k) and C(g, q) defined in Theorems 8.6.1
and 6.6.5 respectively are equivalent as braided tensor categories.
The number m appears in the denominator to compensate for the difference in
normalizations of the bilinear form (,} used in the theory of affine Lie algebras and
the bilinear form ({,)} used in the theory of quantum groups.
The proof of this theorem is rather complicated, and we refer the reader to the
original papers or their exposition in [CP2]. Roughly speaking, Drinfeld proved
that the category of representations of g has a unique deformation as a braided
tensor category (over C[[^j]), and thus any two deformations must be equivalent.
Kazhdan and Lusztig constructed the isomorphism almost explicitly: they showed
that the isomorphism can be obtained by iterations of the formulas for the sim-
simplest possible case, when the isomorphism can be written explicitly in terms of
T-functions (similar to Theorem 8.4.4), and then analyzed the zeroes and poles of
these formulas.
Corollary 8.6.5. Under the assumptions of Theorem 8.6.1, let W be an ir-
irreducible finite-dimensional representation of Q, and let W be corresponding repre-
representation ofUq{o). Then the representations of the braid group Bn on the subspace
of singular vectors V~ C V = W®N, given by monodromy of the KZ equations
with the parameter k (see Theorem 8.1.3 and Corollary 8.4.3) and on the space
(y?)M,(n-) c yq _ (wi)®N, given by the universal R-matrix for Uq(g) (see Theo-
Theorem 6.4.3), are isomorphic.
This is a special case of the Drinfeld-Kohno theorem, which claims that under
the assumptions of Corollary 8.6.5, the representations of Bn on V and V are
isomorphic. This theorem was proved in some special cases by Kohno [Kohl], using
deformation arguments; the general proof can be deduced from Corollary 8.6.5.
Finally, let us note that there exists a highly nontrivial modification of The-
Theorem 8.6.4 for k € -hv - Q+, also due to Kazhdan and Lusztig [KL]. In this
case, the category C($, k) should be taken to be a category of g-modules of level
k satisfying certain conditions, and the tensor product is usually called the "fu-
"fusion tensor product". This category is no longer semisimple and thus it can't be
defined in terms of the representation theory of g. Using this result, Finkelberg
[Finj proved an analogue of Theorem 8.6.4 for k € Z+. In this case, the category
C(g, k) is replaced by the category of integrable modules of level k with the fusion
tensor product, and the category C(g,<j), which in this case is not semisimple, is
replaced by a certain semisimple quotient. We refer the reader to [Fin) for the
exact formulation and for the proof of this theorem.
8.7. GEOMETRIC APPROACH TO EQUIVALENCE OF CATEGORIES 127
8.7. Geometric approach to equivalence of categories
In this section, we will outline another approach to the equivalence of the
categories C(g,«) and C(g, q). We will only consider it for g = sh, which is the only
case admitting explicit formulas and not requiring the use of intersection homology.
Till the end of this section, let g = sl2 and fix arbitrary weights m,...,nN
(which may be integers or not) and a number m € Z+. Denote A = /ii H
2m. As in Lecture 7, denote
where M~ is the lowest-weight Verma module with lowest weight —/j,. Similarly,
replacing sI2-modules by W,(sI2)-modules, we define
Recall the symmetrized homology space Hm(Yx,m; ?*)s™ (see Section 7.4);for
brevity, we will denote it by -ff(z). Then we have the following result, obtained by
Varchenko in [VIj.
THEOREM 8.7.1. Let z± > ¦ ¦ ¦ > zN, q = ewi/K. Then for almost all k, there
exists an isomorphism <pz : W*14iN -> H(z) such that
1. For every w € W^ )iN, (p^(w) is a flat section over the region zx > ¦ ¦ ¦ >
zn of the bundle H(z) with respect to the Gauss-Manin connection (see
Section 7.5).
2. The following diagram is commutative:
pi
H(z)
where Oi(z) = (zx,... ,zi+i,Zi,... ,zn) and Ti is given by the monodromy of
the Gauss-Manin connection on H(z) along the path in Figure 8.1.
We refer the reader to [Vlj for the construction of <pz and for the proof of this
theorem, giving here only the simplest example.
Example 8.7.2. Let m = 1, i.e. A = ^ + h fj,N - 2. Choose a point z0 € R
such that zQ > Zl, and denote C = {c € d(C\{zi,... ,zN},C*)\dc € C-[z0]}, where
? = Am, «) is the local system defined in Section 7.4, and Ci(X, ?*) denotes the
space of singular 1-chains with coefficients in the local system ?*, see Section 7.2.
In other words, we are considering 1-chains such that their boundary is a multiple
of the point z0.
For any i = 1,... ,7V, choose a chain Dt € C as in the following figure (this
defines D; up to isotopy):
with the section of local system chosen by the condition that for real fii,K this
section is positive for zi > ¦¦¦> zt > t > zi+l > ... zN. It is easy to check that this

128 LECTURE 8. MONODROMY OF KZ EQUATIONS
definition can be analytically continued in parameters /ij, k to give a well defined
section of C for all values of k ^ 0, fii (the pedantic reader may find this definition
in [VI] or in [FKV]).
Denoting for brevity the lowest-weight Verma module over Uq(sl2) by Mi =
M~, define the map
*C,
*Dt.
Proposition 8.7.3. For almost all k, restriction of the map 4>za,--,zN *° the
subspace of singular vectors gives an isomorphism WJ,,...l/iA, — H(z), which is
independent of the choices in defining z0 and the cycles Dt.
Proof. If we define the numbers q by A/(t)i ® ... ® evi <S> ¦ ¦ ¦ ® v^) = <HVi <8>
... ® Vn, then explicit calculation shows that dDi = (q - g-1)Ci[ao]- This proves
that the map <j> gives rise to a map Wj>14iN —> H(z). The proof of the remaining
statements is straightforward. d
Now it is easy to check by calculation, using the explicit formula F.29) for the
.R-matrix, that such a map <j>z : W^ ^N satisfies all the properties formulated in
Theorem 8.7.1.
Now, let us recall that there is an isomorphism between the bundle H(z) with
the Gauss-Manin connection and the trivial bundle Wkz with fiber W = W/J1 ^N
with Knizhnik-Zamolodchikov connection (see Theorem 7.5.2). Let us define iso-
isomorphism of vector spaces W^ ...iMw —• W/Jli,,.^N as the composition
W2,.....mw "^ rf(D,H(*)) -> Tt{D, WKZ) -> W^,...,^,
where Tf(D, Wkz) denotes the space of VF-valued solutions of KZ equations on
the domain D = {z|zi > • • • > zN}, and similarly for Tf(D,H(z)), and the last
arrow is the isomorphism of taking asymptotics at z\ 3> •• • ^> zn (see (8.7)).
Then it follows from Theorem 8.6.3 that this isomorphism identifies the action of
.R-matrix on Wq with the half-monodromy operators Af+(oo) on W. Moreover,
this isomorphism also identifies the associativity constraints in C(jj, q) and C(g, k)
(see [VI, V2]). Finally, this isomorphism can be written explicitly in terms of the
asymptotics of certain integrals, which for g = SI2 coincide with the Selberg integral
and thus can be calculated explicitly. This calculation is done in [V2].
This is not yet a construction of the equivalence of categories C(g, q) and C(jj, k),
since this result concerns Verma modules rather than finite-dimensional ones, but
it is clearly very close.
It turns out that the relative homology spaces Hm(Cm; Dx,m;C*) (see Sec-
Section 7.6), which we will for brevity denote H\(z), can also be naturally interpreted
in terms of representations of quantum groups. Denote
- r ®.
where (M^)c is the module contragredient to the lowest-weight Verma module (see
Section 6.2). Then we have the following analogue of Theorem 8.7.1, also proved
in [VI].
8.7. GEOMETRIC APPROACH TO EQUIVALENCE OF CATEGORIES 129
Theorem 8.7.4. Under the assumptions of Theorem 8.7.1, there exists an
isomorphism ipx : {Wc)(>ii ^N -> H,{z) such that
1. For every w € (Wc)^l^IJN, tpz(w) is aflat section over the region zx >
¦ ¦ ¦ > zn of the bundle Hi (z) with respect to the Gauss-Manin connection
[see Section 7.5).
2. The following diagram is commutative:
(wcy
Again, we refer the reader to [Vlj for the construction of ipz and the proof of
the theorem. Construction of ^>z can also be found in [FKV].
Finally, let us discuss the canonical map H(z) —» Hi(z) (see Section 7.6). The
following theorem can be found in [VI].
Theorem 8.7.5. Under the isomorphisms H(z) ~ W,Hi(z) ~ {Wc)i, con-
constructed in Theorems 8.6.3, 8.6.5, the map H{z) —> H\(z) is identified with the map
S : W —> (Wc)q given by the restriction of the tensor product of canonical maps
Therefore, one sees that the canonical map between the usual and relative
homology is an isomorphism if and only if the restriction of S is an isomorphism
(provided that k is generic); thus, it will be so if fi{ ? Z+.
If we recall that usual and relative homology can be interpreted as homology of
Cm with coefficients in the sheaves Rj,?,j\S (see the end of Section 7.6), then these
theorems can be interpreted as a certain correspondence between representations
of Uq(sl2) and various sheaves on Cm extending the original local system on Yz<m.
It turns out that this correspondence exists for much more general representations
than Verma modules or contragredient Verma modules, which are two "extreme
cases" of highest-weight modules. Similarly, both usual and relative homology
are extreme cases of so-called intersection homology, introduced by Goresky and
MacPherson; the corresponding dual notion, generalizing the sheaves j\S,Rj,S,
is the notion of perverse sheaf. Detailed exposition of this theory goes beyond
the scope of this book; we refer the reader to Borel's excellent paper [Bor] for
an introduction to intersection homology and perverse sheaves. It was recently
shown in [BFS] that the category of all highest-weight representations of Uq(g) can
be identified with a certain category of perverse sheaves on configuration spaces;
moreover, this holds for nongeneric q as well. For example, irreducible highest-
weight modules correspond to the intermediate (or Goresky-MacPherson) extension
juS. This is a very far-reaching generalization of Theorems 8.7.1 and 8.7.4, and
can be used to give another construction of the equivalence of categories.

Lecture 9. Quantum Affine Algebras.
Our purpose now is to generalize the theory of KZ equations to the quantum
case. We will consider the representation theory of quantum groups Uq(g), corre-
corresponding to afflne Kac-Moody Lie algebras. While the highest-weight representa-
representations are similar to the undeformed case [LI], the evaluation representations are
more subtle. In the case when g = sin, M. Jimbo [J2j constructed a homomorphism
p : Uq(g) —> Mq(s), which yields a class of evaluation representations, further studied
in [CP1, CP2]. The intertwining operators between a highest-weight module and
a tensor product of another highest-weight module and an evaluation representation
for quantum affine algebras were first studied in [FR.].
In the last section of this lecture we give a quantum analogue of the Sugawara
construction in degree zero, which will be used in the next lecture in the derivation
of the quantum KZ equations.
To simplify the exposition, we assume that g is simply-laced, so that (a, a) = 2
for all roots a.
9.1. Definition of quantum afflne algebras.
In Lecture 6 we introduced the quantum group W?(a) corresponding to an ar-
arbitrary Kac-Moody Lie algebra a and defined Verma modules and highest-weight
representations of liq(a). Prom now on we will consider the case when a is an afflne
Lie algebra: a = g, where g is a finite-dimensional simple complex Lie algebra. The
Hopf algebra Uq (g) is called the quantum affine algebra. It will also be convenient
to consider a bigger Hopf algebra Uq(g), which corresponds to the extended Kac-
Moody algebra g (see Section 2.6). By definition, Wg(g) is obtained from Uq(s) by
adjoining elements qad,a € C. The relations for qad can be easily derived from the
following formulas:
(9-1) [d, e0] = e0, [d, f0] = -/<,, [d, qh°] = 0,
A(d) = d ® 1 + 1 ® d, e(d) = 0, 7(<i) = -d.
Note that these formulas do not make sense formally, since d is not an element of
Uq{o); rather, they should be considered as a shorthand for the relations involving
As in the classical case, every highest-weight module over the algebra Uq(g)
extends to a module over Mq(jj): the action of d on a homogeneous vector w of
degree mis dw = (—A — m)w, where A is an arbitrary constant. The most natural
choice of A, as in the classical case, is given by formula B.25).

132 LECTURE 9. QUANTUM AFFINE ALGEBRAS.
We will use the notation Ug(g+) for the subalgebra in Uq(g) generated by ei
and qhi, and the notation Uq(g~) for the subalgebra in Uq(g) generated by /, and
qhi. We denote by ^(g*) the Hopf algebras obtained by adding the elements qad
to the Hopf algebras W?(g±).
Define the Hopf subalgebra Uq(g-°) C Uq(g) generated by qhi,eu i > 0, and
fiti > 1, and qad.
For every irreducible highest-weight module Lqx over Uq(g) we define the induced
module V^k = W,(fl) ®Mg(g>°) ?*> where the action of Uq(g-°) in L\ is defined by
the relations e0 |x,i= 0, d \l\— —A, gA° |lj = gfe-(flV'A>, where 0 is the maximal
root of g.
As before, we denote by (,} the invariant bilinear form on I) defined by B.17).
We define p = p + AvAo € t); then (/5, a/) = 1,» = 0,..., n. As usual, we identify
lj ~ fj* using the bilinear form; under this identification, we will have p = p + hy d.
9.2. Evaluation representations of quantum affine algebras.
The next ingredient that we need to develop a quantum analogue of the theory
of KZ equations is the construction of evaluation representations of Uq(g).
Let us recall the construction of evaluation representations in the classical case
(see Section 2.8). We took an arbitrary representation nv : U(g) —» End(V), and
then composed it with the evaluation homomorphism
(9.2)
pz :U(g)
: pz{a® tm) = zma, a€g,2€<C*
pz(c) = 0.
The resulting representation irv a pz was called an evaluation representation.
In the quantum case, the evaluation homomorphism has to be defined differ-
differently since the elements a <g> ?n do not make sense any more. One of the acceptable
definitions is as follows.
For z € Cx, define the "shift" automorphism Dz ofUq(g) by
(9.3) Dz(e0) = ze0, Dz(fo) = z~Y fa, Dz = Id on the rest of generators.
(One can think of Dz as conjugation by zd).
Now assume that p : Uq(g) —» Uq(g) is a homomorphism of algebras (not of
Hopf algebras!) which acts as the identity on ei,fi,qhi with i > 0. Then we can
construct a family of homomorphisms
(9.4)
pz =poDz.
It is natural to call them evaluation homomorphisms: in the classical case this
construction gives exactly the homomorphisms (9.2) (for p = pi).
Jimbo [J2] proved the existence of a homomorphism p for g = sin.
Theorem 9.2.1. There exists a homomorphism p : Uq(sln) —> Wq(sln) which
acts as the identity on ej, /j, qhi with i > 0.
Proof. The proof of the theorem is by an explicit construction.
Recall that the Lie algebra sin is spanned by the elements En — Ei+i<i+]_,
1 < i < n — 1, and By, i ^ j, 1 < i, j < n, where Etj is the n x n matrix whose
entries are all zero except the ij-th. entry, which equals 1. Let us define elements
similar to ?y in the quantum group Uq(sln).
9.2. EVALUATION REPRESENTATIONS OF QUANTUM AFFINE ALGEBRAS. 133
The definition uses induction. First of all, one sets Eiti+1 = et, Ei+iti = fa.
Next, assume that Eij have been defined for 0 < \i — j\ < k. Now let i, j be such
that \i — j\ = k. Then one sets
(9.5)
(9.6)
It can be checked that the elements E^ thus defined satisfy the relations
E^ = EuEij - qEijEa, i < l< j,
Ei:i = EuEij - q-'EijEu, i > I > j.
Define the map p by
n-l
(9.7) p(e0) = Enl, p(/0) = Eln, p(qh'>) =
Now we need to check that p extends to a homomorphism of algebras. For the
sake of brevity, we give a complete proof of this fact only for the case of al2.
If n = 2 then the definition of p reduces to
(9-8) p(e0) = f, p{fo) = e, p{qh°) = q'h,
and we have
(9-9) pz(e0) = zf, pz(f0) = z~le, pz{qh") = q~h
(here we abuse the notation by writing e, f,qh instead of ei, /i, qhl).
All the relations are immediate except
(9.10)
= 0,
/o3/i - [3],/02/i/o + [3]q/o/i/o2 - /i/o3 = 0.
Clearly, it is enough to check only one of them. To check that p respects the
second relation in (9.10), it is necessary to show that
(9.11) [e3,/] = [3],e[e,/]e.
Using the Leibnitz rule, we have
¦ +
q-q-
q-q-1
q-q-1
?
The existence of the homomorphism p implies that any highest-weight rep-
representation of the "finite-dimensional" quantum group Uq(g) extends to a repre-
representation of the corresponding quantum affine algebra. Furthermore, using Jimbo
homomorphisms, we can assign to every representation iry : Uq{g) —> End(F) a

134 LECTURE 9. QUANTUM AFFINE ALGEBRAS.
one-parameter family of representations of Uq(g) in the same space V parameter-
parameterized by a nonzero complex number z: ftv(z) = "V °J2. We denote this family of
representations by V(z).
If g is not sin, then the homomorphism p with the required properties does
not exist, so it is not true that a representation of the quantum group Uq(g) can
always be extended to the quantum affine algebra. Therefore, we need to develop
an analogue of the evaluation representations for general quantum affine algebras.
A natural candidate is just the finite-dimensional representations. We quote here
without proof two results regarding such representations. As before, we only con-
consider representations with weight decomposition.
Proposition 9.2.2.
1. Every finite-dimensional representation of the quantum affine algebra Uq(g)
has level 0.
2. Every irreducible finite-dimensional representation V ofUq(g) is generated
by a "highest-weight vector" V\ with the following properties: as a Uq(g)-
module, V = 0(i<AVr*', andVx = Cv\. (This, in particular, implies e^A =
We refer the reader to [CP2, Chapter 12.2B] for the proofs of these statements,
which is based on the so-called Drinfeld's realization of Uq(g) (see [DR5]).
Note that it is not true that an irreducible finite-dimensional representation
of Uq(g) is uniquely defined by the highest weight A; this is not true even for
3B (compare with Theorem 9.6.3 below). The complete classification of finite-
dimensional representations is known (see [CP2j), but it is more complicated.
Since we want to allow analogues of the modules which are obtained by pullback
from arbitrary highest-weight modules, not necessarily from finite-dimensional ones,
we adopt the following definition.
Definition 9.2.3. A representation V of the quantum affine algebra Uq(g) is
an evaluation representation if it has finite length over Uq(g), and all the Uq(g)-
irreducible subfactors are highest-weight representations of Uq(g).
It can be verified that for such representations the statement of Proposition 9.2.2
still holds.
Let irv '¦ Uq(g) —> End(F) be an evaluation representation of the quantum affine
algebra. Then we certainly can twist this representation by the automorphism Dz
and thus create a family of representations: nv(z) = ^v ° Dz. We will denote
this family of representations by V(z). We also need to use Laurent polynomial
representations z~AV[z,z~1}, where V is an evaluation representation and A is a
complex number. As a vector space, this representation is V <S> z~AC[z, z~x\, and
the action of Uq(g) on it is given by the same formulas as in V(z), except that now
z is no longer a number but a formal variable. The element d acts in z~AV[z, z]
in the usual way: d = z-^.
Using Drinfeld's realization, we can prove the following fact (see [KS]).
PROPOSITION 9.2.4. Let V,W be irreducible evaluation representations of the
quantum affine algebraUq(g). Then for almost all z eCx (i.e., for all but countably
many), the tensor product V(z) <g> W is irreducible.
We remark that if V is an evaluation representation then the properties of the
family V(z) are practically the same as those of the family constructed with the
9.3. INTERTWINING OPERATORS. 135
help of the homomorphisms pz. Therefore, the absence of such homomorphisms for
g 7^ sin does not actually create a serious problem.
9.3. Intertwining operators.
As soon as we have defined evaluation representations we can introduce the
quantum intertwining operators: $(z) : V?l fe —» VXo k<S>V^(z), where the hat, as
before, denotes the completion with respect to the grading.
For such intertwining operators we have an analogue of Theorem 3.1.1, first
obtained by I. Frenkel and N. Reshetikhin in [FRj.
THEOREM 9.3.1. LetL\o, LqXi be irreducible highest-weight modules overUq(g),
V any module over Uq(g), and let k be a complex number for which the Uq(g)-
modules V? u, V? t are irreducible. Let g : L\ —> L\ ® V be aUa(g)-homomor-
phism. Then there exists a unique Uq(g)-intertwining operator
(9.12) <S>g(z) : Vx\ k -> Vx\ k®V(z)
such that for every vector w € V* k of degree zero (i.e. w € L\ ) the degree
zero component of <&a(z)w is equal to gvi. The same is true if we replace V(z) by
Proof. The proof is analogous to that of Theorem 3.1.1. Let us construct
the operator $3 using the information we have. First of all, we have to reconstruct
$g(z)w, where w G V^ k is a vector of degree zero.
We know two facts about the vector Q9(z)w:
1) It is annihilated by the elements of positive degree in Uq(g+);
2) Its zero degree component equals gw.
These conditions determine the vector uniquely, which can be proved in abso-
absolutely the same way as in the proof of Theorem 3.1.1, replacing g®tC[i] by elements
of positive degree in Uq(g-°).
The action of $9(z) on the whole module can now be defined in a unique way.
Let wo = xw, where x is a product of fj's and w is of degree zero. It is obvious
that vectors of this form span the module V^ k. The intertwining property forces
us to set
This gives a well defined operator because Vx k can be obtained by inducing L\
to the algebra Uq(g~) from its subalgebra of elements of degree zero.
Thus we have constructed an intertwining operator (9.12). Since no choices
could be made in the process of construction, such an operator is unique. D
Now let us formulate an analogue of Proposition 3.2.1.
Proposition 9.3.2. The operator
is a Uq(g)-homomorphism if and only if
(9.13)
A = A(Ai) - A(A0).

136 LECTURE 9. QUANTUM AFFINE ALGEBRAS.
Proof. Analogous to the proof of Proposition 3.2.1.
?
Our next goal is to deduce a quantum analogue of the operator KZ equation
for <&(z) (see Section 3.2). For this we need the notion of a quantum current. This
notion is introduced with the help of the quasitriangular structure of the quantum
affine algebra.
9.4. Quasitriangular structure in quantum affine algebras.
By the results of Section 6.6, the Hopf algebra Uq(g) admits a quasitriangular
structure (after passing to a certain completion), and the universal it-matrix has
the following form:
(9.14) n= ~
where Xi is an orthonormal basis of 1), aj is a basis of Uq(n+), aJ is the dual basis
of Ug(h~), and cw = kw in V^k.
We cannot define the action of 1Z in evaluation representations, since the oper-
operator d does not act in them. However, since c acts by zero in evaluation represen-
representations, it is natural to throw the factor qc®d+d®° away and consider the truncated
R-matrix
(9.15)
cn.
For a complex number 2^0, define the element TZ(z) = (Dz ® ld)(lZ) =
(ldt8Dz-i)(R.), where Dz was defined by (9.3). Also define
Az(x) = (Dz ® Id)A(z), A°p(x) = (Dz ® Id)Aop(x),
It is easy to check that the element TZ(z) has the following properties (compare
with F.9)):
TZ(z)Az(x) = (q-c®d-d®cA°p(x)qc®d+dts>c)il(z),
(A®U)(R.(z))=n\z(z)K23(z),
(9.16)
The second and third equations are sometimes referred to as "the fusion laws for
the .R-matrix", and the last equation is nothing but a modification of the quantum
Yang-Baxter equation F.14).
Let Wi and W2 be evaluation representations of Uq($). Consider the projection
of fl(z) on the tensor product Wi®W2:
(9.17)
= {*Wl ® ttW2)(R.(z)) = (ttWi(z)
This expression is a power series in z. It is obvious that if z\jz2 = z, then
RmW2(z) = (TrWl(zi) ® ^w^(z2))(Ti-)- Later we will show that this series in fact
converges in a neighborhood of z = 0 (see Corollary 9.5.4).
9.5. FACTORIZATION OF THE H-MATRIX.
Note that (9.16) immediately implies that if Vi, V2, V3 are evalution represen-
representations (and therefore, by Proposition 9.2.2, c acts by zero in them), then we have
the following Yang-Baxter equation with spectral parameter.
(9.18)
9.5. Factorization of the it-matrix.
In this section we are going to describe the fi-matrix Rvw(z) for two irreducible
evaluation representations V, W of Uq(g). We will show that this it-matrix is a
product of a scalar transcendental function given by an infinite product, and a
rational matrix-valued function.
First let us prove a property of Rvw(z) called the crossing symmetry. Before
doing so, we need to find an analogue of the isomorphism V ~ V** (see F.8)) for
evaluation representations.
Proposition 9.5.1. Let V be an evaluation representation ofUq(g). Then
V{z)** is isomorphic to V(zq2h ), and the isomorphism V(zq2h^) —> V{z)** is
given by the operator q2p.
Proof. This proposition directly follows from 72(x) = q2p+2hvdxq~2P-2hvd
(see F.7)). D
li A : V <S> W —> V <S> W is a. linear operator, A = Y,<H ® &i, then let ATl :
V* ® W -> V* ® W and AT* : V ® W -> V ® W* denote the operators ? a? ® 64
and Yl ai ® Ki respectively.
Proposition 9.5.2.
l)RVW(zq2h-V)(q-2" d, 1).
2").
2. R
3. (crossing symmetry 1)
(9.19) (((Rvw(z)-1)T>)-1)Tl = (I2"
4. (crossing symmetry 2)
(9.20) vwTT
Proof. A), B) follow from identity F.12). C), D) are obtained by applying
A), B) twice (which gives an expression for Rv">w and Rv""w), and then using
Proposition 9.5.1.
Proposition 9.5.3. LetV,W be irreducible evaluation representations of the
algebra Uq(g). Then we have the factorization
(9.21)
Rvw(z) =
fVWr
where the matrix elements of Rvw(z) are rational functions of z regular at z =
0, R(v° <g> w°) = v° ® w°, where v°, wa are the highest-weight vectors in V, W
respectively (see Proposition 9.2.2), and fvw is a scalar function meromorphic

LECTURE 9. QUANTUM AFFINE ALGEBRAS.
in C which is regular at 0 and such that fvw@) ^ 0. The function f can be
represented in the form
(9.22)
where A, /j, are the highest weights ofV, W considered as Uq(g)-modules, and pvw (z)
is a rational function such that pvw @) — 1.
Proof. Let us consider the representation V((z)) (that is, formal Laurent
series with coefficients in V) with the action of Uq{g) defined in the same way as in
V[z,z-1] (see Section 9.2). Clearly, the operator PRvw(z) : V((z)) <g> W -+ W ®
V((z)) is well defined and is an isomorphism of representations. On the other hand,
it can be proved that V((z))® W is an irreducible module over Uq(g){{z)) - compare
with Proposition 9.2.4. Therefore, an intertwiner V((z)) <g> W —> W <g> V{{z)) is
unique up to a constant from C((z)). Let us define f{z) € C((z)) by the condition
Rvw{z)(v° ®w°) = f(z)v° ®w°; it is easy to see that f(z) ^ 0.
Let Rvw(z) = f-1(z)Rvw(z). Then R(z)(v° <g> w°)_= v° <g> w°, and R is
uniquely defined by this condition and the condition that PR is an intertwiner. The
latter condition can be written as a system of linear equations whose coefficients
are polynomials in z. Thus, the coefficients of R are rational functions in z.
Now, let us compute f(z). Consider the operator
(compare with (9.19)). Since V**(z) = V(q2^z), we must have
(9.23)
pVW(z)RV"'W(z) = (q2"
for some scalar function pvw(z). This is again an intertwiner. Since Rvw has
rational coefficients, pvw(z) has to be a rational function regular at z = 0, and it
is obvious that pvw@) = 1.
Comparing (9.19) with (9.23), we see that
which implies that
(under the assumption \q\ < 1). The last identity shows that fvw analytically
continues to a meromorphic function on the complex plane. Finally, to calculate
/@), note that Rvw@) coincides with the it-matrix for the "finite-dimensional"
quantum group Uq{g). Q
COROLLARY 9.5.4. Let Wi,W2 be irreducible evaluation representations of
Uq{Q). Then the power series RWlW?(z) converges in a sufficiently small neigh-
neighborhood of z = 0 and defines a holomorphic function in this neighborhood with
values in
Note that the operator RWlW*(z) preserves weight, so it can be represented by
a direct sum of finite-dimensional matrices, and the question of what convergence
means does not arise.
9.6. EVALUATION REPRESENTATIONS AND fl-MATRIX FOR Uq(Sl2). 139
PROPOSITION 9.5.5. Let U,V,W be irreducible evaluation representations of
Uq(9) and let R be as in the Proposition 9.5.3. Then we have
1. Quantum Yang-Baxter equation:
(9.25) Ruv{z)Ruw{zw)Rvw{w) = Rvw(w)Ruw(zw)Ruv(z).
2. Unitarity:
(9.26)
PRwv(z)PRvw(z'1) =
Proof. Part A) immediately follows from the Yang-Baxter equation (9.18) for
Rvw{z). As for part B), it is easy to see that the left-hand side is an intertwiner.
Since V{z) <g> W is irreducible for generic values of z, this intertwiner must be a
constant. Calculating the action on the product of highest-weight vectors, we see
that this constant is one. ?
9.6. Evaluation representations and K-matrix for W?(sl2)-
Let Uq(s\2) be the quantum group corresponding to the Lie algebra 5X2 (see
Example 6.2.2), and let Uq(sli) be the corresponding affine quantum group. It can
be described as the algebra generated by by the elements eo,fa,q±ho,ei, fi,q±hl,
satisfying the defining relations F.4), where i, j take values 0,1 and the Cartan ma-
/ 2 -2\
trix is given by A = (atJ) =1 . Recall that there exists a homomorphism
\ ^ l J
of algebras pz : W,(sB) ->M,(sl2) defined by (9.9).
Let Vn be the (n+ l)-dimensional irreducible representation of Uq(sl2) having
the highest weight n. Then Vn has a homogeneous basis u",..., v™ such that
(9.27)
vl = [n — i + l]ti*'
„»+!
qhvi = ,
where [m] = ? _^_1 and we let v l = vn+1 = 0 (when there is no ambiguity, we
will drop the subscript n).
For any complex number z ^0, we can define the evaluation representation
Vn(z) of Uq($l2) to be Vn{z) = pt(Vn). We can get more complicated finite-
dimensional representations of Uq EB) by tensoring such representations. It turns
out that the structure of these tensor products can be described explicitly, which
was done by Chari and Pressley (see [CP1, CP2]).
We will need the notion of a string. By definition, a string in Cx is a set of
points of the form {z,q~2z, ...,q~2rz}, r € Z+, z G Cx. Let Sn(z) be the string
{qn~1z, qn~'sz,..., q~n+1z}.
We say that two strings Si, S2 are in general position if either S\ U S2 is not a
string, or Si is a subset of 52, or S2 is a subset of S\.
Proposition 9.6.1. Any finite set E of complex numbers with multiplicities
can be written uniquely as a union of strings, any two of which are in general
position to each other.

LECTURE 9. QUANTUM AFFINE ALGEBRAS.
Proof. It is clear that we can restrict our attention to the case when our set
? is a subset of the set {q2mz : m e Z}, where z e Cx is fixed. Let y.{m) be the
multiplicity of the point q2mz. Consider the set Sj = {g2mz : fi{m) > i], with
all points having multiplicity 1. This set splits uniquely into a union of strings in
general position, which correspond to intervals on which fj, > i. It is easy to show
that two strings obtained in this way, even for two different values of i, are also in
general position, and the union of Si is L (counting multiplicities). ?
Now we can formulate the classification theorem, stated in [CP1] (compare
with Proposition 9.2.4).
Theorem 9.6.2. A tensor product Vi^zi) ® ... ® Viw(z/v) is an irreducible
representation ofUq{sh) if and only if the strings S^Czi),..., S^zn) are in general
position.
The proof of this theorem can be found in [CP1, CP2].
The converse result to Theorem 9.6.2 is also true. Recall that we only consider
representations with weight decomposition as defined in Lecture 6.
Theorem 9.6.3. Every finite-dimensional irreducible representation ofUq{si2)
is isomorphic to a tensor product of evaluation representations Vi(z). Two such
tensor products having no one-dimensional factors are isomorphic iff they differ
from each other by the order of factors.
The proof of this theorem, which is based on the technique of Drinfeld's real-
realization of quantum affine algebras, can also be found in [CP1].
Let us now consider in more details the case of two factors Vm(x) ® Vn(jr),
x,y e C*. First we note that the strings STO(x) and Sn(y) are not in general
position if and only if y/x = g±(m+™-2J>+2) for some 0 < p < min(m, n). Consider
the tensor product Vm(x) ® Vn($r) when this equation is satisfied, i.e. when the
tensor product is reducible. Then the structure of the tensor product is described
by the following nonsplit exact sequence:
(9.28) 0 -* Wi -* Vm(x) ® Vn{y) -* W2 -* 0,
where the modules Wi,W2 are irreducible and have the following decomposition
over the subalgebra Uq(sl2) C Uq(sh) generated by ei, fi,qhl:
(9.29)
(9.30)
2
Vm+n_2p © Vm+n-2p-2 © • ¦ • © V|m_n|,
Vm+n © Vm+n-2 © • ¦ • © Vm+n-2p+2,
_ / Vm+n © Vm+n_2 © • • • © Vm+n_2P+2, x/y = <
Vm+n-2p © Vm+n-2p-2 © ¦ • • © Vjm-n|. X/V = '
Let us consider the extreme case p = 1, x/y = qm+n. Then Vm{x) ® Vn{y) D
Vm+n{qmy), i.e. Vm(x) ® Vn(q"m-nx) D Vm+n{q-nx). Therefore
V1(x)®V1(q-2x)DV2(q-1x),
9.6. EVALUATION REPRESENTATIONS AND ii-MATRIX FOR «,(s[2). 141
This shows that
(9.31)
-1) ® V1{xq
D Vm(x).
Now, let us calculate explicitly the .R-matrix for the tensor product of two
evaluation representations of Uq(sl2). Let us start with the calculation of the renor-
malized .R-matrix
(9.32)
PR{x/y) -. Vm(x) ® Vn(y) -* Vn(y) ® Vm(x).
Since the tensor product Vm(x) ® Vn{y) is genericaUy irreducible, this map, for
generic x, y, is uniquely determined by the intertwining property and the normal-
normalization condition R(x, y)v°, ® v° = v%, ® v°. As was discussed above, up to a scalar
factor which we will calculate later, R coincides with the quantum .R-matrix.
In order to find R, let us consider the Clebsch-Gordan basis of Vm ® Vn con-
considered as W,(s[2)-modules. Let us denote by wr the highest-weight vector of the
irreducible component in Vm ® Vn isomorphic to Vm+n_2r. We normalize the vec-
vectors wr so that wr~1 = (e ® l)wr and w° = v^ ® v%. Let wr denote the vectors
constructed in the same way as wr in the tensor product Vn ® Vm, i.e. in the other
order. Let
be the intertwiner for Uq(sl2) defined by the equations
(9-33) pvV? = wr, Prvs =0,r^s
(the Clebsch-Gordan projection).
Since any intertwining operator for Uqfsh) has to be an intertwining operator
for the subalgebra Uq(s\2) generated by eu fuqhl, we can write PR in the form
min(m,n)
(9-34) PR{X/V)= tl Cr(*>V)Pv
r=0
In order to find the constants c,., let us use the condition that T must commute
with /o. We have
A(fo)PR(x,y)wr = PR(x,y)A(fo)wr.
Since /o = eix and ho = —hi in Vm(x), using the formula for comultiplication,
we have
(9.35) PR(x, y){x~xe ® 1 + jtV ® e)wT = (y~xe ® 1 + x^ ® e)PR(x,y)wr.
Now, since wr is a highest-weight vector with respect to Uq(sl2), we have
A ® e)wr = -(e ® qh)wr = -(eq~h ® l)(qh ® qh)wr = -(q~he ® l)qm+"-2r+2wr
Multiplying both sides of this identity by qh ® 1, we get
(9.36) {qh ® e)^ = _g™+"-2'-
Substituting (9.36) into (9.35), and taking into account the definition of {wr}, we
get
(9.37)
lcr_i
= f«-l _ ;r-lnm+n-2r+2
!)cr,

LECTURE 9. QUANTUM AFFINE ALGEBRAS.
which allows us to compute
the following proposition.
recursively, starting with Co = 1, and thus obtain
Proposition 9.6.4. The intertwining operator (9.32) is given by the formula
(9.38)
PR{x/y) =
¦ _ llnm+n-2
where Pr : Vm ® Vn
is defined by (9.33).
Now assume that m,n are arbitrary complex numbers, and replace finite-
dimensional modules Vm,Vn by Verma modules M^,M^. Each of them has a
basis v*, i = 0,1,..., and the action of Uq(sl2) is given by the same formulas (9.27).
Using analytic continuation arguments, we find that formula (9.38) can be extended
to this case:
(9.39)
r=0 j=0
It follows from formula (9.38) that R is defined and invertible iff the strings
Sm(x) and Sn{y) are in general position, i.e. when Vm(x) ® Vn(y) is irreducible.
When x/y = q-(m+"-2r+2)^ the strings are not in general position, and PR is
defined but degenerate: its image is the irreducible subspace Wi in Vn(y) ® Vm(x).
Example 9.6.5. Let us calculate the action of R on the space
This space is two-dimensional with the basis fvm ® «„,«,« ® fvn, where vm =
»S,,ti, = u° are the highest-weight vectors in Vm,Vn respectively. We denote
x/y = z.
In this case, PR{z) = Po 4- ^Ch-" pi +¦¦¦, where z = x/y, and
1 - —f
Vn - "j—r«m ® JVn,
n
which gives
P«(.) ( R/,
[n]
Also, applying A/ to the identity PR(vm ® i;n) = -yn ® -ym, we get
PR(fVm ® ¦[»„ 4- q~Jnvm ® /«„) = /¦«„ ®t)m + q~n'yn ® /«m-
Combining two previous idenitites, we finally get
on(gm-q"m) zqn-qm
PR(z)(fvm ® wn) = -2-^2 m^n /vn ® wro 4- ——^jz^^n
(9'40) ,. _^_*r-V,__ W-^-)_
9.6. EVALUATION REPRESENTATIONS AND .R-MATRIX FOR Uq(Sl2). 143
Let us nowj;ompute the function fvw{z) for the case when V, W are represen-
representations of Uq(sl2) with highest weights m, n. First assume that m = n = 1. In this
case it follows from Example 9.6.5 that
(9.41)
Rvw(z)=Eu®Eu + E22<
, -9(9-9)
'(fil:
4- ^S2i i
E
where the ??„- are elementary 2x2 matrices in the basis v°, v1 = fv°.
Now it is easy to compute pvw{z) and fvw(z) directly, using the method in
the proof of Proposition 9.5.3. The answer is
P{ '
(9.42)
A -
j=\
Now let us use fusion to obtain the answer for general m, n. Recalling inclusion
(9.31), and the law (9.16) of fusion for the universal .R-matrix, we get
(9.43)
where
(9.44)
Rvw(z)(ym®vn) =4>mn{z){vm®vn),
m— 1 n — 1
n n
3=01=0
and f(z) is given by (9.42) Therefore, we get the following result.
Proposition 9.6.6. The universal R-matrix Tl forUq(si~2) acts on the tensor
product Vm(z) ® Vn(l) by
(9-45) Rv<"v»(z) = <j>mn(z)Rv<»v»(z),
where R is as in Proposition 9.6.4 and <t>mn is given by (9.44).
Let us now extend these results to the case when m,n are arbitrary com-
complex numbers, replacing the finite-dimensional modules Vm, Vn by Verma modules
M^,M«, as before.
Define </>mn for generic m, n by formula (9.43). Let us compute it. Clearly, <j>mn
can be written in the form
CO
<t>mn{z) =9m"(l + ^a»nJZJ),
where amn3 are rational functions of qm and qn.
Set
(9.46)

144 LECTURE 9. QUANTUM AFFINE ALGEBRAS.
Then the function 4>mn(z) for integer m, n can be written in the form
(9.47)
= <r
Observe that the right hand side of (9.47) makes sense for any complex m, n, and
its Taylor coefficients are qmn times rational functions in qm, q". Therefore, (9.47)
must hold for all m,n, since a rational function is completely determined by its
values at an infinite sequence of points.
With this new definition of <j>mn, formula (9.45) holds for arbitrary m, n, where
R is defined by (9.39).
9.7. Quantum currents.
Let V be an evaluation representation oiUq(g). Define quantum currents for
Uq(g) as follows:
(9.48)
L+(z) = (Id®irv)(Kop(z)),
We will regard the expressions Ly(z) as formal power series in z±l with values
in the algebra W9(jj)® End(K), where ® denotes a suitable completion of the tensor
product.
Let us define two operations applicable to quantum currents. Let a = ai ® a2 €
Uq{g) ® End(V), 6 = h ® 62 € «,(jj) ® End(W). Define a "product" of a and 6 by
a*b = ai61®a2®62 € Wg(fl)®End(V)®End(W). Also, if V = W, define a "tensor
product" of a and 6 by a ® 6 = 61 ® a\ ® a2&2 G W9(J)®2 ® End(V). It is important
to distinguish these operations from the usual product and tensor product.
Let us now write down the commutation relations for currents.
Proposition 9.7.1. The following relations between power series with values
in End(V?fc ® Wi ® W2) hold true:
(9.49)
(
(9.50)
(9.51)
where
^1 fa) * L^(z2) = PWlW,
^ w2 denotes the permutation of the
^2(z2) * L+1 (
and W2 factors.
Proof. To prove this proposition, it is enough to apply the maps n
"¦y 17rv ^iif, ®itw2 > and ttwi ®irvi ®7rw2 to the quantum Yang-Baxter relation
for n (the last formula in (9.16)). ?
(9.52)
Proposition 9.7.2. The following relations are valid:
(A® Id)(i+ (z)) =qc®d®1L+v{z) ® L+r(z)q-
(A®Id)(?^(z))
9.8. QUANTUM SUGAWARA CONSTRUCTION IN DEGREE ZERO. 145
Proof. To prove this proposition, it is enough to apply the map Id®Id®7rjy
and certain permutations of factors to the second and third equations in (9.16). D
Now we are in a position to deduce the relations between quantum currents
and quantum intertwining operators.
Proposition 9.7.3. Let <S>wAz) ¦ V?k -* V.,% ® Wi(¦*) be the intertwining
operator defined in Section 9.3. Then the relations
(9.53)
are satisfied in Homc(VA9;A., V*k ® Wi) ® End(W2).
Proof. Using the intertwining property and relations (9.52), we obtain
for +,
for -•
=Lw2 B2J3^^ B2I3*^1 (
D
9.8. Quantum Sugawara construction in degree zero.
In Section 2.9 we described the Sugawara construction, which introduces the ac-
action of the Virasoro algebra in highest-weight representations of affine Lie algebras.
The idea of this construction was to express Virasoro elements as semi-infinite series
involving affine Lie algebra elements. Recently a natural candidate for a "q-Virasoro
algebra" was introduced in [SKAO]. Unfortunately, the quantum analogue of the
Sugawara construction is not known. Nevertheless, the degree operator, which cor-
corresponds to the element Lo of the Virasoro algebra, is perfectly well defined in the
quantum case, and therefore it is legitimate to ask how to express it using only the
elements of Wg(fl). The answer is given by the following theorem, which we call the
quantum Sugawara construction in degree zero.
Recall the notation k = k + hv.
Theorem 9.8.1. Let V^k be a highest-weight module overUq(g). Then in
(9.54)
Proof. Let u be the element of a completion of Uq(g) defined by F.30). Using
Theorem 6.7.2, we see that in V*k we have u = g-<A>A+2P)+2P, where, as before,
p = p + hvd and A = A 4- kd — Ac. But it is easy to compute that (A, A + 2/5) = 0
(because we agreed that d multiplies the highest-weight vector by the special value
given by B.25)). Thus, we have
(9.55) u =

LECTURE 9. QUANTUM AFFINE ALGEBRAS.
On the other hand, from the definition of u we have u = m(Y) and Y =
). Let H = Y.i a'i ® b'i- Then> ^^S (9-15)' we get
Y = q-ctsd d®c
(here g = e*). Therefore,
= ?
I)])-
Therefore, in VA9fc we have
(9.56) u = ^T^ i{b'i)q~2kda'i = q~2 m{[
i
This equation, together with (9.55), implies the theorem. ?
Theorem 9.8.1 gives a canonical way to extend a highest-weight representation
Lecture 10. Quantum
Knizhnik-Zamolodchikov Equations.
In this lecture we will deduce the quantum analogue of the Knizhnik-Za-
Knizhnik-Zamolodchikov equations - a consistent system of difference equations satisfied by
the matrix elements of products of intertwining operators. In a representation-
theoretic context this system of equations was first discovered by I. B. Frenkel and
N. Yu. Reshetikhin [FH.]. In a special case of zero central charge it has appeared in
the work of F. Smirnov [Smj as an equation on form factors in completely integrable
models of quantum field theory. The derivation of the quantum KZ equations in
the first three sections is based on the quantum Sugawara construction and is some-
somewhat different from the original approach of [FR]. In the last section we give an
alternative proof, closely following [ITIJMNj.
Linear difference equations in one variable were studied by Birkhoff [Bi], Car-
michael [Ca], Trjitzinsky [Trj, Adams [Ad], and others in the early 20-th century.
Some of their results were generalized by K. Aomoto [A2] to consistent systems of
linear difference equations.
We also consider the classical limit of the quantum KZ equations and obtain
the trigonometric form of the classical KZ equations introduced in Lecture 3.
10.1. Operator quantum Knizhnik-Zamolodchikov equation.
V^k®z-AW[z,z~1}, with A cho-
choIn this section, we deduce a difference equation for the intertwining operators,
which is a (/-analogue of the operator KZ equation of Section 3.2.
Let Q(z) denote a W9(g)-intertwiner: V?k
sen as in Proposition 9.3.2.
Let u be the element in a completion ofWg(g) defined as in F.27), and let
A0.1) U = 92(cd"p)«.
Introduce the notation
A0.2) Q = q , p — q
in V^k. Also, the
We assume that \p\ < 1. It follows from (9.55) that Q = U^1 i
operator Q acts in the Laurent polynomial representation z~AW[z, z~1} as follows:
Qv{z) = v(pz).
Since $ is an intertwiner, we have the following relation between Laurent series
in z:
A0.3)

146 LECTURE 9. QUANTUM AFFINE ALGEBRAS.
On the other hand, from the definition of u we have u = m(Y) and Y =
~.°p). Let 1Z = Yli K®K- Then, using (9.15), we get
Y = q~ct
(here q = e*). Therefore,
i,l,n '
Therefore, in V?k we have
(9.56) u = E7Fi)^2fcd^
i
This equation, together with (9.55), implies the theorem.
1)]).
?
Theorem 9.8.1 gives a canonical way to extend a highest-weight representation
oiUq(g)toUg(g).
Lecture 10. Quantum
Knizhnik-Zamolodchikov Equations.
In this lecture we will deduce the quantum analogue of the Knizhnik-Za-
Knizhnik-Zamolodchikov equations - a consistent system of difference equations satisfied by
the matrix elements of products of intertwining operators. In a representation-
theoretic context this system of equations was first discovered by I. B. Frenkel and
N. Yu. Reshetikhin [FH.]. In a special case of zero central charge it has appeared in
the work of F. Smirnov [Smj as an equation on form factors in completely integrable
models of quantum field theory. The derivation of the quantum KZ equations in
the first three sections is based on the quantum Sugawara construction and is some-
somewhat different from the original approach of [FRj. In the last section we give an
alternative proof, closely following [ITIJMNj.
Linear difference equations in one variable were studied by Birkhoff [Bi], Car-
michael [Ca], Trjitzinsky [Trj, Adams [Ad], and others in the early 20-th century.
Some of their results were generalized by K. Aomoto [A2] to consistent systems of
linear difference equations.
We also consider the classical limit of the quantum KZ equations and obtain
the trigonometric form of the classical KZ equations introduced in Lecture 3.
10.1. Operator quantum Knizhnik-Zamolodchikov equation.
In this section, we deduce a difference equation for the intertwining operators,
which is a g-analogue of the operator KZ equation of Section 3.2.
Let *(z) denote a Uq(g)-intertwiner: V?fc -> V*M®z~^W[z, z~1}, with A cho-
chosen as in Proposition 9.3.2.
Let u be the element in a completion of Uq(g) defined as in F.27), and let
A0.1)
Introduce the notation
A0.2)
Q =
We assume that \p\ < 1. It follows from (9.55) that Q = U~l in Vj*t. Also, the
operator Q acts in the Laurent polynomial representation z~^W[z, z~x] as follows:
Qv{z) = v{pz).
Since $ is an intertwiner, we have the following relation between Laurent series
in z:
A0.3)

148 LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
Using the identities A ® Q-1)#B) = ^{p~1z) and Q = U'1, we obtain
A0.4) ^(p-1*) = (Q ® 1 1
where U is defined by A0.1).
For a ® 6 € W,(fl) ® End(W) and * €
W[z, z'1]), denote
A0.5)
Lemma 10.1.1.
* • (a ® 6) =
Proof. We have {/ = uq2<-cd-/>'> = ]>V 7Fj)«j92(cd~'j), where ? =
Therefore, since $(z) is an intertwiner,
A0.6)
j a., ® 6,-.
Following Drinfeld ([Dr2]), we introduce the notation {X <g> F ® Z) o $ =
(l() > 7B))#X, X, y, Z e Wg(fl). This defines a right action of the tensor cube
of the quantum affine algebra on the space Horned*., Vj^z'^Wlz^z'1]).
Using this notation, we can write A0.6) as follows:
A0.7) *(z)«7 = ^ 2^'
Applying F.9), we get
in A0.8). Using the
Let us separately consider the expression X — TZi2 °'
intertwining property of $(z), we obtain
A0.9) X = Y$(z), Y = m31 ((A <8
where 77131 (o ® 6 ® c) = ca <S) b.
Applying F.9), we find that
A0.10) Y =
Thus we have
• (u® 1).
A0.11)
1 ® I)*(z)i7 = (i/ ® 1) [TC13 o G ® Id)(R°p)(
(using F.31)
= [{(<
T2(cd-p)
10.1. OPERATOR QUANTUM KZ EQUATION
149
Using F.7) and Theorem 6.7.2, we see that in the space of operators on the product
V?k®z~AW[z, z^1] we have the following equality:
A0.12)
= (q-2p ® l)^-1(9-2/>V)(l ® q-kd)(q2p ® 1).
Using A0.12) and the expression of il in terms of 71, we can rewrite A0.11) in
the form
A0.13)
V ){<i2
)(g2M ® 1)}
® q~2kd)Hz)}
1)}
{{q2p)
{{q2p)
Lemma 10.1.1 is proved.
?
The lemma together with equation A0.4) implies the following difference equa-
equation for #(z), first derived by I. Frenkel and N. Reshetikhin in [FRj.
Theorem 10.1.2. The intertwining operator $(z) satisfies the difference equa-
equation
A0.14)
T/iis equation is called the operator qKZ equation.
This theorem is the quantum analogue of Theorem 3.2.2 for classical affine
algebras. Indeed, let us choose in W a basis such that the action of ei,ft,qh' in
this basis is given by analytic functions of q regular at q = 1. Then it is easy to
show, using (9.14) (cf. [FR]), that as q -* 1,
= ?+ (
A0.15)
L%{z) = 1 ® 1 + (, - q'1) J2 3?{z) ® 7TW(a) + O((q - q-1J),
where B is an orthonormal basis of g, and J*(z) are the (modified) classical cur-
currents:
A0.16) J±(z) =
n>0
a =
,a0 e Jj.a* e n±.

150
LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
Therefore, if we take the derivative of equation A0.14) with respect to q at
9 = 1, we will obtain, after some simplification, equation C.11).
10.2. Quantum correlation functions.
The definition of the correlation functions for a quantum afHne algebra is com-
completely analogous to their definition in the classical case.
Let L\t, 0 < i < N, be highest-weight irreducible Uq(g)-modvXes, and let V,
1 < i < N, be evaluation representations of Uq(g).
A correlation function is a function of the form
A0.17)
*?,¦,:.¦.;& (*!»¦••.*") = {u0M\(z1)...Q»(zff)uN+1),
where u0 e (VA90 k)* is the lowest-weight vector, mjv+i G V\N,k is t^ie highest-
weight vector, U{ e V* are arbitrary vectors, gt are Wg(g)-homomorphisms VXi —>
LqXii ® V\ &9'(zi) : LqXi -> i^_i®2,"AiVi[^,^1] are the Wg(g)-intertwiners cor-
corresponding to gi, and iw(z)v = (ld®w)(it(z)v).
Since the operator $ut~^\(zi+i) takes values in the completion V^ k while the
operator i^\(zi) is denned only on Vx k, it is not obvious why the product A0.17)
makes any sense at all. We have already had to deal with this problem in Lecture 3.
The best we can say about the product A0.17) is that it is defined as a formal series
in z\,..., zn. It is easy to find that this series belongs to
Z\ ZJV-1
where A4 = A(A4) - A(Aj_i).
Define \p9i.—,9w (zi,..., zn) to be the power series with coefficients in the space
V = V1 ® ... ® VN such that
A0.18) <6r9i,-,9» (Ul,...,uN) = %l\-;:::9uNN(zi,..., zN).
For brevity we will often drop the superscripts g\,..., giy, assuming that the ho-
momorphisms g{ have been fixed. For instance, we will write <& instead of xpsi.—.sw.
Because of the Wg(jj)-invariance of the operators l>, the coefficients of 9 actually
lie in a finite-dimensional subspace of V, namely, in the space yA«~A° of vectors
of weight Xn — Xo with respect to Uq(g).
Note that in Lecture 3 we considered slightly more general correlation functions,
allowing «o, «w+i to be arbitrary vectors in L\o,L\N respectively. This allowed us
to deduce the rational form of the KZ equations. If we choose uq, un+i to be
the lowest-weight vector (respectively, the highest-weight vector), then we get the
trigonometric KZ equations. For the classical case, it didn't make much difference,
since in this case the rational and trigonometric form of the KZ equations are equiv-
equivalent. However, it turns out that only the trigonometric form admits generalization
when we replace g by Ug(g): if we let uq, un+i be not necessarily highest-weight
vectors then our methods do not allow us to deduce any equations on the corre-
correlation functions for Ug(g). The rational form is related to another deformation of
), known as the double Yangian.
10.4. A FUNDAMENTAL SET OF SOLUTIONS OF qKZ EQUATIONS
10.3. Quantum Knizhnik-Zamolodchikov
equations for correlation functions.
Let us now deduce the difference equations for quantum correlation functions.
Using Theorem 10.1.2, we obtain
A0.19)
= (tio, i1^)... &-1(zi-1)L+, (pqkzj)
Let us now drag L+ to the left and L~ to the right, using the commutation
relations (9.53). Taking into account the relations
A0.20)
= u0 ® qXo+2"v,
(*)"
V) = UN+l
which follow from the definition of the quantum currents, we get the following
equation, derived by I. Frenkel and N. Reshetikhin in [FR].
Theorem 10.3.1. The quantum correlation functions satisfy the following sys-
system of linear difference equations.
A0.21)
Z\
'¦Nj
,,..., ZN).
These equations are called the quantum Knizhnik-Zamolodchikov equations.
10.4. A fundamental set of solutions of the
quantum Knizhnik-Zamolodchikov equations.
Let W be a finite-dimensional vector space, p e C*, \p\ < 1. Consider a system
of difference equations
A0.22)
where
u ... ,zh..., zN), 1 < j < N,
* ? zrAl ¦ • ¦ z-A"W[[z2/Zl,..., zjv/zjv-i]],
A, ? EndW[[z2/zu • • •, zs/zs-r}].
Let A® be the leading coefficient of Aj.
Assume that system A0.22) has a set of m — dimVF solutions *!,...,*m
(with possibly different values of A* for each of them), such that the leading terms
y/°j of tyj in the limit zi+i/zi —*0,l<i<N — l, are linearly independent for
j = 1,..., m. Such a set is called a fundamental set of solutions. Then {*°} is a
basis in W. Let {*°*} be the dual basis of W*. Consider the EndVF-valued series
A0.23)

152 LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
This solution is called the fundamental matrix solution of the system A0.22). It is
the difference analogue of the fundamental solution C.25) of a system of differential
equations.
The fundamental solution can be written in the form
A0.24)
i,..., zN) = F0(zi,..., zN)z1 *
Si .#2
where pB< = A° (from now on, we denote px = e 2tKX, where X is a number or
an operator and t = logg), {Bj} are diagonal in the basis {*°}, and the leading
term of Fo is the identity operator on W. Moreover, the fundamental solution is
determined by this condition (up to the nonuniqueness resulting from the fact that
the solution of the equation pB = A is not unique).
Let us now construct a fundamental set of solutions for the quantum KZ equa-
equations A0.21). First of all, let us notice that A0.21) defines a separate system
of difference equations in each weight subspace in V1 ® ... ® V^. This follows
from the fact that the .R-matrices R(z) preserve weight. Therefore, we can set
W = (V1®...® V^)**, where /xis a fixed weight, and regard A0.21) as a W-valued
system. In this case, we have Ajv = Ao + ?i.
We assume that Ao,fc are chosen generically, so that for every j such that
1 < j 5 ^i and any Wg(g)-weights m,...,Hj occurring in the representations
V1,.. •, Vj, the Verma module Mjfc over Uq(g) with highest weight u e Ao + Hi +
¦ ¦ - + Hj + Q (where Q is the lattice of roots of g) and central charge k is irreducible.
Then the fundamental set of solutions for A0.21) can be constructed as follows.
Choose a basis of W of the form Wi = «M1(i) ®. ¦. ® tiw(j), 1 <i<m, where «w(»)
are vectors in VJ of weight jUj(i) with respect to Uq(g). Clearly, J^MjW = H-
Define \j(i) = Ao + J^r=i Cr(*) (-^jv(i) = A^ = Ao + m).
Let &i(z) : Af* ,;> k —» M*. ,s> fc®z~AjWV*[z,z] be Wg(g)-intertwining op-
operators such that 4j(«)vA,-(i),fc = •z~Aj'l''yAi_i(i) ® uMi@ + lower order terms (here
•ua,*: is the highest-weight vector of the Verma module M*fc over Wg(fl)). Such
operators are unique because of the genericity assumption and the arguments of
Section 9.3. Set
A0.25) <Bri(zi,...)zjv) = (wJ0,fc,*i(zi)---*iV(zJv)t'Aw,fc)
Then, according to Theorem 10.3.1, {*i} is a fundamental system of solutions of
the quantum KZ equations, and the function F defined by A0.23) is its fundamental
solution.
10.5. Holonomic systems of difference equations
If a system of the form A0.22) has a fundamental set of solutions, it is called
consistent. For example, as we showed in the previous section, for generic values of
parameters the quantum KZ equations are consistent.
There is a simple necessary condition for consistency, which we will now de-
describe.
Let A0.22) be consistent, and let F be its fundamental solution. Let 1 < i <
j < N. Then F(zi,.. .,pzi,... ,pzj,...,zn) can be calculated from A0.22) in two
ways: we can first change Zi and then Zj, or we can do it the other way:
10.6. ANALYTICITY OF THE FUNDAMENTAL SOLUTION
A0.26)
2li • • • i Zii . . . , Zjt • . • , Zn)F\Z\, .. . , Zjv)
= Aj{zi,... ,pzj, ...,zj,.. .,zN)Aj(zi, ...,Zi,...,Zj,..., zN)F(zi,.. .,zN).
Since F isinvertible in the ring of power series, we can multiply equation A0.26)
from the right by F~r and get
A0.27) Mzi,-.-,zu...,pzj,...,zN)Aj(z1,...,zi,...,zj,...,zN)
We will call system A0.22) holonomic if equation A0.27) is satisfied. Thus,
every consistent system is holonomic.
Proposition 10.5.1. The quantum KZ equations are a holonomic system.
Proof. The above arguments show that this proposition is true for generic
Ao; by continuity, it is true for non-generic values as well. ?
Remark. Of course, we could check the relation A0.27) directly, using the
quantum Yang-Baxter equation for .R-matrices, quite similarly to what we did in
Section 3.5. We just wanted to show how to deduce A0.27) without any computa-
computations.
10.6. Analyticity of the fundamental solution of
the quantum Knizhnik-Zamolodchikov equations.
In this section we will show that the fundamental solution of the quantum KZ
equations extends to a multivalued meromorphic function on (CX)N.
First of all let us prove a technical lemma.
Lemma 10.6.1. Let
A0.28) F(pz) = A(z)F(z)
be a difference equation, such that A e Maim(C)[[2]], \p\ < 1, and A@) is invertible.
Assume that
(i) The series A converges in the region \z\ < R, and
(ii) The series F{z) = F0{z)zB, where Fo e Matm(C)[[z]], is a fundamental
solution of A0.28).
Then the series F0{z) converges in the region \z\ < e for a suitable s > 0.
Proof. Let us write the series expansions for A and Fo:
A0.29)
A(z) =
, F0(z) =
J=0
3=0
It follows from condition (i) that for any r < R there exists M > 0 such that
Aj|| < Mr'' for any j. Now, from A0.28) and A0.29) we have
A0.30)
V1 fil^ =YtAifi-i.

154 LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
Taking into account that Ao = pB, from A0.30) we get the following recursive
relation for fj (valid for large enough j):
A0.31)
where the superscripts (I) and (r) mean that the action of the corresponding matrix
is on the left and on the right, respectively. Let A" be a constant such that we have
Wip'A^ - A^y^l < K for j large enough. Let K = KM. Let us prove that
li/j|| < C(r/A")~J for a suitable constant C. This will imply the lemma.
Taking the norm of both sides of A0.31), we get (for large enough j)
A0.32) ||/j|| < KQ^r-'Wfj-iW).
i=l
This implies that the sequence \\fj\\ is dominated by a sequence gj which satisfies
the equation gj = K{^i=1 r~'gj-i) for large enough j. But the sequence gj satisfies
the equation rgj+i/K — gj/K = gj, so it is a geometric progression of the form
^(k+i)~'• T^e lemma is proved. ?
Let us extend this result to functions of several variables.
Proposition 10.6.2. Suppose that the coefficients Ai(z) of a system of dif-
difference equations A0.22) are convergent series in the region \zi+i/zi\ < R, 1 <
i < N — 1, and analytically continue to meromorphic functions in (CX)N. Let
F(z) = FQ(z)z^1 ...zBn be the fundamental solution of this system (see section
10.4). Then the series Fq(z) is convergent in the region \zi+1/zi\ < R' for a suit-
suitable constant R', and continues to a meromorphic function on (CX)N.
Proof. Fix a vector z = (zlt..., zjv), and consider the function of one vari-
variable F(u) = F(zuuz2,...,uN-1zN). Then F(u) = F0(u)uB, where B = B2 +
2B3-i h (N - 1)BN, F0(u) = F0{zuuz2,...,uN-1zN)zf1 ...z%N. Also, the
function F(u) satisfies the difference equation F(pu) = A(u)F(u), where A is given
by formula A0.33) below. It follows from the conditions of the proposition that
A is holomorphic near u = 0 and extends to a meromorphic function in C. By
Lemma 10.6.1, this implies that the series Fo is also holomorphic near u = 0. It
is easy to show that the estimate on the radius of convergence of the series Fo
given in the proof of Lemma 10.6.1 is uniform with respect to z when the ratios
Zi+i/zi axe all sufficiently small. This implies that the series -Fo(z) is convergent
in the region \zi+\/zi\ < R' for a suitable constant R'. The fact that it contin-
continues to a meromorphic function on (CX)N follows immediately from the difference
equations. d
THEOREM 10.6.3. Let Ao, k be generic, and let q be such thatp = q~2K satisfies
\p\ < 1. Let F(z) = Fo{i)zfl ... z3,*1 be the fundamental solution of the quantum
KZ equations A0.21). Then for some constant K > 0 the series Fo converges
in a region \zi\ > K\z^\ > ¦¦¦ > ii''"'!^!, and its sum defines a holomorphic
function in this region which extends to a meromorphic function in (CX)N. In
particular, quantum correlation functions *(z) always (for any Ao, k) have the form
'E'(z) = z7^ .. . ,z7Viw\E'o(z), where 'S/o is meromorphic in (Cx)".
10.8. CLASSICAL LIMIT OF THE QUANTUM KZ EQUATIONS
Proof. The first statement follows from Section 10.4, Proposition 10.6.2, and
the fact that the coefficients of A0.21) satisfy the conditions of this proposition.
This, in turn, is a consequence of the fact that the .R-matrices Rv'v3 (z) are mero-
meromorphic functions on C with no pole at z = 0 (see Corollary 9.5.4).
The statement that *0(^) is meromorphic for generic Ao, k follows from the
fact that *(z) = F(z)w, where w is a suitable vector in W (independent of z). For
arbitrary Ao, k, the statement follows by continuity. ?
Note that correlation functions were not meromorphic in the classical case
(q — 1): the hyperplanes z< = Zj were their branching loci.
10.7. The noncommutative product
formula for the fundamental solution.
Consider a system of difference equations of the form A0.22) with the coeffi-
coefficients satisfying the conditions of Proposition 10.6.2. Let F(z) = F0(z)zfl ... z^N
be the fundamental solution of this system. Here we present a formula for compu-
computation of this solution.
We will construct the fundamental solution as a limit of finite products. Let
z denote a vector (zu...,zN). Define_F(u), F0(u), u e C, as in the proof of
Proposition 10.6.2. Then F(pu) = A(u)F(u), where A(u) is given by the formula
A0.33)
A{u) =AN(zupuz2,p2u2z3,... ,pN-2uN-2zN^,pN-2uN-1zN)
., uN~lzN)
... A3(z1,puz2,pu2z3,..., uN'1z
¦ A2(z1,uz2...,uN~1zN).
(this product contains N(N — l)/2 factors).
For any m > 0 we have
A0.34) F{u) =
Therefore, we have
A0.35) F0{u) = A
. ..A~1(pm-1u)F(pmu).
... A-1(pTn~1u)F0(pmu)p
Set Gm(u) = p-™BF0{pmu)pmB. Recall that pB = A@). Assume that p is
small enough so that ||J4di4@)^1|| < IpI. Then it is clear that linim^oo Gm(u) = 1
uniformly in u from a neighborhood of 0. Therefore, passing to the limit as m —> oo
in A0.35), we obtain an explicit product formula for the fundamental solution:
A0.36)
F(u)= lim {A-1(u)A-1{pu)...A-1(pm-1u)A{0)n>)zi
In particular, formula A0.36) allows us to compute the fundamental solution
of the quantum KZ equations if p is sufficiently small.
10.8. Classical limit of the quantum
Knizhnik-Zamolodchikov equations.
In the limit q —> 1 the quantum affine algebra Uq(g) tends to the universal
enveloping algebra U(g). Therefore, the correlation functions for the quantum

156 LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
case converge to the classical correlation functions. Hence, the quantum Knizhnik-
Zamolodchikov equations should converge to the classical (trigonometric) Knizhnik-
Zamolodchikov equations. Let us see how this happens.
Let us choose a basis in each of representations V3 such that the action of
ei> fi,Qh m this basis is given by analytic functions of q regular at q = 1. This
is always possible, for example, if V3 are irreducible over Uq(g) and are obtained
by pulling back to Uq(g) by the homomorphism p : Uq{%) —> Uq(g) denned in
Theorem 9.2.1 (for g = sln).
Let V, W be evaluation representations of g. Then on V ® W we have
A0.37)
?J+(*)®a=i;
z; ® Xi
+ n-
f°
a ® a
a?B
X-Z
where Xi is an orthonormal basis of t), the currents J* are defined by A0.16), and
n± are defined by C.44).
Therefore, projecting the relation A0.15) (for the 4- sign) to a finite-dimensional
representation V ofUq($), we get the following relation between operators on W®V:
A0.38)
R(z) = 1 + (q-1 - q)r(loSz) + O((q~l - qJ), <f - 1,
where t(z) is the trigonometric r-matrix defined by C.47).
Now, it is easy to see that, on substituting A0.38) into A0.21), in the limit as
q —* 1 we get precisely the trigonometric KZ equations C.46) with Xi = logzj.
Therefore, Theorem 10.3.1 should be regarded as a quantum version of Theo-
Theorem 3.8.1.
10.9. Modified quantum Knizhnik-Zamolodchikov equations.
Let Rvw(z) be the renormalized R-matrix as defined in Proposition 9.5.3, and
let p = q~2K. Consider the modified quantum Knizhnik-Zamolodchikov equations
A0.39)
, dV'V1
I . . ¦ H^
X U
V
O'K
where V3 = V)ij are highest-weight representations of W,(s[2). The difference
between these equations and A0.21) is that we now have the rational functions
Rv'v'_{z) instead of Rvty3(z). Therefore, the fundamental solutions F of A0.21)
and F of A0.39) are connected by the formula
A0.40)
F(z) =
10.10. ANOTHER PROOF OF THE QUANTUM KZ EQUATIONS 157
where G(z) is the scalar-valued function satisfying the system of difference equations
A0.41)
Z\
j,..., ZN),
where <fcy(z) is defined by RViVi(z) = Rv'vi(z)(pi:j(z), and G -> 1 when all the
Zj/zj+1 —» oo simultaneously.
Equations A0.41) can be solved explicitly using the technique of Section 10.7,
after which the function G will be represented as an infinite product (in this case the
product is commutative since the function G is scalar). Therefore, the nontrivial
information about the solutions of the quantum KZ equations is concentrated in the
fundamental solution of system A0.39) with rational coefficients. Prom now on we
will work with system A0.39) and will call it the modified quantum KZ equations.
10.10. Another proof of the quantum
Knizhnik-Zamolodchikov equations.
In this section we give a simpler, and more direct, proof of the quantum KZ
equations for quantum correlation functions than that given above. The advantage
of the above proof (which is very close to the original proof in [FR] and practically
identical to the proof in [ITIJMN]) is that it gives the operator KZ equation as a
by-product.
Let us first assume that N = 2, and we have a correlation function
A0.42) *(zi,z2) = (wo.sHsi)*2^)^)
(see section 10.3).
Let us consider the expression
A0.43) *"(«!,%) = <uo,*1(
where U = q2<-cd~p)u, and u is the quantum Casimir element, which acts in the
representation V?uk in A0.43). Using (9.55), we find that
A0.44) Vu{zuz2) = q~{{)
Now let us consider the .R-matrix 7Z which was denned by (9.15). We have
A0.45)
n =
where E(i) is a basis of Uq(h+), and F(i) is the dual basis (with respect to the
Drinfeld pairing) of Wg(n~) (E{0) = F(Q) = 1). We assume that {E(i)} is a ho-
homogeneous basis with respect to the Cartan subalgebra, i.e. qhE(i)q~h =

158
LECTURE 10. QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS.
It follows from the definition of «,(§) that E{i) and F(i) behave in the following
way with respect to the comultiplication:
A0.46)
= F(i) ® 1
where Ts(i,j) e «,(§), s = 1,2, the degree of Ti(i, j) is lower than the degree of
E(i), and the degree of T2(i, j) is higher than the degree of F(i).
Now let us compute the element U in terms of E{i), F(i). We have (see Theo-
Theorem 9.8.1)
A0.47)
U =
ld)(q2kd ® l)Kop(q
op(q-2kd
1)))
Since the elements et annihilate the vector «3, and since $2 is an intertwining
operator, we have A(ej)$2(«2)«3 = 0, which implies
A0.48) {ei <8> 1)$2(«2)«3 = —A <8> q
Therefore,
A0.49) (E(i) ® 1)*2(«2)«3 :
Similarly, since $1(^i) is an intertwiner, and «o is annihilated by /,, using
A0.46), we get
<w} = (u0,{'y®'y){Aop(F(i)
A0.50) v "' v
Using A0.47), A0.49), A0.50), we can write A0.43) in the form
A0.51) i
Now we have to compute the expression {uo,^1{z1)q~^ixi^2{z2)u3). We have
A0.52)
Substituting A0.52) in A0.51), comparing it with A0.44), and using the iden-
identity 7~2(z) = q-z^xq2?, we get
A0.53)
Z:?
10.10. ANOTHER PROOF OF THE QUANTUM KZ EQUATIONS
Let us now use the fact that the total weight of W is \i = A2 — Ao. This implies that
(h ® 1)* = n(h)y - A ® ft)*, ft e I). Applying this to A0.53), we get
A0.54) *(p-1«1,«2)
Now it remains to observe that
A0.55) ^2q-2Kl3'Wl'y-1(F
(the last equality is because of F.26)). Therefore, from A0.54) we get
A0.56)
or equivalently,
A0.57)
which coincides with A0.21) for N = 2 (two evaluation representations) and j = 1.
Now let us apply A0.56) to the case where the two evaluation representations
involved are Wx = V1^) ® ... ® Vj{zj), and W2 = VJ+1(zj+1)
This yields
A0.58)
N.
It is easy to show, using the fusion properties of the .R-matrix F.9) and the
Yang-Baxter equation, that system A0.58) is equivalent to A0.21). This gives a
new proof of Theorem 10.3.1.
Of course, a similar method of proof applies to the classical KZ equations. We
omitted its discussion for brevity.

Lecture 11. Solutions of the Quantum
Knizhnik-Zamolodchikov Equations for s\
In this lecture, we will give explicit formulas for the solutions of the quantum
KZ equations for g =5B. These formulas are natural (/-analogues of the formulas
of Lecture 4. The simplest solutions of the quantum KZ equations were given in
the original paper [FR]. The general formula for the solutions in the case g = sl2
was conjectured by A. Matsuo [Mat3], who proved it in certain special cases (see
[Mat2, Mat3]). A complete proof was given by A. Varchenko [V3]. Except for
the simplest case, we omit the proofs, since they are much more technical than in
the classical case.
As a preliminary, we define analogues of some classical special functions and
study their properties. The most important of these functions is the (/-hypergeomet-
ric function, also called the basic hypergeometric function, which was introduced
by E. Heine in the middle of the last century. We refer the reader to the monograph
[GaRaJ for further details.
Finally, let us note that generalization of these formulas to other simple Lie
algebras is a difficult problem. For example, for Lie algebras other than sln there
is no quantum version of the "evaluation homomorphism" g —> B, and therefore the
structure of the evaluation representations over affine quantum groups is much more
complicated than in the classical case (see Lecture 9). So, the best we can hope for
is to get integral formulas for the solutions of the quantum KZ equations for g = sln.
Such formulas were found by Tarasov and Varchenko in [TV1]. Unfortunately, even
the formulation of these results is very cumbersome. Therefore, in this lecture we
only consider g = sl2, referring the reader to [TV1] for the discussion of the sln
11.1. (/-analogues of classical special functions.
This section is devoted to (/-deforming various special functions. This subject
was created by Euler and is called <7-analysis. Throughout this section we assume
that q is a complex number with \q\ < 1.
Let us start with the power function
(ii.D (i -,)- = ± «(« + D
71=0
Its (/-analogue is the (/-power function
n=0

162
LECTURE 11. SOLUTIONS OF THE qKL EQUATIONS FOR 5I2
where we use the notation {a} = 3^-, and {n}\ = {l}{2}...{ra}.
It is clear that the g-power function converges to the power function in the
region \z\ < 1 when q —* 1. However, the <jr-power function continues to a function
meromorphic in z for z ? C (see below), while this is not the case for q = 1.
This property is typical in <jr-analysis: in the process of <jr-deformation a branching
singularity disappears and is replaced by an infinite sequence of poles and zeros.
We will see more examples of such behavior in the subsequent sections.
The <jr-power function has an infinite product expansion, which was found by
Cauchy.
Proposition 11.1.1 (q-binomial theorem).
A1.3)
n=0
PROOF. Denote the right hand side of A1.3) by G(z). Then we have
A - zqa)G(qz) = A - z)G(z).
This implies that if we write G{z) =
the recursive relations
qnAn - <f+n-1
l^=o Anzn, then the coefficients At satisfy
n.1 = An - An_,,
Solving this recursive relation and taking into account that Aq = 1, we get An =
{a}{a+%)\a+n~1}, and thus G(z) = {A - z)~a}. ?
As an immediate corollary of the (/-binomial theorem we note the following
useful identity:
A1.4)
Ui *> s~ {A-xf)<>y
Now let us consider the exponential function
Its most obvious ^-analogue would be Y^L
nient to choose another generalization, namely
l • However, it is more conve-
conveObviously, as q —» 1 we have Eq(z(l — q)) —» exp(z).
This function also admits an infinite product expansion, which is due to Euler:
oo
A1.6) Eq(-z) = T[(l-zqn).
11.2. JACKSON INTEGRAL.
Indeed, to obtain this formula, it is enough to set u = zqa in A1.3), and con-
consider the limit as qa —> 00. This procedure is the <jr-analogue of the formula
artoo(l-«/a)"a = e2.
Now let us qr-deform the T-function. Recall that the T-function is given by
(The first formula is valid when Re a > 0.) The second formula is Euler's formula
(see, for example, [WW]). Since T(a + 1) = ar(a), this implies that
Let us define the <jr-gamma function by
Clearly, r, -> V as q
We have
Thus,
1.
l-qn
A1.9)
In particular, if n is a nonnegative integer, then Tq(n + 1) = {n}\, similarly to
F(ra +1) = n\. Note also that it immediately follows from the definition that
11.2. Jackson integral.
The g-analogue of integration in g-analysis is a certain summation which is
called Jackson integration.
Let f(t) be a function of a complex variable. Let a e C. The Jackson integral
of / is defined by the formulas
f
Jo
A1.10)
1"
Ja
n=0
f(t)dqt = a(l-q) f^ f(aqn)qn,
n=~co
3 -1
f(t)dqt = a(l-q) 22 f(°4n)qn-
n=—oo
It is easy to see that the Jackson integrals A1.10) coincide with the Riemann sums
for /(*) corresponding to the partition of the interval of integration by points of
the geometric progression aqn. Therefore, if / is a holomorphic function, then in

164 LECTURE 11. SOLUTIONS OP THE gKZ EQUATIONS FOR S[2
the limit as q —> 1,0 < g < 1, the Jackson integrals converge to the usual integrals
of f(t) over [0, a], [0, oo], and [o, oo], respectively.
As an example, let us consider the <jr-analogue of Euler's beta-function. Recall
that B(x,y) = IX32+^ ¦ Analogously, define
A1.11)
Bq
Tg(x)Tg{y)
Recall that the classical S-function has an integral representation
A1.12) B(x,y) =
Similarly, the <?-beta function has a Jackson integral representation.
Proposition 11.2.1.
. -t
Proof. It follows from the definition of the ?-gamma function that Tq{x+1) ¦¦
Lrg(x). Therefore, we have the difference equations
(n.14) 7_y
Bq{x,y+l)= x_ ;+yBq{x,y).
Combining these two equations, we get
A1.15) qvBq(x + l,y) + Bq(x,y + l) = Bq(x,y).
On the other hand, denote by b(x, y) the right hand side of A1.13). Then it is very
easy to see that b(x,y) satisfies A1.15). Also, it is obvious that
b{x, 1) =
*-U-
Therefore, we have
Bq{x,n) = b(x,n), n
Now, it is easy to see that both Bq(x,y) and b(x,y) are holomorphic functions
of s = qy near s = 0 once a value of x such that Re x > 0 has been fixed. By
the analytic continuation principle, we have the equality Bq — b for Re x > 0,
Re y > 0. ?
11.3. The g-hypergeometric function.
The q-hypergeometric function 2</>i(<?a, qb; qc; 1, z) was introduced by Heine and
is defined by the formula
11.4. SOME SECOND ORDER DIFFERENCE EQUATIONS.
It is clear that this function is a <jr-deformation of the Gauss hypergeometric function
given by D.19). It is easy to check directly that this function satisfies the following
second order difference equation:
A1.17) (z{d + a}{d + 6} - {d}{d + c-
6; <7C; q, z) = 0,
where d =
A1.18)
and
This equation is a <jr-analogue of the hypergeometric equation of Gauss. It is easy
to show that 2^>i (qa, qb; qc; q, z) is the only solution of this equation regular at z = 0
6
As we know, the Gauss hypergeometric function has an integral representation
Analogously, the ^-hypergeometric function has a Jackson integral representation.
Proposition 11.3.1.
Jo
di iQ") & ( a b c \ — q ¦
) 29l{q'q''q;q'z>-rq(b)vq(c-b)J0" {(i_t)»-c}i-t-
This proposition is proved by verifying that the right hand side of A1.19)
satisfies the difference equation A1.17) and has value 1 at z = 0. The complete
proof can be found in [GaRa].
11.4. Some second order difference equations.
Consider a difference equation of the form
A1.20) (Aoz + B0)f(q2z) + {AlZ + B^fiqz) + {A2z + B2)f{z) = 0,
where /(z) is a scalar function, and AitBi are complex numbers. In this section we
will see that a generic equation of this form reduces to Heine's ^-hypergeometric
equation A1.17).
First of all, Heine's g-hypergeometric equation can be written in the form
A1.20), as follows:
A1.21) (qa+bz - <?c-1)/(<?2z) + {-{qa + qb)z + q^1 + l)f{qz) + (z - l)f(z) = 0.
Now assume that we are given any equation of the form A1.20). If B2 ^ 0, A2 ^ 0,
we can set w = — z^, g(w) = /(z), and reduce A1.20) to the form
A1.22) (A'ow + B'0)g(q2w) + {A\w + B^qw) + (w - l)g(w) = 0,
where A'- = 4*-, B1, = — -f1-. Let s be a solution of the equation
A1.23) B[qs + B'oq23 = 1.
In general, there are two solutions of this equation (up to adding integer multiples of
2th/ log q). After making a change of variable g(w) = wsh(w), we get the following
equation for h(w):
A1.24) {A'o-w - B?)h{q2w) + (A'{w + 1 + B^)h(qw) + (w - l)h{w) = 0,

166
LECTURE 11. SOLUTIONS OF THE gKZ EQUATIONS FOR 5[2
where Aq = q2sA^, Bo' = —q2sB'Q, and A" = qsA'1. This is an equation of the form
A1.21), where
A1.25) A%=qa+b, A'{ = -(qa + qb), Bq = g^1.
For generic Ai,Bi, these equations have a unique solution (a,b,c), up to adding
integer multiples of 27ri/logg and permutation (a, 6) —> F,a).
We have proved the following proposition.
Proposition 11.4.2. Let AitBi be generic, and let Sj,aj,bj,Cj,j = 1,2, be
the two distinct solutions of the system of equations
q2sB0 + q"Bi + B2 = 0,
A1.26)
B2
Then the functions
'(z) =
a', qb'; q°»; q, -zA2/B2)
are the only solutions of A1.20) which have the form f(z) = zsg(z) for some
function g(z) regular in a neighborhood ofO and satisfying g@) = 1.
11.5. The simplest solution of the quantum Knizhnik-Zamolodchikov
equations and the g-hypergeometric function.
In this section we will show that the simplest nontrivial solution of the quantum
KZ equations, which arises when the space of solutions is two-dimensional, is given
by the g-hypergeometric function. Again, these computations are parallel to the
ones in the classical case (see Section 4.2).
Recall the modified quantum KZ equations, which we will write in the following
form:
A1.27)
Zj+1
where W takes values in the tensor product VIX1 <g>... <g> V^N of highest-weight repre-
representations of Uq (sk), and<jr'2A'J = itj{q2X), where q2X acts on a subspace of weight fi
by 9<2A'**>. All other notation is as in Lecture 10. These equations coincide with the
modified <?KZ equations A0.39) if we let A = (Ao + Ajv + 2p)/2 and use the unitarity
of the renormalized .R-matrix: R\j{x)Rji{x~1) = 1 (see Proposition 9.5.5).
As before, we identify the dual of the Cartan subalgebra for sfe with C so that
ah»2; then {X,fj,} = A/i/2.
11.5. SIMPLEST SOLUTION AND g-HYPERGEOMETRIC FUNCTION
Let us consider the modified quantum KZ equations A1.27) for JV = 2, fix =
m, fi2 = n. Then they take the form
A1.28)
Let us restrict our attention to the case where $ takes values in the subspace
of weight m + n — 2 (in Lecture 4, this case was called "level 1 solutions"). Then
it follows from A1.28) that *(p^i,p^2) = qx(-m+n^2)9(zuz2). If we additionally
assume that W has the form 9(zi,z2) = zf1 z%*(a meromorphic function of z\,z2)
for some c*i, a2 ? C, then it is easy to see that
A1.29) 9{zi, z2) =
and ip(z) satisfies the following equation:
A1.30)
It is convenient to renormalize ip by introducing
i
tp(z) = z *• i>{z).
Then equation A1.30) can be rewritten as follows:
A1.31) i/,(pz) = gBA)l+A(m-n>fi(z)V'(^).
Since the weight subspace of weight m + n — 2 is two-dimensional, ip can be
written as i/>(z) = /4)i{z)fvm.®vn+ilJ{z)vrn®fvn, where ip\, ijJ are scalar functions,
and vm,vn are the highest-weight vectors in Vm, Vn respectively.
Using the formulas for the .R-matrix in this space from Example 9.6.5, we get
the following equations for ^l, ife:
f 11321
From this we get the following second order difference equation for ipi(z):
A1.33)
(pz - qm+n)i>i(p2z)
We have seen in Section 11.4 that for generic values of parameters, any equation
of this form is equivalent to the hypergeometric equation, and therefore the solutions
are given by ^-hypergeometric functions (note that now p plays the role of the
parameter q). The values of parameters are determined from the following system

168 LECTURE 11. SOLUTIONS OF THE qKZ EQUATIONS FOR
of equations (compare with A1.26)):
- qrn+np2s + (pqn+X
-2s 1 _-*+b
m+A , n-X
- p = 0,
Recalling that p = q~2K, we see that, up to adding integer multiples of 27rilogp and
interchanging a <-> b, these equations have two sets of solutions:
A1.34)
n + X n m + n + 2A n — m + 2A
= -, &i = c
x" 2« ' ~ «' "x" 2«
7Ti — A to m + n — 2X
S2 = H — , O2 = 1 H , 02 = 1 +
2k
2«
2k
= 2+
m - n - 2\
2~k '
This gives a solution for tpi{z) in terms of the <?-hypergeometric function. Since
the first equation in A1.32) allows us to express ip2{z) as a linear combination of
i/>i(z) and ip\(pz), we have the following result.
Proposition 11.1. Let N = 2,fii = m,fj,2 = re. T/ien <te solutions of the
quantum KZ equations A1.28) in the subspace of weight m + n —2 are given by
A1.35)
fvn),
where z = z\/z2, and for j = 1,2,
J; pCi; p,
vrith Sj,aj,bj,Cj given by A1.34).
11.6. Integral formulas for solutions
In this section we give integral formulas for the solutions of the qKZ equations
A1.1) for g = 3B, following Matsuo's paper [Mat 3].
We are considering solutions 9>(zi,..., zjv) of A1.1) taking values in the sub-
space of weight Y^Pi ~ 2m °f ^e tensor product VM1 ® ... <g) V^. The number
m e Z+ will be called the level of the solution. These solutions will be given by
integral formulas similar to D.29).
11.6. INTEGRAL FORMULAS FOR SOLUTIONS
Let us start by defining the analogue of the local system and its homology.
Denote for brevity z = (zu ..., zN), t = (ti,..., tm), and let
A1.37)
n (?)"" fp -«-^f ¦} n
Here {A — z)~a}p is the g-analogue of the power function, defined by A1.2), with
q replaced by p.
We will be interested in integrals of the form / <6(z, t)p(z, t)dPt\... dptm, where
p is a rational function of z, t. In principle, integrals like this could be realized as
certain Jackson integrals. However, it is difficult to do so explicitly. Therefore,
in this section we will treat the integral purely formally: we will use the symbol
/ f(t, z)dpti... dptm to denote any functional on functions of the form /(z, t) =
4>(z, t)p(z, t) satisfying the condition
where {dtf}p = ^p 'Z{ is the quantum analogue of the operator ijj (compare
with A1.18)). We will not discuss the question of explicitly constructing such
integrals, dimension of the corresponding "homology space", or even existence of
them, referring the reader to [A2].
Finally, recall that in the classical case, the rational function p was Sm-sym-
metric. It turns out that this symmetry condition must also be deformed in a
nontrivial way.
Lemma 11.6.1. Let o~i,i = l,...,m — 1, be the standard generators of the
symmetric group Sm. Define
A1.38)
= f(h,..., ti+1,tit ...,
U - q2ti+i
Then this can be extended to an action of S^ on the space of rational functions of
t\, ¦ ¦ ¦ ,tm.
The proof of this proposition is straightforward.
Now we can formulate the main theorem of this section, conjectured in [Mat3]
and proved in [V3].
Theorem 11.2.2. The following function >t is a solution of the quantum KZ
equations A1.1) with values in (VMl ® ... ® VMN)^-''i'~ m':
A1.39)
h
U
where $ is given by A1.11), the sum is taken over all m ? Z^ such that^rrii = m,
ai = rn\ + • ¦ • + m.i, and
v(m) _
fmNVnN
[17.AT]!

LECTURE 11. SOLUTIONS OF THE gKZ EQUATIONS FOR
i=l aj-i+l^J^a
Finally,- pm is given by
A1.40) Pm = <7Ei
where
AL41) Aii = 1,-^v fly= s,-gW
The proof of this theorem is quite technical, and we refer the reader to [V3].
The proof is significantly simplified if we only consider level one solutions (m = 1),
which is done in [Mat2]. In particular, if we let N = 2, m = 1, then the solutions
can be written in terms of the g-hypergeometric function (see Proposition 11.1.1),
and the integral formulas given by A1.13) coincide with the integral formula A1.19)
for the <jr-hypergeometric function.
Lecture 12. Connection Matrices for
the Quantum Knizhnik-Zamolodchikov
Equations and Elliptic Functions
In this lecture we will study the connection matrices for the quantum KZ equa-
equations, which belong to the general class of linear difference equations. Linear differ-
difference equations in one variable were studied by G. D. BirkhofffBi], R. D. Carmichael
[Ca], W. J. Trjitzinsky [Tr], C. R. Adams [Ad] and others in the early 20-th cen-
century. Their results were extended to the case of commutative systems of several
variables by K. Aomoto [A2]. It follows from the general theory of difference equa-
equations that the connection matrices, which relate the fundamental solutions with
fixed asymptotics in different asymptotic zones, are expressed in terms of elliptic
functions. We will give these matrices explicitly in the simplest case, when they
coincide with the connection matrices for the q hypergeometric equation.
The important property of the connection matrices for the quantum KZ equa-
equations is that they can be interpreted as exchange matrices for intertwining operators
between representations of the quantum affine algebra. This fact is a (/-analogue
of the relation between monodromy of the KZ equation and exchange relations for
intertwining operators between representations of an affine Lie algebra, which was
established in Lecture 8. This property yields a large class of solutions of the star-
triangle relations. In the simplest case one recovers Baxter's original solution of
the XYZ model in statistical mechanics [Bax]. These results were originally dis-
discovered by I. B. Frenkel and N. Yu. Reshetikhin in [FR]. More recently, the general
connection matrices for the quantum KZ equations in the sh case were explicitly
identified by V. Tarasov and A. Varchenko in [TV5].
12.1. Linear difference equations for functions of one complex variable.
We will start with first order scalar equations.
Recall first that the power function f(z) = A — z)
solution of the differential equation
can be defined as the
A2.1)
/' =
1- zJ
with the initial condition /@) = 1. Analogously, the g-power function {A — z) "}
defined by A1.3) can be characterized as the solution of the difference equation
A2.2)

170 LECTURE 11. SOLUTIONS OP THE gKZ EQUATIONS FOR s[2
Finally,- pm is given by
N
A1.40)
where
A1.41)
Pm =
The proof of this theorem is quite technical, and we refer the reader to [V3].
The proof is significantly simplified if we only consider level one solutions (m = 1),
which is done in [Mat2]. In particular, if we let N = 2, m = 1, then the solutions
can be written in terms of the g-hypergeometric function (see Proposition 11.1.1),
and the integral formulas given by A1.13) coincide with the integral formula A1.19)
for the <jr-hypergeometric function.
Lecture 12. Connection Matrices for
the Quantum Knizhnik-Zamolodchikov
Equations and Elliptic Functions
In this lecture we will study the connection matrices for the quantum KZ equa-
equations, which belong to the general class of linear difference equations. Linear differ-
difference equations in one variable were studied by G. D. Birkhoff[Bi], R. D. Carmichael
[Ca], W. J. Trjitzinsky [Tr], C. R. Adams [Ad] and others in the early 20-th cen-
century. Their results were extended to the case of commutative systems of several
variables by K. Aomoto [A2]. It follows from the general theory of difference equa-
equations that the connection matrices, which relate the fundamental solutions with
fixed asymptotics in different asymptotic zones, are expressed in terms of elliptic
functions. We will give these matrices explicitly in the simplest case, when they
coincide with the connection matrices for the q hypergeometric equation.
The important property of the connection matrices for the quantum KZ equa-
equations is that they can be interpreted as exchange matrices for intertwining operators
between representations of the quantum affine algebra. This fact is a ^-analogue
of the relation between monodromy of the KZ equation and exchange relations for
intertwining operators between representations of an affine Lie algebra, which was
established in Lecture 8. This property yields a large class of solutions of the star-
triangle relations. In the simplest case one recovers Baxter's original solution of
the XYZ model in statistical mechanics [Bax]. These results were originally dis-
discovered by I. B. Frenkel and N. Yu. Reshetikhin in [FR]. More recently, the general
connection matrices for the quantum KZ equations in the sl2 case were explicitly
identified by V. Tarasov and A. Varchenko in [TV5].
12.1. Linear difference equations for functions of one complex variable.
We will start with first order scalar equations.
Recall first that the power function f(z) = A — z)
solution of the differential equation
can be defined as the
A2.1)
cl
1-Z
f
with the initial condition /@) = 1. Analogously, the g-power function {A — z) a]
defined by A1.3) can be characterized as the solution of the difference equation
A2.2)

172 LECTURE 12. CONNECTION MATRICES FOR ?KZ EQUATIONS
with the same initial condition. We can rewrite this equation as
/(«*) - f{z) _
A2.3)
;/(*).
which obviously goes to the differential equation A2.1) for the power function as
q —> 1. This is another way to see that the <jr-power function tends to the power
function as q —> 1.
Now let us consider the connection problem. The power function is well defined
for \z\ < 1. We want to extend it outside of this disk, in particular, find its values
for z ? R+, \z\ > 1. Let 7+, 7_ be the paths which connect 0 with a given point
2o > 1 and pass the branching point z =¦ X from above (respectively below). Let
A+fn, A_f0 be the analytic continuations of fo{z) = A — z)~a to the neighborhood
of zq along 7+,7_. Then we have A+fo(z) = e~2niaA-fo(z), when z is near zq.
We also have a solution fao(z) defined for \z\ > 1, z ^ R~, such that f<x>(z) ~ z~a,
z —> +oo. It is clear that A±fa = e:F'rla/<x,. This identity, connecting two solutions
fo, /oo defined by their asymptotics at 0, oo with each other, is called the connection
identity. The constant C± = e^7"" is called the connection coefficient.
This construction can be generalized to the q-case. In this case we are in an
even better position, since we do not have to choose a path of analytic continuation
- our functions are single-valued.
Consider the difference equation for the <?-power function. Aside from the
solution /o(z) = {A — z)~a}, it has another solution /oo(z), meromorphic in z q-
R~ U {0}, and defined uniquely by the asymptotics f<x>(z) ~ z~a, z —» +oo. This
solution can be constructed as the product fco{z) = z~aY[^-i t!.p «-.¦ Thus,
we can write down the connection equation fn(z) = C(z)fao(z), where
A2.4)
where
A2.5)
- zqn
a)(l -
A - zqn){X -
Q(z;q)
is the standard elliptic theta-function, up to a constant.
Let a(z) be a multivalued function of the form zsao(z), where ao{z) is a mero-
meromorphic function in a region Q, c CUoo. Then we say that a(z) is quasimeromorphic
in Q.
The function C(z) is quasimeromorphic in Cx. Also, since it is obtained from
the connection equation, it satisfies the condition C(qz) = C{z). We call functions
satisfying such conditions pseudoconstants.
Let us now consider a general picture. Let q = e*. Let A(z) € GLn{C(z)).
Assume that A{Q) = A$ and A(oo) = A^, and both of these matrices are invertible.
We also assume that each of the matrices j4o, A^ has distinct eigenvalues such that
the ratio of two eigenvalues is never equal to an integer power of q.
Consider the difference equation
A2.6) F(qz) = A{z)F(z),
where F is a vector-function taking values in Cn. Then we have the following result
(see, e.g., [Tr]).
12.1. LINEAR DIFFERENCE EQUATIONS FOR SCALAR FUNCTIONS
173
Theorem 12.1.1. Let V\,... ,vn be an eigenbasis of An, and let ui,... ,un be
an eigenbasis of A^. Fix complex numbers a\,..., an, bi,...,bn such that Aqvj =
e^Vj, AxUj = etb'Uj. Then there exist unique solutions Fi,...,Fn, Gi,...,Gn
of A2.2) of the form
A2.7) Fj(z) = z"'F°(z), Gj(z) = z»iG°(z),
where F° are regular at 0 and meromorphic in Cx, G° are regular at oo and mero-
meromorphic in C*, and F°@) = Vj, G°(oo) = Uj.
The solutions Fi will be called asymptotic solutions near 0, corresponding to
the vectors Vi.
Now we can write down the connection identity. Since the functions {Fj} and
{Gj} form bases of Cn at a generic point, there exists a unique matrix C{z) of
quasimeromorphic functions in Cx such that
A2.8) F,(z) = Y.Cik
The matrix C(z) is called the connection matrix. It satisfies the equality C{qz) =
C(z). Thus, the entries of C(z) can be expressed as a product of a constant, a power
of z, and a ratio of products of theta-functions. The problem of finding the matrix
C(z) explicitly is not less difficult than the problem of finding the monodromy of a
system of differential equations, and in general cannot be solved. However, in the
special case of the quantum KZ equations C(z) can in principle be found explicitly.
In the next section, we do it for the case considered in Section 11.3, i.e. when the
space of solutions is two-dimensional.
This construction can be slightly generalized. Suppose that A@) and A(oo)
are diagonalizable, and none of them has two eigenvalues whose ratio is a nonzero
integer power of q (i.e. multiple eigenvalues are allowed). In this case, it can be
shown that Theorem 12.1.1 still holds, and thus a connection matrix can be defined.
This generalization is important for the quantum KZ equations, since in this case
A@) and A(oo) will often have multiple eigenvalues.
Remark 12.1.1. Morally, a connection matrix introduced in this section is the
monodromy "from zero to infinity" for ^-difference equations with one variable. We
should emphasize that we do not know an adequate notion of monodromy between
any other points than 0 and oo.
Remark 12.1.2. It is necessary to note that from the point of view of the
monodromy theory the classical limit q —> 1 is quite nontrivial. As we have already
mentioned, one of the most subtle issues is the transformation of a condensing
sequence of zeros and poles into branching, which becomes transparent when one
considers closely the limit {A — z)~a} —> A — z)~". As q —t 1, the zeros and poles
of the function {A — z)~a}, which are q~n,q~~n~a, n > 0 (as seen from formula
A1.3)), accumulate and in the limit fill the entire ray {z > 1}. As a result, outside
of this ray the function {A — z)~a} converges to A — z)~a uniformly on every
compact set, while the ray z > 1 becomes the branching locus in the limit.
Now consider the equation A2.2) and its solutions fo(z) = {A — z)~a} and
fao{z). The function fo(z) is a single-valued meromorphic function, while f<x{z) is
a quasimeromorphic function, i.e. it is a product of the power function z~a and a
meromorphic function. Thus we will understand /„ as a single-valued function on

174 LECTURE 12. CONNECTION MATRICES FOR qKZ EQUATIONS
C* \ R_ and having a cut at R_. In the Emit as q —* 1, the zeros and poles of
/oo fill the interval @,1], but outside of (—oo, 1] the function /oo(z) converges to
its classical limit (z — l)~a, uniformly on every compact set.
Let us now consider what happens with monodromy in the limit as q —> 1.
According to our definition, the monodromy is the connection matrix, C = /o/^,1,
so it has a limit as <jr —> 1 only away from the real axis (i.e. where both /o and
/oo are regular). The set C \ R is disconnected and consists of two connected
components - the upper and the lower half-plane. It is easy to check that C(z) has
two different limits in these two regions: enm in the lower half-plane and e~nia in
the upper half-plane. This corresponds to the fact that in classical theory (q = 1)
we have two ways to define the monodromy, corresponding to two paths of analytic
continuation.
12.2. Connection relation for the <jr-hypergeometric equation.
Consider the function f(z) = 2(f>i(qa,qb,qc',q,z~x). This function satisfies the
difference equation
A2.9) A - q2z)f(q2z) + ((qc+1 + q2)z - qa - qb)f(qz) + (qa+b - qc+1z)f(z) = 0,
which is obtained from A1.21) by replacing z with z'1. Set g(w) = f(qa+b~c~1w).
Then g(w) satisfies the difference equation
A2.10) {q1-cw-q-a-b)g{q2w)+(-{l+q1-c)w+q-b+q~a)g{qw)+{w-l)g{w) = 0.
Set g(w) = wsh(w). If s = a or s = b, this transformation reduces A2.10) to the
^-hypergeometric equation A1.21), with new parameters (a,b,c), given by
A2.11)
(a',6',c') = (a, a- c+ l,o — 6 + 1) for s = a,
(a", 6", c") = F - c + 1,6,6 - a + 1) for s = b.
This implies the following proposition.
Proposition 12.2.1. The functions
A2.12)
are solutions of A1.21) quasimeromorphic on Cx U oo.
It is clear that for generic values of the parameters the functions «i, «2 are
linearly independent. Therefore, the function 2<t>i(qa,qb,<lc',<l,z), which is also a
solution of A3.21), should be expressed as a linear combination of «i, «2 with pseu-
doconstant coefficients. This representation is given by the following proposition,
which generalizes the result of Gauss on the monodromy of the hypergeometric
function.
Proposition 12.2.2.
°; q, z) = a(z)Ul(z) + C(z)u2(z)
A2.13)
r,F)r,(c-a) 9(z;q) '
Tq(c)Tq(a-b)z"e(zqb;q)
rq(a)Tq(c-b) Q(z;q) '
12.3. THE CONNECTION MATRIX IN THE SIMPLEST CASE
The proof of this theorem is based on the integral formula A1.19). It can be
found in [GaRa].
12.3. The connection matrix for the quantum
Knizhnik-Zamolodchikov equations in the simplest case.
In this section we will compute the connection matrix for the difference equation
A1.32), which is equivalent to the quantum KZ equations in the simplest nontrivial
case. In this case the matrix A(z) equals
A2.14)
A{z) =
z— g"+w
Recall that the basic solutions of A2.6) for such A(z) which are quasimeromorphic
in C have the form
A2.15)
where
A2.16) ^j2)W
and ^2 = ¦2'V'i i where the operator E is defined by
A2.17) Ef(z) =
Similarly, using Proposition 12.2.1, we can write the basic solutions which are
quasimeromorphic in C* U oo: they have the form
A2.18)
where
A2.19) V]^Z)
and <p2'} = Eip^
Let B(z) = (Bij(z)), i,j = 1,2, be the connection matrix between the solutions
ip(%) and </>(*', i.e.
A2.20)
have
A2.20)
Using Proposition 12.2.2, we can find the pseudoconstants By B). Indeed, we

176
LECTURE 12. CONNECTION MATRICES FOR gKZ EQUATIONS
Substituting A2.16), A2.19) into this equation, and using A2.13), we get
rp(l + 2^±gA)Fp(^g±gA) ziQ{qm-nz.p)
B\\ =
A2.21)
Q(q"+mz;p) '
^ff^) 2 "+"~+2A 9(g~2Az; p)
e(qn+mz;p)
-B22 —-
Using the g-Euler identity F,(x)r,(l — x) = C(q)/Q(qx;q), it is easy to show
that the cross-ratio B11B22/B12B21 can be expressed purely in terms of theta-
functions (i.e., the F-terms cancel). This implies that one can renormalize the
functions i>^\ <f>^ in such a way that the matrix B will contain no F-factors.
For g = 3B and general m, n it was proved in [TV5] that such a renormalization is
always possible. This renormalization is closely related to the vertex-IRF transform
of statistical mechanics (see [P]).
The corresponding renormalized solutions Ft have a geometric meaning - they
are integrals over certain cycles of g-hypergeometric differential forms. The matrix
B can be interpreted in terms of elliptic quantum groups [FV2]. We do not discuss
this theory here, and recommend reading the original papers [TV5, Fel, FV2,
FV3].
12.4. The connection matrix and the
exchange matrix for intertwining operators
We want to generalize to the quantum affine case our previous definitions of
the exchange matrices: the definition of Lecture 6 for the quantum group Uq(g)
and the definition of Lecture 8 for the affine Lie algebra jj.
As before, let us assume that the level k is generic, q is not a root of unity, and
|p| < 1, where p = q~2K. Let V, W be finite-dimensional representations of Uq{$)
and let A, v G P+. It follows from Theorem 9.3.1 that we have an isomorphism of
spaces of intertwining operators:
A2.22)
, LI ® V ® W) cz
® V(Zi) ® W(z2)),
which can be described as follows. Note that because of complete reducibility,
l, LI ® V ® W) =* 0 H %
where, as before, H»v = Homw,,(s)(L?, L» ,
set
A2.23) l>9l®92B1,z2) = (I>9I(
>!/). Then for ^Gi^.sa
i)®Id)o*^(z2),
H*w we
where <&9(z) is as in Proposition 9.3.2. This intertwiner is well-defined as an analytic
function of zi,Z2 for \z\\ S> \z2\. It follows from Theorem 10.6.3 that this intertwiner
12.4. THE EXCHANGE MATRIX FOR INTERTWINING OPERATORS
can be analytically continued to a meromorphic function in the region \z2\ > \z\\;
we will denote this analytic continuation by A#9(zi,z2). Note that unlike the
classical case, we do not need to choose a path of analytic continuation. Thus,
the product R(z1/z2)A^(z) is a W,(?j)-intertwuier L\k -* L\k ® W(z2) ® V(Zl),
well defined for \z2\ » \z\\. Hence, it must have the form #9'(z2,Zi) for some
g' e HomWii(g)(L^, L« ® W ® V). We define the exchange matrices
B±{z1/z2,\, u) :
by
A2.24)
, L% ® V ® W) ~> HomWg(g)(L«, L% ® W ® V)
R±(z1/z2)Ai3(z1,z2) =
2,21)
(compare with (8.23)). Note that because of the term ^±(z1/z2), the matrix B
nontrivially depends on Z\/z2, unlike the exchange matrix for affine Lie algebras
which did not depend on Zj.
Proposition 12.4.1. The exchange matrices B± satisfy the following rela-
relations:
B$w(z)B*v(z-')= Id,
A2.25)
A2.26)
where we use the same conventions as in Proposition 6.8.1.
This theorem is completely parallel to the relations for the exchange matrices for
quantum groups and affine Lie algebras (see Proposition 6.8.1 and Proposition 8.5.3,
respectively), and is proved in the same way as Proposition 6.8.1.
Now we can formulate the result about the relation of the connection matrices
and exchange matrices. Note that we have a natural map
A2.27)
where Ua,uJ are the highest-weight vector in ?' and the lowest-weight vector in
{L%)*, respectively. It follows from irreducibility of L\, Lv that this map is injective
(compare with the arguments in the proof of Theorem 3.1.1).
Consider the quantum KZ equations A1.21) with two variables, A2 = A, Ao = f,
Vi = V, V2 = W. These equations are with respect to a function with values in
(V® W)x~", and they reduce to an equation with one variable C, — z2/z\. Let C(Q
be the connection matrix for this equation.
PROPOSITION 12.4.2. The restriction of the matrix C(() to the image of the
embedding A2.27) coincides with ByW(Q~^, A, is) (up to multiplication by a constant
nondegenerate matrix from left and right and by a scalar meromorphic function).
Here we denote Byw = ¦
PROOF. The proof follows from the fact that for gi G H?v and g2 G Hxw the
correlation function ^(zl7z2) = (v* k, $91®S3(z1, Z2)vx,k) is an asymptotic solution
of the quantum KZ equations in two variables in the region \z\\ S> \z2\, correspond-
corresponding to the vector <u*, C, ® g2)v\) G (V ® W)x~". This statement follows from the
results of Lecture 10. d

178 LECTURE 12. CONNECTION MATRICES FOR <zKZ EQUATIONS
Thus, Proposition 12.4.2 shows that the connection matrix for the 2-variable
quantum KZ equations can be interpreted as an exchange matrix. This fact leads
to a factorization of connection matrices for the AT-variable quantum KZ equations,
similarly to how it happens in the classical case (Lecture 8).
COROLLARY 12.4.3. The matrices ByW(Q are elliptic functions, i.e.
Finally, we note that we can allow the weights A, v to be generic while keep-
keeping the representations V, W finite-dimensional, similarly to the construction in
Section 6.8. In this case the embedding A2.27) is an isomorphism, and thus the
exchange matrix can be considered as an operator
Taking direct sum over all v and denoting B = PB, we get operators B±(z,X) :
V ® W —> V ® W. Note that the operator B preserves weight, but is not a Uq(o)-
homomorphism. In this case, Proposition 12.4.1 takes the following form.
PROPOSITION 12.4.4. The exchange operator B±(z,X), defined above, satisfies
the following relations.
1. The modified (or dynamical) Yang-Baxter equation: for any three finite-
dimensional representations V, W, U
BfivWzt, A - h^fBtf^/zs, XfBftuizt/za, A -
A2.28) = B§u(z2/z3, X)±B1v3u(z1/z3, A - h&)±B1v2w(z1/z2, A)=\
where B\,2w(z,X - hS^^iy <g> w <g> u) := B]^w(z,X - w^m))*^ <g> w <g> u),
v G V, w 6 W, u 6 U if u is homogeneous, and wt(u) denotes the weight of
u (the other notation is defined similarly).
2. The unitarity condition:
A2.29)
Byw(z , A)T =
Note that if V,W are irreducible, then R+(z) and R (z) satisfy the condi-
condition R+(z) = ?(z)R~(z), where ? is a scalar function. Therefore, ByW(z,X) =
Z(z)ByW(z,X).
Lecture 13. Current Developments
and Future Perspectives
In this lecture we discuss problems related to the Knizhnik-Zamolodchikov
equations which naturally arise from our exposition. Some of these problems were
first formulated in [FR], where the general program of study of the quantum case
was outlined. Since the appearance of [FR], this program has been developed in
numerous papers. Many questions have been answered, and many of the problems
mentioned below are being intensively studied while we write these lines. However,
the reader will be able to find those that are still waiting their turn.
13.1. Knizhnik-Zamolodchikov equations: quantum versus classical
The exposition of our book is largely based on the parallel between the represen-
representation theory of affine Lie algebras and quantum affine algebras, and consequently
on the analogy between the Knizhnik-Zamolodchikov equations and their quantum
counterpart. However, finding a g-deformation of various structures that appear
in the representation theory of affine Lie algebras in relation to the KZ equations
usually requires ingenuity and often resists straightforward generalizations. In this
and the next two subsections, we will indicate some missing constructions in the
quantum theory.
In Lectures 4 and 11 we presented an explicit form of solutions of the KZ
[SV2] and quantum KZ [Mat3, V3] equations for s^ at arbitrary level. They
are expressed in terms of certain hypergeometric integrals and their ^-analogues,
which encode a lot of structure. In particular, the integrands in the nondeformed
case were factored via the free bosonic fields and vertex operators. This naturally
leads to the free field realization of SI2, as we explained in Lecture 5. Conversely,
the free field realization of sl^ representations and intertwining operators yields
Schechtman-Varchenko integral formulas for solutions of KZ equations.
The remarkable analogy between the undeformed and deformed cases of the
KZ equations and their solutions naturally leads to the question about the free
field realization of Uq(sl2) representations and factorization of Matsuo's formulas.
By now, a number of free field realizations of quantum affine algebra Uq(s[2) have
been produced [AOS, Mat4]. Also, intertwining operators in these realizations
were constructed explicitly, and therefore one can deduce certain expressions for
solutions of quantum KZ equations. Unfortunately, the resulting formulas are too
involved and lack the simplicity and at the same time richness of the integral for-
formulas discovered by Matsuo. To establish the relation between the two types of
expressions seems to be a rather hopeless task except for the most elementary cases
[Mat4].

178
LECTURE 12. CONNECTION MATRICES FOR gKZ EQUATIONS
Thus, Proposition 12.4.2 shows that the connection matrix for the 2-variable
quantum KZ equations can be interpreted as an exchange matrix. This fact leads
to a factorization of connection matrices for the AT-variable quantum KZ equations,
similarly to how it happens in the classical case (Lecture 8).
COROLLARY 12.4.3. The matrices ByW(Q are elliptic functions, i.e.
B$w(p() = B$W(C).
Finally, we note that we can allow the weights A, v to be generic while keep-
keeping the representations V, W finite-dimensional, similarly to the construction in
Section 6.8. In this case the embedding A2.27) is an isomorphism, and thus the
exchange matrix can be considered as an operator
B±(z, A, v):(V® W)x-" ->(W<3> V)x~"'.
Taking direct sum over all v and denoting B = PB, we get operators B±(z,\) :
V ® W —» V ® W. Note that the operator B preserves weight, but is not a Uq(o)-
homomorphism. In this case, Proposition 12.4.1 takes the following form.
PROPOSITION 12.4.4. The exchange operator B±(z,\), defined above, satisfies
the following relations.
1. The modified (or dynamical) Yang-Baxter equation: for any three finite-
dimensional representations V, W, U
B1v2w(z1/z2,\-
A2.28)
where B^w(z,X -
:= B\?w(z,\ -
w ® u) := tlyW(z, A - «rt(,«))±(i' ®w®u),
v G V, w e W, u 6 U if u is homogeneous, and wt(u) denotes the weight of
u (the other notation is defined similarly).
The unitarity condition:
A2.29)
,W-
-1, A)T = Idv®iv •
Note that if V,W are irreducible, then R+(z) and R (z) satisfy the condi-
condition R+(z) = ?(z)R~(z), where ? is a scalar function. Therefore, ByW(z, A) =
i(z)ByW(z,\).
Lecture 13. Current Developments
and Future Perspectives
In this lecture we discuss problems related to the Knizhnik-Zamolodchikov
equations which naturally arise from our exposition. Some of these problems were
first formulated in [FR], where the general program of study of the quantum case
was outlined. Since the appearance of [FR], this program has been developed in
numerous papers. Many questions have been answered, and many of the problems
mentioned below are being intensively studied while we write these lines. However,
the reader will be able to find those that are still waiting their turn.
13.1. Knizhnik-Zamolodchikov equations: quantum versus classical
The exposition of our book is largely based on the parallel between the represen-
representation theory of affine Lie algebras and quantum affine algebras, and consequently
on the analogy between the Knizhnik-Zamolodchikov equations and their quantum
counterpart. However, finding a (/-deformation of various structures that appear
in the representation theory of affine Lie algebras in relation to the KZ equations
usually requires ingenuity and often resists straightforward generalizations. In this
and the next two subsections, we will indicate some missing constructions in the
quantum theory.
In Lectures 4 and 11 we presented an explicit form of solutions of the KZ
[SV2] and quantum KZ [Mat3, V3] equations for sfe at arbitrary level. They
are expressed in terms of certain hypergeometric integrals and their g-analogues,
which encode a lot of structure. In particular, the integrands in the nondeformed
case were factored via the free bosonic fields and vertex operators. This naturally
leads to the free field realization of s^, as we explained in Lecture 5. Conversely,
the free field realization of sfe representations and intertwining operators yields
Schechtman-Varchenko integral formulas for solutions of KZ equations.
The remarkable analogy between the undeformed and deformed cases of the
KZ equations and their solutions naturally leads to the question about the free
field realization of Uq(sl2) representations and factorization of Matsuo's formulas.
By now, a number of free field realizations of quantum affine algebra Uq(si2) have
been produced [AOS, Mat4]. Also, intertwining operators in these realizations
were constructed explicitly, and therefore one can deduce certain expressions for
solutions of quantum KZ equations. Unfortunately, the resulting formulas are too
involved and lack the simplicity and at the same time richness of the integral for-
formulas discovered by Matsuo. To establish the relation between the two types of
expressions seems to be a rather hopeless task except for the most elementary cases
[Mat4].

LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
On the other hand, Matsuo's formulas do suggest a certain free field realization
based on g-deformation of certain operators and free bosonic fields. As we have
seen in Lecture 5, the factorization of solutions of the KZ equations consists of two
parts, related to the irrational product factor and the rational function factor of
the integrand. The former factor was presented as a matrix element (vacuum ex-
expectation value) of the product of vertex operators, and it admits a g-deformation
via q-vertex operators first introduced in [FJ]. The latter factor of the integrand
comes from the vacuum expectation of the product of bosonic fields in the Fock
representation of the Heisenberg algebra, also called the /37-system. The quan-
quantum analogue of the /37-system is in fact known in the physics literature as the
one-dimensional Zamolodchikov-Faddeev algebra [Fad], [ZZ]. It is generated by
Pn,in, n e Z, satisfying the following relations in the form of the currents E.18):
2 — wq
\z\>\w\
KZ1
/3BO(«7) -
7(«7OB),
\w\>\z\
W
where «(x) = EneZ^n-
Now we are ready to formulate our first problem.
Problem 1: Derive the integral formulas of Matsuo forjsolutions of quantum
KZ equations for Uq{sh) from a free field realization of Uq(sl2) presumably based
on the quantum /Jy-system.
Naturally the above discussion can be generalized from sfe to sin and to an
arbitrary simple Lie algebra. Solutions of KZ equations in the general case reviewed
at the end of Lecture 4 still admit a concise and elegant form [SV2]. This cannot
be said about their derivation from the free field realization (cf. [ATY]), mainly
because of the complexity of the latter construction. We expect that a successful
g-deformation of the general case will first require clearing up our understanding of
the free field realization of arbitrary affine Lie algebras.
13.2. Monodromy of the Knizhnik-Zamolodchikov
equations, tensor categories, and quantum groups
In Lectures 8 and 12 we discussed the connection matrices between certain
special bases of solutions of KZ equations. These bases naturally arise from repre-
representations of the solutions as matrix elements of products of intertwining operators.
In the nondeformed case, the connection matrices satisfied the braid relations and
yielded a description of the monodromy of the KZ equation via certain represen-
representations of the braid group. In the quantum case, the connection matrices were
identified with the elliptic solutions of the star-triangle relations, which can be
viewed as a deformation of the braid relations.
Moreover, in the undeformed case, we encountered another class of connection
matrices that relate the bases of the factored solutions with the bases of solutions
arising from "fusion" of the intertwining operators when their arguments approach
each other. These two types of connection matrices satisfy certain relations that can
be viewed as axioms of a braided tensor category considered in Lecture 8. It turns
13.2. MONODROMY OF KZ EQUATIONS AND QUANTUM GROUPS 181
out that generalization of the fusion bases (and consequently of the second type of
connection matrices) to the quantum case presents serious difficulties. In fact, this
question is related to a more general problem of defining the quantum analogue of
the operator product expansion that will be discussed in the next section. Even if
we manage to define the fusion connection matrices, it is not clear what identities
they satisfy and how one should generalize the notion of braided tensor category.
The hypothetical analogue of the braided tensor category was called in [FR] the
analytic tensor category. We can summarize the above discussion as the following
problem.
Problem 2: Define and construct an elliptic deformation of braided tensor
category from the representation theory of quantum affine algebras and/or solutions
of quantum KZ equations.
One can look at the former part of the problem from a different point of view.
We recall from Lecture 6 that braided tensor categories also appear in the repre-
representation theory of quantum groups. The Drinfeld-Kohno theorem extended by
Kazhdan and Lusztig establishes a relation between braided tensor categories aris-
arising from two different sources. This leads us to a question about generalization of
the notion of quantum group in relation to quantum KZ equation.
The comparison of the connection matrices for KZ and quantum KZ equa-
equations suggests a good candidate for such a generalization. We recall first that the
quantum affine algebra Uq(g) admits a second presentation, discovered by Drinfeld
[Dr5], via the current generators e,(z), fi(z), K^(z), i = 1,... , rank g, and certain
Serre-like relations that involve the structure function
A3.2)
sin(i: - y + t)
sin(a; - y - t)
that also appears in the quantum ,87-system A3.1) if we set z = eix/2, y = eiw/2,q =
elt. In this form, the quantum affine algebra admits an elliptic deformation if we
replace the structure function A3.2) by its elliptic counterpart
A3.3)
9{x-y-t,T)
where <9(x, r) is the Jacobi elliptic theta-function A.8) [DI1]. The resulting algebra
also has a Hopf algebra structure [DF], [DI2] and is called the elliptic quantum
group associated to a simple Lie algebra g. Another version of elliptic quantum
group was introduced by Felder [Fel] based on a solution of the dynamical Yang-
Baxter equation. The relation between two types of elliptic quantum groups asso-
associated to 3B was studied in [EnFe].
As in the case of quantum affine algebra, one can define a class of finite-
dimensional evaluation representations for elliptic quantum groups and consider
the braiding matrices. In the case of the elliptic quantum group associated to sl-2
(and more generally for s(n) there is an ad hoc procedure that allows us to con-
construct the solutions of the star-triangle relation [DJMO]. Combining the results of
[TV5] with [Fel], one can show that these solutions in the s^ case coincide with the
connection matrices for quantum KZ equations. Construction of fusion again leads
to more serious problems, since the category of finite-dimensional representations

182
LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
is far from being semisimple. In spite of these difficulties, in the case g — sfe one
can define by explicit formulas elliptic counterparts of the 6j-symbols that satisfy
relations that enjoy remarkable similarity to_the relationsjor the usual 6j-symbols
defining the structure of a braided tensor category [FT]. This leads to our next
problem:
Problem 3: Define and construct an elliptic deformation of the braided ten-
tensor category from the finite-dimensional representation theory of elliptic quantum
groups.
Now a comparison of Problems 2 and 3 yields
Problem 4: Find a generalization of Drinfeld-Kazhdan-Lusztig equivalence
that relates the elliptic versions of braided tensor categories obtained from the so-
solutions of quantum KZ equations and representations of an elliptic quantum group.
So far, the only result in this direction is the theorem proved in [TV5] which
identifies the connection matrices for the quantum KZ equations for g = 3B with the
R-matrices for the elliptic group corresponding to sfe. This should be considered
as an analogue of the Drinfeld-Kohno theorem discussed in Lecture 8.
Problems 2, 3 and 4 are the outcome of the general philosophy encoded in the
diagram of Section 1.6. The analogy with the corresponding results in the unde-
formed case is encoded in the right lower parallelogram of the diagram. Under-
Understanding of the relations presented by the left lower parallelogram is substantially
more difficult and will be discussed in Section 13.5 below.
13.3. Vertex operator algebras, conformal field
theory in genus zero and their y-deformations
When the central charge k is a natural number, the Knizhnik-Zamolodchikov
equations for the correlation functions of the intertwining operators can be viewed
as an important component of a more fundamental and elaborate structure known
as Wess-Zumino-Novikov-Witten (WZNW) conformal field theory. Conformal field
theory (CFT) was originally introduced in the physics literature [BPZ], and its
global geometric aspects were studied in [FS] and then made transparent in the
beautiful formulation of G. Segal [Seg2]. The theory of vertex operator algebras
was developed in [B], [FLM], [FHL] in relation to realization of the Monster group,
and its geometric interpretation that establishes the link with conformal field theory
at genus zero was found in [F] and extended in [H]. The relation of conformal
field theory with tensor categories was discovered and studied in [MSI], [MS2].
The WZNW model can be regarded as the most important and best studied class
of examples of CFT. It was introduced by Witten [Witl], and soon afterwards
Knizhnik and Zamolodchikov derived their famous equations [KZ]. It turns out
that these equations make sense for arbitrary values of the central charge, except
for k = — hw. In the case of general k, the conformal field theory is defined only in
genus zero, but still has a rich structure.
In spite of the fact that we did not discuss conformal field theory in this book, its
philosophy is certainly present in many of our constructions. The idea of writing
the commutation relations in g in terms of the currents Jx(z) — ^xfrejz^"
(see Section 2.10) is much more than just a technical convention simplifying some
formulas. The same technique also proved to be very useful in the lecture on the
free field realization. In fact, it turns out that this is just a beginning of some
new algebraic formalism in which the basic objects are not just vectors in some
13.3. VERTEX OPERATOR ALGEBRAS AND THEIR 9-DEFORMATIONS
183
vector space (as it is, say, in the theory of Lie algebras) but "fields" on Riemann
surfaces and the product (which is usually called "the operator product expansion")
of the fields satisfies some condition coming from geometry, namely, from gluing
together Riemann surfaces along a boundary component. In the simplest case where
the surface is a formal disk, the fields are written as Laurent series in the local
parameter 2; examples of such fields are the currents Jx(z) or /3(z). This structure
is formalized in the theory of vertex operator algebras [B], [FLM], [FHL] and is
well defined for all values of the central charge k. The vertex operator algebra point
of view to various constructions appearing in the book suggests many interesting
problems and directions for future research. Passage from the formal disk to an
arbitrary curve is best done using the language of jD-modules, as is suggested in
the recent (unpublished) paper [BeDr]; see also [HL].
The very first paper on CFT [BPZ] emphasizes the special role of the Virasoro
algebra, which has become a part of the structure of vertex operator algebras. In
the WZW model the Virasoro algebra is introduced via the Sugawara construction
B.35) and is used in the derivation of KZ equations. The understanding of what
is a meaningful g-deformation of the Virasoro algebra came only ten years after
the definition of quantum affine algebras [SKAO]. The quasiclassical version of
this algebra, i.e. the corresponding Poisson algebra, was previously introduced in
[FrRl]. In our derivation of the quantum KZ equations we avoid the introduction
of the full q-Virasoro algebra, using only its zero component, which is easier to
define. Thus we are led to the following problem:
Problem 5: Find a g-analogue of the Sugawara construction of the full q-
Virasoro algebra.
One of the central structures of CFT and vertex operator algebras is the opera-
operator product expansion. Its presence is hidden in the seemingly unnatural definition
of intertwining operators 4(z) between Verma modules and evaluation representa-
representations. In fact these operators can be defined in a more natural way, which works in a
much more general situation. Namely, for any Riemann surface with marked points,
a choice of local parameters near these points, and a highest-weight jj-module as-
assigned to each of these points (all modules must have the same level), we can define
the space of invariants as the space of all linear maps of the tensor product of these
modules to C, which are invariant under the action of the Lie algebra of g-valued
meromorphic functions on the surface regular outside of these points (see details in
[TUY], [KL]). In physics literature, this space of invariants is called the space of
conformal blocks. In particular, if we take the sphere with 3 marked points 0, z, 00
and denote by Y{z) : Vi ® V2 ® V3 —* C such an invariant map, then its restric-
restriction to the top degree subspace in the module assigned to the point 2 is (up to
dualization) the intertwining operator $(z) of the form considered in Lecture 3.
Conversely, it can be shown that every intertwining operator $(z) gives rise to a
unique invariant map Y(z). This approach also allows us to define a natural flat
connection on the space of invariants considered as a vector bundle over the moduli
space of marked Riemann surfaces. In the case of a sphere with N marked points,
this flat connection coincides with the one given by the KZ equations. Moreover, it
was proved in [TUY] that if k 6 Z+ and we only consider integrable modules, then
these spaces of invariants behave nicely under gluing of Riemann surfaces and thus
define a modular functor in the sense of Segal [Seg2], which is a part of the struc-
structure of CFT. A special case of the intertwining operators YB) gives rise to a vertex

184 LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
operator algebra structure on the g-representation with the highest weight A = 0
(k is fixed) and defines a structure of a module over this vertex operator algebra on
any integrable g-module with central charge k. These intertwining operators were
also studied by algebraic methods of vertex operator algebras in [FZhu]. In the
case when k = 1 and g is simply-laced, the vertex operator algebra and its modules
admit especially simple realization [FK], [Segl], [B], [FLM]. A natural question
is what happens when we consider a g-deformation of these structures. In [FJ] q-
vertex operators were introduced, and were used to construct a quantum analogue
of the g-representations of [FK] and [Segl]. In [FR] the quantum currents that
play a key role in the derivation of quantum KZ equations were also constructed.
However, further insights are needed to solve the following problem:
Problem 6: Give a definition of g-vertex operator algebra, modules and inter-
intertwining operators. Construct examples of these structures using the highest weight
modules for the quantum affine algebras.
This problem was first formulated in [FJ], and since the discovery of the quan-
quantum Virasoro algebra in [SKAO] there has been significant progress in this direction
[FrR2]. The solution of this problem should contain as an important component the
construction of the operator product expansion in the quantum case, thus paving
the way towards solution of Problem 2. Besides the purely algebraic problems
related to deformation of vertex operator algebras, there are no less interesting
geometric questions. In particular, what is the quantum analogue of sewing of
spheres with marked points and given local parameters at these points? Possibly,
the notion of quantum sphere together with the action of the protective quantum
group [Ml] should be an important ingredient of such quantum geometry.
Another important geometric aspect of vertex operator algebras associated to
representations of g is the geometric interpretation of the spaces of intertwining
operators. In the nondeformed case, this is precisely the relation between the
KZ equations and the homology and cohomology of local systems, developed in the
classical case by Schechtman and Varchenko [SV2], [VI]. We have seen in Lecture 7
that the cycles of integration appearing in solutions of KZ equations should be taken
from the twisted homology groups of certain configuration spaces. Moreover, the
homology groups themselves can be identified with the spaces of singular vectors in
the tensor product of representations of quantum groups (for generic values of q).
On the other hand, we have seen in Lecture 11 that the solutions of quantum KZ
equations are given in terms of g-hypergeometric functions expressed by Jackson
integrals with discrete "cycles of integration". The comparison of the undeformed
and deformed cases raises a natural question about a geometric interpretation of
the quantum integrals.
The first step towards constructing a g-analogue of the de Rham cohomology
associated with Jackson integrals was done by Aomoto [A2] before the discovery of
the quantum KZ equations. The study of the solutions of quantum KZ equations
led Tarasov and Varchenko to a substantial development of g-homology theory (see
[TV5]). In particular, in the sh case they were able to characterize the linear span
of discrete cycles of integration for the solutions of the quantum KZ-equations as
certain homology groups, and find a natural isomorphism of these spaces with spaces
of tensor products of representations of Felder's elliptic quantum group associated
to s^.
Another approach to quantum de Rham cohomology was suggested in [M2].
13.4. ELLIPTIC KZ EQUATIONS AND SPECIAL VALUES OF k 185
13.4. Elliptic Knizhnik-Zamolodchikov equations
and special values of the central charge
In our book, we consider primarily generic values of the central charge, namely,
k = k + hv g Q. The theory becomes more involved when the central charge
becomes rational, and it drastically changes its behavior at the critical level when
k = 0. We remarked in the previous subsection that when k is a natural number
the Knizhnik-Zamolodchikov equations arise from WZNW conformal field theory.
The algebraic structure of solutions of the KZ equations in this case was studied
in [FSV]. In fact, it comes naturally from the geometry of loop groups and their
representations [PS]. The understanding of a noncommutative version of various
aspects of this rich theory is a difficult program, and at the present moment we
have to be content with the existing algebraic approaches.
Another aspect of the Wess-Zumino-Witten model for a natural value of the
central charge is that the conformal field theory is defined for an arbitrary genus.
In particular, for every Riemann surface one can define the space of invariants
(conformal blocks), which carries a natural flat connection which can be called
"higher genus KZ equations" (see [I]), and try to generalize the genus zero case
considered in our book. The most studied generalization corresponds to Riemann
surfaces of genus one, i.e. a torus with marked points. The resulting differential
equation on the forms is called the elliptic Knizhnik-Zamolodchikov equation, or the
Knizhnik-Zamolodchikov-Bernard equation. It was derived by Bernard in [Berl,
Ber2]. A purely mathematical exposition can also be found in [EK1, EK2],
[FW2]. The derivation of the elliptic KZ equation in [EK1] is also based on the
operator KZ equation and follows the argument of Lecture 3, but instead of matrix
elements of products of intertwining operators it uses the traces of these products.
The integral formulas for the solutions of the elliptic KZ equations can be studied
directly as it was done in Lecture 4. It is also possible to write these formulas using
a technique similar to the one used in Lecture 5, which is done in the case g = sfe
in [FV1], [EK1].
A generalization of the quantum KZ equations to surfaces of higher genus is
unknown in general. However, in the case of genus one, we have an appropriate
quantum analogue of elliptic KZ equations: this is the quantum KZ equations with
trigonometric ii-matrices replaced by their elliptic counterparts. The properties of
these equations were studied in [E], [FTV1], [FTV2], and the solutions in the case
g = 3B were constructed explicitly in [FTV1]. The strong parallel between the
deformed and undeformed case suggests the following problem:
Problem 7: Derive elliptic quantum KZ equations for traces of products of
intertwining operators from the operator quantum KZ equations.
Assuming that we know the solution to Problem 1, one can try to obtain the
formulas for the solutions of the elliptic quantum KZ equations in the case g = sfe
[FTVI] from a free field realization of W,(sB) based on a quantum ,87-system.
When A; is a rational number but no longer a positive integer, the geometric
realization of the Wess-Zumino-Witten model is not known and we have to appeal
to algebraic methods. We refer the reader to the paper [FM] for an account of
conformal field theory in this case. In the last paper of their series, Kazhdan and
Lusztig [KL] were able to extend the equivalence of the braided tensor categories
of affine Lie algebras and quantum groups discussed in Lecture 8 to the case when
k ? —Q+ (with the exception of certain "small values" of k). In this case, the
parameter q of the quantum group becomes a root of unity, and the corresponding

186
LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
braided tensor category of finite-dimensional representations is no longer semisimple
but contains a remarkable semisimple subquotient category [AP].
When k € Q+ the structure becomes even more mysterious, since a direct gen-
generalization of Kazhdan-Lusztig constructions does not yield a braided tensor cat-
category. However, in his thesis (published in [Fin]), Finkelberg found a remarkable
duality between the representation category arising in Wess-Zumino-Witten CFT,
i.e. with k e Z+,k > /iv, and a subquotient category of the Kazhdan-Lusztig
category. Combining this with the results of [KL], he showed that the braided ten-
tensor category appearing in Wess-Zumino-Witten CFT coincides with the semisimple
subquotient of the category C(g,g) denned in [AP]. Comparison of these results
and Problems 2, 3, 4 above naturally suggests the corresponding questions for a
nongeneric k.
In the quantum case, one can also fix a generic k but choose the deformation
parameter q to be a root of unity, and again refer to Problems 2, 3, 4 in this case.
Finally, one can consider both nongeneric parameters k and q and expect even more
subtle arithmetic properties of all those structures.
At the critical value of the central charge when k = 0, it can be shown that suit-
suitably renormalized asymptotics of the correlation functions are the eigenfunctions
of the operators
The problem of finding eigenfunctions of these operators and the relation of this
problem with the Bethe ansatz are discussed in [FFR], [RV]. This deep problem is
closely related with the so-called "geometric Langlands correspondence" considered
by Beilinson and Drinfeld [BeDr].
The behavior of quantum affine algebras at the critical value k — 0, in particular
the center of the completed quantum universal enveloping algebra, is described in
[RS]. See also [FrRl]. It is an interesting problem to construct and find a natural
interpretation of a g-deformation of the geometric Langlands correspondence. It
is tempting to make a wild guess that when q is a root of unity, it should acquire
arithmetic properties not unrelated with the original Langlands program.
13.5. Double loop algebras and quantum afflne algebras
In the above four sections, we discussed g-deformation of various structures
related to affine Lie algebras, quantum groups and Knizhnik-Zamolodchikov equa-
equations. In particular, a g-deformation of quantum groups leads to double deformed
algebras, namely elliptic quantum groups. As we discussed in Section 1.6, there
exists another direction of generalization of the aforementioned structures, by con-
considering their afflne analogues. In this approach, quantum affine algebras replace
the quantum groups associated to simple finite-dimensional Lie algebras, and affine
Lie algebras become double affine or double loop algebras. This direction is much
less explored, although it is at least as promising as the program of g-deformation,
and we will make a few remarks in this connection.
First we note that most of the constructions related to the representation theory
of affine Lie algebras have exact analogues if we replace a simple Lie algebra g by the
affine Lie algebra g. In partucular, one can write "affine" Knizhnik-Zamolodchikov
equations, using a nondegenerate symmetric bilinear form on g. Solutions of these
equations will take values in the product of the highest-weight representations of g.
13.6. QUANTUM KZ EQUATIONS AND PHYSICAL MODELS
187
A straightforward generalization of the Kohno-Drinfeld theorem will yield the affine
quantum group Uq(g) and its highest weight representations. It is substantially
more difficult to extract from this approach the evaluation representations of Uq(g)
and consequently the quantum Knizhnik-Zamolodchikov equation. To achieve this,
one may need to consider "double" evaluation representations of g and restrict
the affine Knizhnik-Zamolodchikov equation to the product of finite-dimensional
modules. In the concise form, we have the following problem:
Problem 8: Derive the quantum Knizhnik-Zamolodchikov equation from the
affine Knizhnik-Zamolodchikov equation.
Two-dimensional central extension of the double loop algebra obtained from
the repeated "affinization" of a simple Lie algebra g, gives rise to a family of one-
dimensional central extensions parametrized by one-dimensional moduli of elliptic
curves [EF]. This point of view also leads to a natural realization of central extension
of the corresponding double loop group [FKh]. Following the parallel between the
loop slgebras and the double loop algebars discussed in [EF], one can conjecture that
the representation theory of the above central extensions of double loop algebras
yields a commutative system of <9-equations, which would naturally lead to the
quantum Knizhnik-Zamolodchikov equations. This can be viewed as a variant of
Problem 7. A solution of either of these two related problems should also shed light
on the geometric nature of the solutions of the quantum KZ equations.
13.6. Quantum KZ equations and physical models
The derivation of the quantum Knizhnik-Zamolodchikov equation from the rep-
representation theory of quantum affine algebras led to a completely new approach to
the theory of the XXZ quantum spin chain. In particular, using the integral formu-
formulas for the simplest correlation functions associated to Uq(sl2), one obtains explicit
representation of the correlators of the XXZ model. This important application of
representation theory was developed by Jimbo, Miwa and their collaborators at the
Research Institute for Mathematical Sciences, in Kyoto, and is described in detail
in the book [JM]. Many remaining problems on XXZ model and representation
theory can be found in that book. Jimbo and Miwa also obtained an application of
the elliptic analogue of the quantum KZ equation to the more general XYZ chain,
though the representation theoretic meaning of the elliptic KZ equation is not fully
understood at the present moment.
Another unrelated application of the quantum KZ equations to completely in-
tegrable (but not conformal) quantum field theory (QFT) is described by Smirnov
in the book [Sm]. He develops a remarkable transformation between nonlinear sys-
systems and g-linear systems. In particular, the quantum KZ equation for zero central
charge turns out to be related to the locality properties of quantum fields. This
special case of quantum KZ equations in a disguised form first arose in Smirnov's
theory. Recently, there appeared further indications of the deep relation between
integrable quantum field theory and g-deformation of vertex operator algebras. One
example is the special role of the Zamolodchikov algebra, which can be viewed as
a g-analogue of the affine Clifford algebra, in a particular integrable QFT [Luk].
However, in spite of long and extensive work on integrable QFT, we are still lacking
the conceptual mathematical picture behind it. We hope that the representation

188
LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
theory of quantum affine Lie algebras and the quantum KZ equation will be instru-
instrumental in helping us to understand the mathematical nature of this important area
of theoretical physics.
References
[Ab] Abe, E., Hopf algebras, Cambridge Univ. Press, 1980.
[Ad] Adarns, C. R-, On the linear ordinary q-difference equation, Ann. of Math. 30 A929),
195-205.
[AGS1] Alvarez-Gaume, L., Gomez, C. and Sierra, G., Hidden quantum symmetries in rational
conformal field theories, Nuclear Phys. B 319 A989), 155-186.
[AGS2] , Duality and quantum groups, Nuclear Phys. B 330 A990), 347-398..
[AP] Andersen, H. H. and Paradowski, J., Fusion categories arising from semisimple Lie
algebras, Comm. Math. Phys 169 A995), 563-588.
[Al] Aomoto, K-, Gauss-Manin connection of integral of difference products, J. Math. Soc.
Japan 39 A987), 191-207.
[A2] , A note on holonomic q-difference system, Algebraic Analysis I (Kashiwara,
M., Kawai, T., eds.), Academic Press, New York, 1988, pp. 25-28.
[AOS] Awata, H., Odake, S., and Shiraishi, J., Free boson realization of ?/,(«(«), Comm.
Math. Phys. 162 A994), 61-83.
[ATY] Awata, H., Tsuchiya, A. and Yamada, Y., Integral formulas for the WZNW correlation
functions, Nuclear Phys. B. 365 A991), 680-696.
[Bat] Bateman, H., Higher transcendental functions, vol. 3 (Erdelyi, A., et al., eds.), McGraw-
Hill, New York, 1955.
[Bax] Baxter, R. J., Exactly solved models in statistical mechanics, Academic Press, London,
1982.
[BeDr] Beilinson, A. and Drinfeld, V., Chiral algebras, unpublished.
[BD] Belavin, A.A. and Drinfeld, V.G., Solutions of classical Yang-Baxter equation and
simple Lie algebras, Punktsional. Anal, i Prilozhen. 16 A982), no. 3, 1-29 (Russian);
English translation in Functional Anal. Applic. 16 A982), 159-180.
[BPZ] Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B., Infinite conformal symmetry
in two dimensional quantum field theory, Nuclear Phys. B 241 A984), 33.
[Berl] Bernard, D., On the Wess-Zumino-Witten model on the torus, Nuclear Phys. B 303
A988), 77-93.
[Ber2] , On the Wess-Zumino-Witten model on Riemann surfaces. Nuclear Phys. B
309 A988), 145-174.
[BF] Bernard, D. and Felder, G., Fock representations and BRST cohomology in SLB)
current algebra, Comm. Math. Phys. 127 A990), 145-168.
[BFS] Bezrukavnikov, R., Finkelberg, M. and Schechtman, V., Factorizable sheaves and quan-
quantum groups, q-alg/9712001 (to appear).
[Bi] Birkhoff, G. D., The generalized Riemann problem for linear differential equations and
the allied problems for linear difference and q-difference equations, Proc. Amer. Acad.
Arts Sci. 49 A913), 521-568.
[Birj Birman, J., Braids, links, and mapping class groups, Ann. Math. Stud., vol. 82, Prince-
Princeton Univ. Press, Princeton, NJ, 1972.
[B] Borcherds, R., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad.
Sci. USA 83 A986), 3068-3071.
[Bor] Borel, A., Introduction to middle intersection cohomology and perverse sheaves, Alge-
Algebraic groups and their generalizations: classical methods (University Park, PA, 1991),
Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc, 1994, pp. 25-52.
[Bour] Bourbaki, N., Groupes et algebres de Lie, Chap. 4-6, Hermann, Paris, 1968.

188
LECTURE 13. CURRENT DEVELOPMENTS AND FUTURE PERSPECTIVES
theory of quantum affine Lie algebras and the quantum KZ equation will be instru-
instrumental in helping us to understand the mathematical nature of this important area
of theoretical physics.
References
[Ab] Abe, E., Hopf algebras, Cambridge Univ. Press, 1980.
[Ad] Adams, C. R., On the linear ordinary q-difference equation, Ann. of Math. 30 A929),
195-205.
[AGS1] Alvarez-Gaume, L., Gomez, C. and Sierra, G., Hidden quantum symmetries in rational
conformal field theories, Nuclear Phys. B 319 A989), 155-186.
[AGS2] , Duality and quantum groups, Nuclear Phys. B 330 A990), 347-398..
[AP] Andersen, H. H. and Paradowski, J., Fusion categories arising from semisimple Lie
algebras, Comm. Math. Phys 169 A995), 563-588.
[Al] Aomoto, K-, Gauss-Manin connection of integral of difference products, J. Math. Soc.
Japan 39 A987), 191-207.
[A2] , A note on holonomic q-difference system, Algebraic Analysis I (Kashiwara,
M., Kawai, T., eds.), Academic Press, New York, 1988, pp. 25-28.
[AOS] Awata, H., Odake, S., and Shiraishi, J., Free boson realization of Uq(sl^), Comm.
Math. Phys. 162 A994), 61-83.
[ATY] Awata, H., Tsuchiya, A. and Yamada, Y., Integral formulas for the WZNW correlation
functions, Nuclear Phys. B. 365 A991), 680-696.
[Bat] Bateman, H., Higher transcendental functions, vol. 3 (Erdelyi, A., et al., eds.), McGraw-
Hill, New York, 1955.
[Bax] Baxter, R. J., Exactly solved models in statistical mechanics, Academic Press, London,
1982.
[BeDr] Beilinson, A. and Drinfeld, V., Chiral algebras, unpublished.
[BD] Belavin, A.A. and Drinfeld, V.G., Solutions of classical Yang-Baxter equation and
simple Lie algebras, Funktsional. Anal, i Prilozhen. 16 A982), no. 3, 1-29 (Russian);
English translation in Functional Anal. Applic. 16 A982), 159-180.
[BPZ] Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B., Infinite conformal symmetry
in two dimensional quantum field theory, Nuclear Phys. B 241 A984), 33.
[Berl] Bernard, D., On the Wess-Zumino-Witten model on the torus, Nuclear Phys. B 303
A988), 77-93.
[Ber2] , On the Wess-Zumino-Witten model on Riemann surfaces, Nuclear Phys. B
309 A988), 145-174.
[BF] Bernard, D. and Felder, G., Fock representations and BRST cohomology in SLB)
current algebra, Comm. Math. Phys. 127 A990), 145-168.
[BFS] Bezrukavnikov, R., Finkelberg, M. and Schechtman, V., Factorizable sheaves and quan-
quantum groups, q-alg/9712001 (to appear).
[Bi] Birkhoff, G. D., The generalized Riemann problem for linear differential equations and
the allied problems for linear difference and q-difference equations, Proc. Amer. Acad.
Arts Sci. 49 A913), 521-568.
[Bir] Birman, J., Braids, links, and mapping class groups, Ann. Math. Stud., vol. 82, Prince-
Princeton Univ. Press, Princeton, NJ, 1972.
[B] Borcherds, R., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad.
Sci. USA 83 A986), 3068-3071.
[Bor] Borel, A., Introduction to middle intersection cohomology and perverse sheaves, Alge-
Algebraic groups and their generalizations: classical methods (University Park, PA, 1991),
Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc, 1994, pp. 25-52.
[Bour] Bourbaki, N., Groupes et algebres de Lie, Chap. 4-6, Hermann, Paris, 1968.
189

REFERENCES
[PJ Pasquier, V., Etiology of IRF models, Corara. Math. Phys. 118 A988), 335-364.
[PS] Pressley, A. and Segal, G., Loop groups, Oxford Univ. Press, Oxford, 1986.
[R] Reshetikhin, N. Yu., Jackson-type integrals, Bethe vectors, and solutions to a difference
analog of the Knizhnik-Zamolodchikov system, Lett. Math. Phys. 26 A992), 153-165.
[RS] Reshetikhin, N.Yu. and Semenov-Tian-Shansky, M.A., Central extensions of quantum
current groups, Lett. Math. Phys. 19 A990), 133-142.
[RT] Reshetikhin, N. Yu. and Turaev, V. G., Ribbon graphs and their inavriants derived
from quantum groups, Comm. Math. Phys. 127 A990), 1-26.
[RV] Reshetikhin, N. and Varchenko, A., Quasiclassical asymptotics of solutions to the KZ
equations, Geometry, topology, h physics, Conf. Proc. Lecture Notes Georn. Topology,
VI, Internat. Press, Cambridge, MA, 1995., pp. 293-322.
[Rol] Rosso, M-, Finite-dimensional representations of the quantum analog of the enveloping
algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 A988), 581-593.
[Ro2] , An analogue of P.B. W. theorem and the universal R-matrix for Uh (s!(JV + 1)),
Comm. Math. Phys. 124 A989), 307-318.
[STV] Schechtman, V., Terao, H. and Varchenko, A., Local systems over complements of hy-
perplanes and the Kac-Kazhdan conditions for singular vectors, J. Pure Appl. Algebra
100 A995), 93-102.
[SV1] Schechtman, V. and Varchenko, A., Integral representations of N-point conformal cor-
correlators in the WZW model, preprint, Bonn, 1989.
[SV2] , Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106
A991), 139-194.
[Segl] Segal, G. B-, Unitary representations of some infinite-dimensional groups, Com. Math.
Phys. 80 A981), 301-342.
[Seg2] Segal, G-, Two dimensional conformal field theories and modular functor, Proc. IX
Congress Math. Phys., Swansea, 1988, Adam Hilger, Bristol, 1989, pp. 22-37.
[SKAO] Shiraishi, J., Kubo, H., Awata, H., and Odake, S., A quantum deformation of the
Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 A996),
33-51.
[SS] Shnider, S. and Sternberg, S., Quantum groups. From coalgebras to Drinfeld algebras.
A guided tour, International Press, Cambridge, MA, 1993.
[Sm] Smirnov, F., Form factors in completely integrable models of Quantum Field Theory,
World Scientific, Singapore, 1992.
[SoVa] Soibelman, Ya. S. and Vaksman, L. L-, An algbera of functions on the quantum group
SVB), Functional Anal. Appl. 22 A988), 170-181.
[Stl] Stolin, A., On rational solutions of the Yang-Baxter equations. Maximal orders in the
loop algebra, Comm. Math. Phys. 141 A991), 533-548.
[St2] , A geometrical approach to rational solutions of the classical Yang-Baxter equa-
equation. I, Proceedings of the 2nd Gauss Symposium. Conference A: Mathematics and
Theoretical Physics (Munich, 1993), de Gruyter, Berlin, 1995, pp. 347-357.
[Sw] Sweedler, M. E., Hopf algebras, Benjamin, New York, 1969.
[T] Tanisaki, T., Killing forms, Harish- Chandra isomorphisms and universal R-matrices
for quantum algebras, Infinite Analysis, part A and part B (Kyoto, 1991), Adv. Ser.
Math. Phys., vol. 17, World Scientific, pp. 941-961.
[TV1] Tarasov, V. and Varchenko, A., Jackson integral representations for solutions of the
quantized Knizhnik-Zamolodchikov equation, St. Petersburg Math. J. 6 A995), 275-
314; Correction, Algebra i Analiz 6 A994), no. 3.
[TV2] , Asymptotic solutions to the quantized Knizhnik-Zamolodchikov equation and
Bethe vectors, Mathematics in St. Petersburg,, Amer. Math. Soc. Transl. Ser. 2, 174,
Amer. Math. Soc, Providence, RI, 1996, pp. 235-273.
[TV3] , Solutions to the quantized Knizhnik-Zamolodchikov equation and the Bethe
Ansatz, Group theoretical methods in physics (Toyonaka, 1994), World Sci. Publishing,
River Edge, NJ, 1995, pp. 473-478.
[TV4] , Geometry of q-hypergeometric functions as a bridge between Yangians and
quantum affine algebras, Invent. Math. 128 A997), 501-588.
[TV5] , Geometry of q-hypergeometric functions, quantum affine algebras and elliptic
quantum groups, preprint, q-alg/9703044 (to appear).
REFERENCES
[Tr] Trjitzinsky, W.J., Analytic theory of linear q-difference equations, Acta Math. 61
A933), 1-38.
[TKJ Tsuchiya, A., Kanie, Y., Vertex operators in conformal field theory on P1 and mon-
odromy representations of braid group, Conformal field theory and solvable lattice mod-
models (Kyoto, 1986), Adv. Stud. Pure Math., vol. 16, Academic Press, Boston, 1988,
pp. 297-372.
[TUY] Tsuchiya, A., Ueno, K. and Yamada, Y., Conformal field theory on universal family of
stable curves with gauge symmetries, Integrable systems in quamtum field theory and
statistical mechanics, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, 1992,
pp. 459-566.
[VI] Varchenko, A., Multidimensional hypergeometric functions and representation theory
of quantum groups, Adv. Ser. Math. Phys., vol. 21, World Scientific, River Edge, NJ,
1995.
[V2J , Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal
base, Comm. Math. Phys. 171 A995), 99-138.
[V3] , Quantized Knizhnik-Zamolodchikov equations, quantum Yang-Baxter equa-
equation, and difference equations for q-hypergeometric functions, Comm. Math. Phys.
162 A994), 499-528.
[Wak] Wakimoto, M., Fock representations of the affine Lie algebra a{X\ Comm. Math. Phys.
104 A986), 605-609.
[WW] Whittaker, E.T., Watson, G.N., Course of modern analysis, 4th ed., Cambridge Univ.
Press, 1958.
[Wic] Wick, G.C., The evaluation of the collision matrix, Phys. Rev. 80 A950), 268-272.
[Witl] Witten, E., Non-abelian bosonization in two dimensions, Comm. Math. Phys. 92
A984), 455-472.
[Wit2] , Physics and geometry, Proc. Internat. Congr. Math. (Berkeley, Calif., 1986),
Amer. Math. Soc, Providence, RI, 1987, pp. 267-303.
[ZZJ Zamolodchikov, A. B. and Zamolodchikov, Al. B., Factorized S-matrices in two dimen-
dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. of
Physics 120 A979), 253-291.

REFERENCES
[Ca] Carmichael, R. D-, The general theory of linear q-difference equations, Amer. J. Math.
34 A912), 147-168.
(CP1] Chari, V. and Pressley, A., Quantum affine algebras, Comm. Math. Phys. 142 A991),
261-283.
[CP2] , A guide to quantum groups, Cambridge Univ. Press, 1995.
[Chi] Cherednik, I., Integral solutions of trigonometric Knizhnik-Zamolodchikov equations
and Kac-Moody algebras, Publ. Res. Inst. Math. Sci. 27 A991), 727-744.
[Ch2] , Generalized braid groups and local r-matrix systems, Dokl. Akad. Nauk SSSR
307 A989), no. 1, 49-53 (Russian); English translation in Soviet Math. Dokl. 40
A990), no. 1, 43-48.
[CF] Christe, P. and Flume, R., The four-point correlations of all primary operators of the
d = 2 conformally invariant SUB) a-model with Wess-Zumino term, Nuclear Phys. B
282 A987), 466-494.
[CL] Coddington, E.A. and Levinson, N., Theory of ordinary differential equations, McGraw-
Hill, New York, 1955.
[CK] De Concini, C. and Kac, V.G., Representations of quantum groups at roots of 1, Op-
Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory
(A. Connes et al, eds.), Birkhauser, 1990, pp. 471-506.
[DJMM] Date, E., Jimbo, M., Matsuo, A., and Miwa, T., Hypergeometric-type integrals and
the sl{2, C) Knizhnik-Zamolodchikov equation, Proceedings of the conference on Yang-
Baxter equations, conformal invariance and integrability in statistical mechanics and
field theory, Internat. J. Modern Phys. B 4 A990), 1049-1057.
[DJMO] Date, E., Jimbo, M., Miwa, T. and Okado, M., Fusion of the eight-vertex SOS model,
Lett. Math. Phys. 12 A986), 209-215; Erratum and Addendum:, Lett. Math. Phys. 14
A987), 97.
[De] Deligne, P., Equations differentielles a points singuliers reguliers, Lecture Notes in
Math., vol. 163, Springer-Verlag, Berlin-New York, 1970.
[DF] Ding, J. and Frenkel, I. B., Isomorphism of two realizations of quantum affine algebra
Uq(gl(n)), Comm. Math. Phys. 156 A993), 277-300.
[DI1] Ding, J. and Iohara, K., Generalization and deformation of Drinfeld quantum affine
algebras, Lett. Math. Phys 41 A997), 181-193.
[DI2] , Drinfeld comultiplication and vertex operators, J. Geom. Phys. 23 A997), 1-11.
[Drl] Drinfeld, V.G., Quantum groups, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 798-
820.
[Dr2] , On almost cocommutative Hopf algebras, Leningrad Math.J. 1 A990), no. 2,
321-342.
[Dr3] , Quasi-Hopf algebras, Algebra i Analiz 1 A989), no. 6, 114-148 (Russian);
English translation in Leningrad Math. J. 1 A990), 1419-1457.
[Dr4] , On quasi-triangular quasi-Hopf algebras and a group closely connected with
Gal(Q/Q), Algebra i Analiz 2 A990), no. 4, 149-181 (Russian); English translation in
Leningrad Math. J. 2 A990), 829-860.
[Dr5] , New realization of Yangian and quantum affine algebra, Soviet Math. Dokl.
36 A988), 212-216.
[EnFe] Enriquez, B. and Felder, G., Elliptic quantum groups ET,T)(sl2) and quasi-Hopf algebras,
preprint, 1997 (to appear).
[ESV] Esnault, H., Schechtman, V. and Viehweg, E., Cohomology of local systems on the
complement of hyperplanes, Invent. Math. 109 A992), 557-561.
[E] Etingof, P., Representations of affine Lie algebras, elliptic r-matrix systems, and spe-
special functions, Comm. Math. Phys. 159 A994), 471-502.
[EF] Etingof, P. and Frenkel, I. B-, Central extensions of current groups in two dimensions,
Comm. Math. Phys. 165 A994), 429-444.
[EK1] Etingof, P. and Kirillov, A., Jr., Representations of affine Lie algebrast parabolic dif-
differential equations, and Lame functions, Duke Math. J. 74 A994), 585-614.
[EK2] ____, Affine analogue of Jack and Macdonald polynomials. Duke Math. J. 78 A995),
229-256.
[Fad] Faddeev, L. D., Quantum completely integrable models infield theory, Soviet Sci. Rev.,
Ser. C: Math. Phys. Rev., vol. 1, Harwood Academic Publ., 1980, pp. 107-155.
REFERENCES
[FRT1] Faddeev, L. D., Reshetikhin, N. Yu., and Takhtadzhyan, L. A., Quantization of Lie
groups and Lie algebras, Leningrad Math. J. 1 A990), no. 1, 193-225.
[FRT2] , Quantum groups, Braid group, knot theory and statistical mechanics, Adv.
Ser. Math. Phys., 9, World Sci. Publishing, 1989, pp. 97-110.
[FG] Falceto, F. and Gawedzki, K., Elliptic Wess-Zumino-Witten model from elliptic Chern-
Simons theory, Lett. Math. Phys. 38 A996), 155-175.
[FZ] Fateev, V. A. and Zamolodchikov, A. B., Operator algebra and correlation functions
of two-dimensional SUB) x SUB) chiral Wess-Zumino model, Sov. J. Nucl. Phys. 43
A986), 657-670.
[FF1] Feigln, B. and Frenkel, E., Representations of affine Kac-Moody algebras and bozoniza-
tion, Physics and mathematics of strings, Knizhnik Mem. Vol., World Sci., Singapore,
1989, pp. 271-316.
[FF2] , Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math.
Phys. 128 A990), 161-189.
[FF3] , Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras,
Infinite Analysis, Part A,B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci.,
River Edge, NJ, 1992, pp. 197-215.
[FF4] , Integrals of motion and quantum groups, Integrable systems and quantum
groups (Montecatini Terme, 1993), Lecture Notes in Math, 1620, Springer, Berlin,
1996, pp. 349-418.
[FF5] , Quantum W-algebras and elliptic algebras, Comm. Math. Phys. 178 A996),
653-678.
[FFR] Feigin, B., Frenkel, E. and Reshetikhin, N., Gaudin Model, Bethe ansatz and critical
level, Comm. Math. Phys. 166 A994), 27-62.
[FSV] Feigin, B., Schechtman, V. and Varchenko, A., On algebraic equations satisfied by
hypergeometric correlators in WZW models. I, Comm. Math. Phys. 163 A994), 173-
184; II, Comm. Math. Phys. 170 A995), 219-247.
[FM] Feigin, B. and Malikov, F., Fusion algebra at a rational level and cohomology of nilpo-
tent subalgebras o/slB), Lett. Math. Phys. 31 A994), 315-325.
[Fel] Felder, G., Elliptic quantum groups, Xlth International Congress of Mathematical
Physics (Paris, 1994), Internat. Press, Cambridge, MA, 1995, pp. 211-218.
[FTV1] Felder, G., Tarasov, V. and Varchenko, A., Solutions of the elliptic qKZB equations
and Bethe ansatz, q-alg/9606005 (to appear).
[FTV2] , Monodromy of solutions of the elliptic quantum Knizhnik-Zamolodchikov-Ber-
Knizhnik-Zamolodchikov-Bernard difference equations, q-alg/9705017 (to appear).
[FW1] Felder, G. and Wieczerkowski, C, Topological representations of the quantum group
Uq{sh), Comm. Math. Phys. 138 A991), 583-605.
[FW2] , Conformal blocks on elliptic curves and the Knizhnik-Zamolodchikov-Bernard
equations, Comm. Math. Phys. 176 A996), 133-161.
[FV1] Felder, G., and Varchenko, A., Integral representations of solutions of the elliptic
Knizhnik-Zamolodchikov-Bernard equations, Internat. Math. Res. Notices 5 A995),
221-233.
[FV2] , On representations of the elliptic quantum group ?T,^(si2), Comm. Math.
Phys. 181 A996), 741-761.
[FV3] , Solutions of the elliptic qKZB equations and Bethe Ansatz, I, preprint q-
alg/9606005 (to appear).
[Fin] Finkelberg, M., An equivalence of fusion categories, Geom. Funct. Anal. 6 A996),
249-267.
[FrRl] Frenkel, E., and Reshetikhin, N., Quantum affine algebras and deformations of the
Virasoro and W-algebras, Comm. Math. Phys. 178 A996), 237-264.
[FrR2] , Towards deformed chiral algebras, preprint q-alg/9706023 (to appear).
[F] Frenkel, I. B., Lectures on vertex operator algebras, Institute for Advanced Study,
February 1988; Yale Univ., Fall 1988.
[FHL] Frenkel, I. B., Huang, Y.-Z., and Lepowsky, J., On axiomatic approaches to vertex
operator algebras and modules., Mem. Amer. Math. Soc. 104 A993), no. 494.
[FJ] Frenkel, I. B. and Jing, N., Vertex representation of quantum affine algebras, Proc.
Nat. Acad. Sci. USA 85 A988), 9373-9377.

REFERENCES
[FK] Frenkel, I. B. and Kac, V., Basic representations of affine Lie algebras and dual reso-
resonance models, Invent. Math. 62 A980), 23-66.
[FKh] Frenkel, I. B. and Khesin, B-, Four-dimensional realization of two-dimensional current
groups, Comm. Math. Phys. 178 A996), 541-562.
[FKV] Frenkel, I. B., Kirillov, A. and Varchenko, A., Canonical basis and homology of local
systems, Internat. Math. Res. Notices 16 A997), 783-806.
[FLM] Frenkel, I. B., Lepowsky, J. and Meurman, A., Vertex operator algebras and the Mon-
Monster, Academic Press, Boston, 1988.
[FR] Frenkel, I. B. and Reshetikhin, N. Yu., Quantum affine algebras and holonomic differ-
difference equations, Comm. Math. Phys. 146 A992), 1-60.
[FT] Frenkel, I. B. and Turaev. V. G., Elliptic solutions of the Yang-Baxter equation and
modular hypergeometric functions, Arnold-Gelfand mathematical seminars (Arnold,
V. I., Gelfand, I. M., Smirnov, M., and Retakh, V. S., eds.), Birkhauser, 1997, pp. 171-
204.
[FZhu] Frenkel, I.B. and Zhu, Y.-C, Vertex operator algberas associated to representations of
affine and Virasoro algebras, Duke Math. J. 66 A992), 123-168.
[FS] Friedan, D. and Shenker, S., The analytic geometry of two-dimensional conformal field
theory, Nuclear Phys. B 281 A987), 509.
[GaRa] Gasper, G. and Rahman, M-, Basic hypergeometric series, Cambridge Univ. Press,
1990.
[GM] Gelfand, S.I. and Manin, Yu.L, Homological algebra, Algebra V, Encyclopaedia Math.
Sci., vol. 38, Springer, Berlin, 1994, pp. 1-122.
[GMMOS] Gerasimov, A., Morozov, A. Olshanetsky, M., Marshakov, A., and Shatashvili, S.,
Wess-Zumino- Witten model as a theory of free fields, Internat. J. Modern Phys. A 5
A990), 2495-2589.
[GG] Goto, M. and Grosshans, F. D., Semisimple Lie algebras, Lecture Notes in Pure and
Applied Mathematics, Vol. 38, Marcel Dekker, Inc., New York-Basel, 1978.
[GH] Griffiths, P. and Harris, J., Principles of algebraic geometry, Wiley-Interscience, New
York, 1978.
[GR] Gunning, R.C. and Rossi, H., Analytic functions of several complex variables, Prentice-
Hall, Englewood Cliff, NJ, 1965.
Par] Hartshorne, R., Algebraic geometry, Springer-Verlag, New York, 1977.
[H] Huang, Y.-Z., Two-dimensional conformal geometry and vertex operator algebras, Birk-
Birkhauser, Boston, 199.
[HL] Huang, Y.-Z., and Lepowsky, J., On the D-module and formal-variable approaches to
vertex algebras. Topics in geometry, Progr. Nonlinear Differential Equations Appl., 20,
Birkhauser, Boston, 1996, pp. 175-202.
[Hu] Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer-
Verlag, New York, 1972.
[ITIJMN] Idzumi, M., Tokihiro, T., Iohara, K., Jimbo, M., Miwa, T., and Nakashima, T., Quan-
Quantum affine symmetry in vertex models, Internat. J. Modern Phys. A 8 A993), 1479-
1511.
[I] Ivanov, D., Knizhnik-Zamolodchikov-Bemard equations on Riemann surfaces, Internat.
J. Modern Phys. A 10 A995), 2507-2536.
[Iv] Iversen, B., Cohomology of sheaves, Springer, Berlin, 1986.
[Jl] Jimbo, M.A., A q-difference analogue ofUa and the Yang-Baxter equation, Lett. Math.
Phys. 10 A985), 62-69.
, A q-analogue of U(qI(N + 1)), Heche algebra and the Yang-Baxter equation,
[J2]
[JM]
[Kac]
[KK]
[Kas]
[KL]
Lett. Math. Phys. 11 A986), 247-252.
Jimbo, M.A., and Miwa, T., Algebraic analysis of solvable lattice models, Conf. Board.
Math. Sci. Regional Conf. Ser. Math., vol. 86, Amer. Math. Soc, 1995.
Kac, V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, 1990.
Kac, V.G. and Kazhdan, D., Structure of representations with highest weight of infinite-
dimensional Lie algebras, Advances in Math. 34 A979), 97-108.
Kassel, C, Quantum groups, Springer, New York, 1995.
Kazhdan, D. and Lusztig, G-, Tensor structures arising from affine Lie algebras. I-IV,
J. Amer. Math. Soc. 6 A993), 905-947, 949-1011; 7 A994), 335-381, 383-453.
REFERENCES
[KS] Kazhdan, D., and Soibelman, Y., Representations of quantum affine algebras, Selecta
Math. 1 A995), 537-595.
[KT1] Khoroshkin, S.M., and Tolstoy, V.N., Universal R-matrix for quantized affine (su-
perjalgebras, Comm. Math. Phys. 141 A991), 599-617.
[KT2] , On Drinfeld's realization of quantum affine algebras, J. Geom. Phys. 11 A993),
445-452.
[Kir] Kirillov, A., Jr., On an inner product in modular tensor categories. I, J. Amer. Math.
Soc. 9 A996), 1135-1169; II, preprint, 1996.
[KR1] Kirillov, A.N. and Reshetikhin, N. Yu., Representations of the algebra Uq(slB)), q-
orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and
groups (Luminy-Marseille, 1988) (V. Kac, ed.), Adv. Ser. Math. Phys., vol. 7, World
Scientific, Teaneck, NJ, 1989, pp. 285-339.
[KR2] , q-Weyl group and a multiplicative formula for universal R-matrices, Comm.
Math. Phys. 134 A990), 421-431.
[KZ] Knizhnik, V.G., and Zamolodchikov, A.B., Current algebra and Wess-Zumino model
in two dimensions. Nuclear Phys. B 247 A984), 83-103.
[Kohl] Kohno, T., Monodromy representations of braid groups and Yang-Baxter equations,
Ann. Inst. Fourier 37 A987), no. 4, 139-160.
[Koh2] , Hecke algebra representations of braid groups and classical Yang-Baxter equa-
equations., Conformal field theory and solvable lattice models (Kyoto, 1986), Adv. Stud.
Pure Math., vol. 16, Academic Press, Boston, 1988, pp. 255-269.
[Ku] Kuroki, G-, Fock space representations of affine Lie algebras and integral representa-
representations in the Wess-Zumino-Witten models, Comm. Math. Phys. 142 A991), 511-542.
[Law] Lawrence, R. J., Homological representations of the Hecke algebra, Comm. Math. Phys.
135 A990), 141-191.
[LS] Levendorskii, S.Z. and Soibelman, Ya. S., Some applications of quantum Weyl groups,
J. Geom. Phys. 7 A990), 241-254.
[Luk] Lukyanov, S., Free field representation for massive integrable models, Comm. Math.
Phys. 167 A995), 183-226.
[LI] Lusztig, G., Quantum deformations of certain simple modules over enveloping algebras,
Adv. Math. 70 A988), 237-249.
[L2] , Introduction to quantum groups, Birkhauser, Boston, 1993.
[L3] , Monodromic systems on affine flag manifolds, Proc. Roy. Soc. London Ser. A
445 A994), 231-246; Erratum, ibid. 450 A995), 731-732.
[Mac] MacLane, S-, Categories for the working mathematician, Graduate Texts in Mathe-
Mathematics, vol. 5, Springer-Verlag, New York, 1971.
[Man] Mandelstam, S., Dual-resonance models, Physics Rep. 13 A974), 259-353.
[Ml] Manin, Yu. I., Quantum groups and non-commutative geometry, Centre de Recherches
Mathematiques, Montreal, 1988.
[M2] , Notes on quantum groups and quantum de Rham complexes, Theoret. and
Math. Phys. 92 A992), 997-1023.
[Matl] Matsuo, A., An aplication of Aomoto-Gelfand hypergeometric functions to the SU(n)
Knizhnik-Zamolodchikov equation, Comm. Math. Phys. 134 A990), 65-77.
[Mat2] , Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamo-
Knizhnik-Zamolodchikov equation, Comm. Math. Phys. 151 A993), 263-273.
[Mat3] , Quantum algebra structure of certain Jackson integrals, Comm. Math. Phys.
157 A993), 479-498.
[Mat4] , Free field representation of quantum affine algebra Uq(sh), Phys. Lett. B 308
A993), 260-265.
[Mat5] , q-deformation of Wokimoto modules, primary fields and screening operators,
Comm. Math. Phys. 160 A994), 33-48.
[MP] Moody, R. V. and Pianzola, A., Lie algebras with triangular decompositions, Canad.
Math. Soc. Ser. Monographs and Adv. Texts, Wiley, New York, 1995.
[MSI] Moore, G., Seiberg, N., Classical and quantum conformal field theory, Comm. Math.
Phys. 123 A989), 177-254.
[MS2] , Lectures on RCFT, Superstrings '89 (Proc. of the 1989 Trieste Spring School)
(M. Green et al, eds.), World Sci., River Edge, NJ, 1990, pp. 1-129.
[OT] Orlik, P. and Terao, H., Arrangements of hyperplanes, Springer-Verlag, Berlin, 1992.

Index
6j-symbols, 95
/37-system, 67
quantum, 180
g-hypergeometric function, 164
Asymptotic solution
for a difference equation, 173
of KZ equations, 117
Asymptotic zone, 117
Braid group, 87
Braided tensor category, 86
Cartan matrix, 16
generalized, 21
symmetrizable, 22, 82
Casimir element, 19
quantum, 92
Central charge, 23
Classical r-matrix, 45
Cohomology with coefficients in a local sys-
system, 99
Conformal blocks, 183
Connection, 97
flat, 98
Connection matrices
for difference equations, 173
for KZ equations, 119
Correlation function, 34
for quantum affine algebras, 150
Coxeter number, 19
dual, 19
Drinfeld associator, 125
Drinfeld category, 124
Drinfeld-Kohno theorem, 126
Elliptic quantum group, 181
Evaluation homomorphisms, 132
Evaluation representation
of affine Lie algebras, 25
of quantum affine algebras, 134
Exchange matrix
for affinwLie algebras, 122
for quantum affine algebras, 177
for'quantum groups, 94
Fock module, 63, 68
Free field realization
of intertwining operators, 73-77
of Verma modules, 70
Gauss-Manin connection, 105
Heisenberg algebra, 63, 68
Hexagon axiom, 86
Highest-weight vector, 17
Holonomic system of difference equations,
153
Holonomy, 98
Homology
relative, 107
singular, 99
Hopf algebra, 79
cocommutative, 81
quasitriangular, 85
Hypergeometric function, 53
Hyperplane arrangement, 101
Jackson integral, 163
Knizhnik-Zamolodchikov equations, 35
elliptic, 185
operator, 32
quantum, 151
modified, 156

INDEX
operator, 149
trigonometric, 44
Knizhnik-Zamolodchikov-Bernard equation,
185
Level, 23
of a solution of KZ equations, 49
critical, 23
generic, 24
Lie algebras
affine, 20
Kac-Moody, 21
extended, 22
simple finite-dimensional, 15
simply-laced, 16
Local system, 98
Monodromy, 114
Normal ordering, 64
Orlik-Solomon algebra, 101
Pentagon axiom, 86
Pochhammer loop, 53
Quantum affine algebra, 131
Quantum double, 89
Quantum groups, 83
Quasimeromorpllic function, 172
Screening operators, 75
Shapovalov form, 17
Singular vectors, 19
Star-triangle relation, 94
Sugawara construcion, 26
Universal ij-matrix, 85
forW,(fle),90
for quantum affine algebras, 136
Vacuum vector, 64
Verma module, 17
contragredient, 18
for affine Lie algebras, 22
contragredient, 24
for quantum groups, 83
Vertex operator, 64
Virasoro algebra, 25
Wakimoto module, 70
Weight decomposition, 17
for quantum groups, 83
Weight subspace, 17
Weyl module, 23
Yang-Baxter equation
classical, 45
dynamical, 96
quantum, 87
with a spectral parameter, 137
Selected Titles in This Series
(Continued from the front of this publication)
27 Nathan J. Fine, Basic hypergeometric series and applications, 1988
26 Hari Bercovici, Operator theory and arithmetic in H°°, 1988
25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988
24 Lance W. Small, Editor, Noetherian rings and their applications, 1987
23 E. H. Rothe, Introduction to various aspects of degree theory in Banach spaces, 1986
22 Michael E. Taylor, Noncommutative harmonic analysis, 1986
21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors,
The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof,
1986
20 Kenneth R. Goodearl, Partially ordered abelian groups with interpolation, 1986
19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984
18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981
17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980
16 O. Timothy O'Meara, Symplectic groups, 1978
15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977
14 V. Guillemin and S. Sternberg, Geometric asymptotics, 1977
13 C. Pearcy, Editor, Topics in operator theory, 1974
12 J. R. Isbell, Uniform spaces, 1964
11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964
10 R. Ayoub, An introduction to the analytic theory of numbers, 1963
9 Arthur Sard, Linear approximation, 1963
8 J. Lehner, Discontinuous groups and automorphic functions, 1964
7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961
7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961
6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951
5 S. Bergman, The kernel function and conformal mapping, 1950
4 O. F. G. Schilling, The theory of valuations, 1950
3 M. Marden, Geometry of polynomials, 1949
2 N. Jacobson, The theory of rings, 1943
1 J. A. Shohat and J. D. Tamarkin, The problem of moments, 1943
(See the AMS catalog for earlier titles)