Introduction to Quantum Groups

  • George Lusztig

Part of the Modern Birkhäuser Classics book series (MBC)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. The Drinfeld-Jimbo Algebra U

    1. Front Matter
      Pages 1-1
    2. George Lusztig
      Pages 2-13
    3. George Lusztig
      Pages 14-18
    4. George Lusztig
      Pages 19-33
    5. George Lusztig
      Pages 34-39
    6. George Lusztig
      Pages 48-54
    7. George Lusztig
      Pages 55-60
  3. Geometric Realization of F

    1. Front Matter
      Pages 61-62
    2. George Lusztig
      Pages 63-67
    3. George Lusztig
      Pages 68-80
    4. George Lusztig
      Pages 81-88
    5. George Lusztig
      Pages 89-91
    6. George Lusztig
      Pages 92-105
    7. George Lusztig
      Pages 113-128
  4. KASHIWARA’S Operators and Applications

    1. Front Matter
      Pages 129-129
    2. George Lusztig
      Pages 130-131
    3. George Lusztig
      Pages 132-141

About this book


The quantum groups discussed in this book are the quantized enveloping algebras introduced by Drinfeld and Jimbo in 1985, or variations thereof. It is shown that these algebras have natural integral forms that can be specialized at roots of 1 and yield new objects, which include quantum versions of the semi-simple groups over fields of positive characteristic. The theory of quantum groups has led to a new, extremely rigid structure, in which the objects of the theory are provided with canonical bases having rather remarkable properties. This book contains an extensive treatment of the theory of canonical bases in the framework of perverse sheaves. The theory developed in the book includes the case of quantum affine enveloping algebras and, more generally, the quantum analogs of the Kac–Moody Lie algebras.

Introduction to Quantum Groups will be of interest to mathematicians working in the representation theory of Lie groups and Lie algebras, knot theorists, theoretical physicists, and graduate students. Since large parts of the book are independent of the theory of perverse sheaves, the work may also be used as a textbook.


There is no doubt that this volume is a very remarkable piece of work...Its appearance represents a landmark in the mathematical literature.

—Bulletin of the London Mathematical Society

This book is an important contribution to the field and can be recommended especially to mathematicians working in the field.

—EMS Newsletter

The present book gives a very efficient presentation of an important part of quantum group theory. It is a valuable contribution to the literature.

—Mededelingen van het Wiskundig

Lusztig's book is very well written and seems to be flawless...Obviously, this will be the standard reference book for the material presented and anyone interested in the Drinfeld–Jimbo algebras will have to study it very carefully.


[T]his book is much more than an 'introduction to quantum groups.' It contains a wealth of material. In addition to the many important results (of which several are new–at least in the generality presented here), there are plenty of useful calculations (commutator formulas, generalized quantum Serre relations, etc.).

—Zentralblatt MATH


Fourier-Deligne transform Kac-Moody Lie algebras Kashiwara's operators Permutation Representation theory Weyl group algebra binomial braid group relations complete reducibility theorems homomorphism integrable U-module perverse sheaves

Authors and affiliations

  • George Lusztig
    • 1
  1. 1.MITCambridgeUSA

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media, LLC 2010
  • Publisher Name Birkhäuser Boston
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-0-8176-4716-2
  • Online ISBN 978-0-8176-4717-9
  • About this book