Table Of Content155
Graduate Texts in Mathematics
Editorial Board
I.H. Ewing F.W. Gehring P.R. Halmos
Graduate Texts in Mathematics
TAKEUTIIZARING. Introduction to Axiomatic 33 HIRSCH. Differential Topology.
Set Theory. 2nd ed. 34 SPITZER. Principles of Random Walk. 2nd ed.
2 OXTOBY. Measure and Category. 2nd ed. 35 WERMER. Banach Algebras and Several
3 SCHAEFFER. Topological Vector Spaces. Complex Variables. 2nd ed.
4 HILTON/STAMMBACH. A Course in 36 KELLEy/NAMIOKA et al. Linear Topological
Homological Algebra. Spaces.
5 MAC LANE. Categories for the Working 37 MONK. Mathematical Logic.
Mathematician. 38 GRAUERT/FRITZSCHE. Several Complex
6 HUGHES/PIPER. Projective Planes. Variables.
7 SERRE. A Course in Arithmetic. 39 ARVESON. An Invitation to C*-Algebras.
8 TAKEUTI/ZARING. Axiomatic Set Theory. 40 KEMENy/SNELL/KNAPP. Denumerable Markov
9 HUMPHREYS. Introduction to Lie Algebra~ Chains. 2nd ed.
and Representation Theory. 41 ApOSTOL. Modular Functions and Dirichlet
10 COHEN. A Course in Simple Homotopy Series in Number Theory. 2nd ed.
Theory. 42 SERRE. Linear Representations of Finite
11 CONWAY. Functions of One Complex Groups.
Variable. 2nd ed. 43 GILLMAN/JERISON. Rings of Continuous
12 BEALS. Advanced Mathematical Analysis. Functions.
I3 ANDERSON/FULLER. Rings and Categories of 44 KENDIG. Elementary Algebraic Geometry.
Modules. 2nd ed. 45 LoEVE. Probability Theory I. 4th ed.
14 GOLUBITSKy/GUILEMIN. Stable Mappings and 46 LoEvE. Probability Theory II. 4th ed.
Their Singularities. 47 MOISE. Geometric Topology in Dimensions 2
15 BERBERIAN. Lectures in Functional Analysis and 3.
and Operator Theory. 48 SACHSlWu. General Relativity for
16 WINTER. The Structure of Fields. Mathematicians.
17 ROSENBLATT. Random Processes. 2nd ed. 49 GRUENBERGIWEIR. Linear Geometry. 2nd ed.
18 HALMos. Measure Theory. 50 EDWARDS. Fermat's Last Theorem.
19 HALMos. A Hilbert Space Problem Book. 51 KLINGENBERG. A Course in Differential
2nd ed. Geometry.
20 HUSEMOLLER. Fibre Bundles. 3rd ed. 52 HARTSHORNE. Algebraic Geometry.
21 HUMPHREYS. Linear Algebraic Groups. 53 MANIN. A Course in Mathematical Logic.
22 BARNES/MACK. An Algebraic Introduction to 54 GRAVERIW ATKINS. Combinatorics with
Mathematical Logic. Emphasis on the Theory of Graphs.
23 GREUB. Linear Algebra. 4th ed. 55 BROWN/PEARCY. Introduction to Operator
24 HOLMES. Geometric Functional Analysis and Theory I: Elements of Functional Analysis.
Its Applications. 56 MASSEY. Algebraic Topology: An
25 HEWITT/STROMBERG. Real and Abstract Introduction.
Analysis. 57 CROWELL/Fox. Introduction to Knot Theory.
26 MANES. Algebraic Theories. 58 KOBUTZ. p-adic Numbers, p-adic Analysis,
27 KELLEY. General Topology. and zeta-Functions. 2nd ed.
28 ZARISKIISAMUEL. Commutative Algebra. 59 LANG. Cyclotomic Fields.
Vol.l. 60 ARNOLD. Mathematical Methods in Classical
29 ZARISKIISAMUEL. Commutative Algebra. Mechanics. 2nd ed.
Vol.lI. 61 WHITEHEAD. Elements of Homotopy Theory.
30 JACOBSON. Lectures in Abstract Algebra I. 62 KARGAPOLOVIMERLZJAKOv. Fundamentals of
Basic Concepts. the. Theory of Groups.
31 JACOBSON. Lectures in Abstract Algebra II. 63 BOLLOBAS. Graph Theory.
Linear Algebra. 64 EDWARDS. Fourier Series. Vol. I. 2nd ed.
32 JACOBSON. Lectures in Abstract Algebra III.
Theory of Fields and Galois Theory. continued after index
Christian Kassel
Quantum Groups
With 88 Illustrations
Springer-Science+Business Media, LLC
Christian Kassel
Institut de Recherche Mathematique Avancee
Universite Louis Pasteur-C.N.R.S.
67084 Strasbourg
France
Editorial Board
J.H. Ewing F. W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Indiana University University of Michigan Santa Clara University
Bloomington, IN 47405 Ann Arbor, MI 48109 Santa Clara, CA 95053
USA USA USA
Mathematics Subject Classification (1991): Primary-17B37, 18DlO, 57M25, 81R50;
Secondary-16W30, 17B20, 17B35, 18D99, 20F36
Library of Congress Cataloging-in-Publication Data
Kassel, Christian.
Quantum groups/Christian Kassel.
p. cm. - (Graduate texts in mathematics; voI. 155)
Includes bibliographical references and index.
ISBN 978-1-4612-6900-7 ISBN 978-1-4612-0783-2 (eBook)
DOI 10.1007/978-1-4612-0783-2
1. Quantum groups. 2. Hopf algebras. 3. Topology.
4. Mathematical physics. 1. Title. II. Series: Graduate texts in
mathematics; 155.
QC20.7.G76K37 1995
512'.55-dc20 94-31760
Printed on acid-free paper.
© 1995 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc in 1995
Softcover reprint of the hardcover 1s t edition 1995
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission ofthe publisher (Springer Science+Business Media, LLC), except for brief excerpts in
connection with reviews or scholarly anaJysis. Use in connection with any form of information storage
and retrievaJ, electronic adaptation, computer software, or by similar or dissimilar methodology now
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
Production managed by Francine McNeill; manufacturing supervised by Genieve Shaw.
Photocomposed pages prepared using Patrick D.F. Ion's TeX files.
987654321
ISBN 978-1-4612-6900-7
Preface
{( Eh bien, Monsieur, que
pensez-vous des x et des y ?»
Je lui ai repondu :
{( C'est bas de plafond. »
V. Hugo [Hug51]
The term "quantum groups" was popularized by Drinfeld in his address to
the International Congress of Mathematicians in Berkeley (1986). It stands
for certain special Hopf algebras which are nontrivial deformations of the
enveloping Hopf algebras of semisimple Lie algebras or of the algebras of
regular functions on the corresponding algebraic groups. As was soon ob
served, quantum groups have close connections with varied, a priori remote,
areas of mathematics and physics.
The aim of this book is to provide an introduction to the algebra behind
the words "quantum groups" with emphasis on the fascinating and spec
tacular connections with low-dimensional topology. Despite the complexity
of the subject, we have tried to make this exposition accessible to a large
audience. We assume a standard knowledge of linear algebra and some
rudiments of topology (and of the theory of linear differential equations as
far as Chapter XIX is concerned).
We divided the book into four parts we now briefly describe. In Part I
we introduce the language of Hopf algebras and we illustrate it with the
Hopf algebras SLq(2) and Uq(.s((2)) associated with the classical group
8L2. These are the simplest examples of quantum groups, and actually the
only ones we treat in detail. Part II focuses on two classes of Hopf algebras
that provide solutions of the Yang-Baxter equation in a systematic way. We
review a method due to Faddeev, Reshetikhin, and Takhtadjian as well as
Drinfeld's quantum double construction, both designed to produce quan
tum groups. Parts I and II may form the core of a one-year introductory
course on the subject.
Parts III and IV are devoted to some of the spectacular connections
alluded to before. The avowed objective of Part III is the construction of
isotopy invariants of knots and links in R3, including the Jones polynomial,
Preface
VI
from certain solutions of the Yang-Baxter equation. To this end, we intro
duce various classes of tensor categories that are responsible for the close
relationship between quantum groups and knot theory. Part IV presents
more advanced material: it is an account of Drinfeld's elegant treatment of
the monodromy of the Knizhnik-Zamolodchikov equations. Our aim is to
highlight Drinfeld's deep result expressing the braided tensor category of
modules over a quantum enveloping algebra in terms of the corresponding
semisimple Lie algebra. We conclude the book with the construction of a
"universal knot invariant". This is a nice, far-reaching application of the
algebraic techniques developed in the preceding chapters.
I wish to acknowledge the inspiration I drew during the composition of
this text from [Dri87] [Dri89a] [Dri89b] [Dri90] by Drinfeld, [JS93] by Joyal
and Street, [Tur89] [RT90] by Reshetikhin and Turaev. After having become
acquainted with quantum groups, the reader is encouraged to return to
these original sources. Further references are given in the notes at the end
of each chapter. Lusztig's and Turaev's monographs [Lus93] [Tur94] may
complement our exposition advantageously.
This book grew out of two graduate courses I taught at the Department
of Mathematics of the Universite Louis Pasteur in Strasbourg during the
years 1990-92. Part I is the expanded English translation of [Kas92]. It is a
pleasure to express my thanks to C. Bennis, R. Berger, C. Mitschi, P. Nuss,
C. Reutenauer, M. Rosso, V. Turaev, M. Wambst for valuable discussions
and comments, and to Raymond Seroul who coded the figures. lowe special
thanks to Patrick Ion for his marvellous job in preparing the book for
printing, with his attention to mathematical, English, typographical, and
computer details.
Christian Kassel
March 1994, Strasbourg
Notation. - Throughout the text, k is a field and the words "vector
space", "linear map" mean respectively "k-vector space" and "k-linear
map". The boldface letters N, Z, Q, R, and C stand successively for the
nonnegative integers, all integers, the field of rational, real, and complex
numbers. The Kronecker symbol l5ij is defined by l5ij = 1 if i = j and is
zero otherwise. We denote the symmetric group on n letters by Sri' The
sign of a permutation u is indicated by c(u).
The symbol 0 indicates the end of a proof. Roman figures refer to the
numbering of the chapters.
Contents
Preface v
Part One Quantum 8L(2) 1
I
Preliminaries 3
1 Algebras and Modules . 3
2 Free Algebras ...... 7
3 The Affine Line and Plane 8
4 Matrix Multiplication ... 10
5 Determinants and Invertible Matrices 10
6 Graded and Filtered Algebras 12
7 Ore Extensions . . 14
8 Noetherian Rings 18
9 Exercises 20
10 Notes ....... 22
II
Tensor Products 23
1 Tensor Products of Vector Spaces 23
2 Tensor Products of Linear Maps 26
3 Duality and Traces ........ 29
4 Tensor Products of Algebras .. 32
5 Tensor and Symmetric Algebras 34
6 Exercises 36
7 Notes ............... 38
viii Contents
III
The Language of Hopf Algebras 39
1 Coalgebras.. 39
2 Bialgebras ............ 45
3 Hopf Algebras . . . . . . . . . . 49
4 Relationship with Chapter I. The Hopf Algebras GL(2)
and SL(2). . . . . . . . . . . 57
5 Modules over a Hopf Algebra. . . . . . . . . . . . . . . 57
6 Comodules......................... 61
7 Comodule-Algebras. Coaction of SL(2) on the Affine Plane 64
8 Exercises 66
9 Notes.............................. 70
IV
The Quantum Plane and Its Symmetries 72
1 The Quantum Plane . . . . . . . . . . . . . 72
2 Gauss Polynomials and the q-Binomial Formula 74
3 The Algebra Mq(2) ......... . 77
4 Ring-Theoretical Properties of Mq(2) . 81
5 Bialgebra Structure on Mq(2) . .... . 82
6 The Hopf Algebras GLq(2) and SLq(2) 83
7 Coaction on the Quantum Plane 85
8 Hopf *-Algebras 86
9 Exercises 88
10 Notes ..... . 90
V
The Lie Algebra of SL(2) 93
1 Lie Algebras . . . . . 93
2 Enveloping Algebras . . 94
3 The Lie Algebra .5[(2) . 99
4 Representations of .5[(2) 101
5 The Clebsch-Gordan Formula. 105
6 Module-Algebra over a Bialgebra. Action of .5[(2) on the
Affine Plane ....................... 107
7 Duality between the Hopf Algebras U(.5[(2)) and SL(2) 109
8 Exercises 11 7
9 Notes............................ 119
VI
The Quantum Enveloping Algebra of .5[(2) 121
1 The Algebra Uq(.5[(2)) . . . . . . . . . . . . . 121
2 Relationship with the Enveloping Algebra of .5[(2) 125
3 Representations of Uq . . . . . . . . . . . . . . . . 127
4 The Harish-Chandra Homomorphism and the Centre of U 130
q
Contents ix
5 Case when q is a Root of Unity. 134
6 Exercises 138
7 Notes .............. . 138
VII
A Hopf Algebra Structure on Uis[(2)) 140
1 Comultiplication............. 140
2 Semi simplicity . . . . . . . . . . . . . . 143
3 Action of Uq(.s[(2)) on the Quantum Plane 146
4 Duality between the Hopf Algebras Uq(.s[(2)) and SLq(2) 150
5 Duality between U (.s[(2))-Modules and SLq(2)-Comodules 154
q
6 Scalar Products on U (.s[(2))-Modules 155
q
7 Quantum Clebsch-Gordan . 157
8 Exercises 162
9 Notes............ 163
Part Two Universal R-Matrices 165
VIII
The Yang-Baxter Equation and (Co)Braided Bialgebras 167
1 The Yang-Baxter Equation . . . . . . . . . . . . 167
2 Braided Bialgebras. . . . . . . . . . . . . . . . . . . . . 172
3 How a Braided Bialgebra Generates R-Matrices .... 178
4 The Square of the Antipode in a Braided Hopf Algebra 179
5 A Dual Concept: Cobraided Bialgebras 184
6 The FRT Construction . . . . . . 188
7 Application to GLq(2) and SLq(2) 194
8 Exercises 196
9 Notes................ 198
IX
Drinfeld's Quantum Double 199
1 Bicrossed Products of Groups ..... . 199
2 Bicrossed Products of Bialgebras . . . . . 202
3 Variations on the Adjoint Representation 207
4 Drinfeld's Quantum Double . . . . . . . . 213
5 Representation-Theoretic Interpretation of the
Quantum Double . . . . 220
6 Application to Uq(.s[(2)) . 223
7 R-Matrices for U q 230
8 Exercises 236
9 Notes ...... . 238
