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  • © 2010

Introduction to Quantum Groups

Birkhäuser

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  • A classical introduction to quantum groups

  • Exercises and open problems included

  • The standard reference book for the material presented

  • Includes supplementary material: sn.pub/extras

Part of the book series: Modern Birkhäuser Classics (MBC)

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  • ISBN: 978-0-8176-4717-9
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Table of contents (42 chapters)

  1. Front Matter

    Pages i-xiv
  2. The Drinfeld-Jimbo Algebra U

    1. Front Matter

      Pages 1-1
  3. THE DRINFELD JIMBO ALGERBRA U

    1. The Algebra f

      • George Lusztig
      Pages 2-13
    2. Weyl Group, Root Datum

      • George Lusztig
      Pages 14-18
    3. The Algebra U

      • George Lusztig
      Pages 19-33
    4. The Quasi-\(\mathcal{R}\)-Matrix

      • George Lusztig
      Pages 34-39
    5. Complete Reducibility Theorems

      • George Lusztig
      Pages 48-54
    6. Higher Order Quantum Serre Relations

      • George Lusztig
      Pages 55-60
  4. Geometric Realization of F

    1. Front Matter

      Pages 61-62
  5. GEOMETRIC REALIZATION OF F

    1. Review of the Theory of Perverse Sheaves

      • George Lusztig
      Pages 63-67
    2. Quivers and Perverse Sheaves

      • George Lusztig
      Pages 68-80
    3. Fourier-Deligne Transform

      • George Lusztig
      Pages 81-88
    4. Periodic Functors

      • George Lusztig
      Pages 89-91
    5. Quivers with Automorphisms

      • George Lusztig
      Pages 92-105
    6. The Signed Basis of f

      • George Lusztig
      Pages 113-128
  6. KASHIWARA’S Operators and Applications

    1. Front Matter

      Pages 129-129
  7. KASHIWARAS OPERATIONS AND APPLICATIONS

    1. The Algebra \(\mathfrak{U}\)

      • George Lusztig
      Pages 130-131
    2. Kashiwara’s Operators in Rank 1

      • 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.

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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.

—ZAA

[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

Keywords

  • 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

Reviews

From the reviews:

"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."

—ZAA

"[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

“George Lusztig lays out the large scale structure of the discussion that follows in the 348 pages of his Introduction to Quantum Groups. … A significant and important work. … it’s terrific stuff, elegant and deep, and Lusztig presents it very well indeed, of course.” (Michael Berg, The Mathematical Association of America, January, 2011)

Authors and Affiliations

  • MIT, Cambridge, USA

    George Lusztig

Bibliographic Information

Buying options

eBook
USD 79.99
Price excludes VAT (USA)
  • ISBN: 978-0-8176-4717-9
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD 99.99
Price excludes VAT (USA)