**Graduate Studies in Mathematics**

Volume: 42;
2002;
307 pp;
Hardcover

MSC: Primary 17;
Secondary 81; 82

Print ISBN: 978-0-8218-2874-8

Product Code: GSM/42

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

**Electronic ISBN: 978-1-4704-2093-2
Product Code: GSM/42.E**

List Price: $60.00

AMS Member Price: $48.00

MAA Member Price: $54.00

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# Introduction to Quantum Groups and Crystal Bases

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*Jin Hong; Seok-Jin Kang*

The notion of a “quantum group” was introduced by V.G.
Dinfel&dacute; and M. Jimbo, independently, in their study of the quantum
Yang-Baxter equation arising from 2-dimensional solvable lattice models.
Quantum groups are certain families of Hopf algebras that are deformations of
universal enveloping algebras of Kac-Moody algebras. And over the past 20
years, they have turned out to be the fundamental algebraic structure behind
many branches of mathematics and mathematical physics, such as solvable lattice
models in statistical mechanics, topological invariant theory of links and
knots, representation theory of Kac-Moody algebras, representation theory of
algebraic structures, topological quantum field theory, geometric
representation theory, and \(C^*\)-algebras.

In particular, the theory of “crystal bases” or “canonical
bases” developed independently by M. Kashiwara and G. Lusztig provides a
powerful combinatorial and geometric tool to study the representations of
quantum groups. The purpose of this book is to provide an elementary
introduction to the theory of quantum groups and crystal bases, focusing on the
combinatorial aspects of the theory.

#### Readership

Graduate students and research mathematicians interested in nonassociative rings and algebras.

#### Reviews & Endorsements

Book by Hong and Kang is the first expository text which gives a detailed account on the relationship between crystal bases and combinatorics. This book provides an accessible and “crystal clear” introduction and overview of the relatively new subject of quantum groups and crystal bases, … It will be an indispensable companion to the research papers.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Introduction to Quantum Groups and Crystal Bases

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction xi12 free
- Chapter 1. Lie Algebras and Hopf Algebras 120 free
- Chapter 2. Kac-Moody Algebras 2140
- Chapter 3. Quantum Groups 3756
- Chapter 4. Crystal Bases 6382
- Chapter 5. Existence and Uniqueness of Crystal Bases 91110
- Chapter 6. Global Bases 119138
- Chapter 7. Young Tableaux and Crystals 149168
- Chapter 8. Crystal Graphs for Classical Lie Algebras 169188
- §8.1. Example: U[sub(q)](B[sub(3)]-crystals 170189
- §8.2. Realization of U[sub(q)](A[sub(n–1)]-crystals 179198
- §8.3. Realization of U[sub(q)](C[sub(n)]-crystals 181200
- §8.4. Realization of U[sub(q)](B[sub(n)]-crystals 190209
- §8.5. Realization of U[sub(q)](D[sub(n)]-crystals 197216
- §8.6. Tensor product decomposition of crystals 203222
- Exercises 207226

- Chapter 9. Solvable Lattice Models 209228
- Chapter 10. Perfect Crystals 229248
- §10.1. Quantum afflne algebras 229248
- §10.2. Energy functions and combinatorial R-matrices 235254
- §10.3. Vertex operators for U[sub(q)](sl[sub(2)]-modules 239258
- §10.4. Vertex operators for quantum affine algebras 242261
- §10.5. Perfect crystals 247266
- §10.6. Path realization of crystal graphs 252271
- Exercises 260279

- Chapter 11. Combinatorics of Young Walls 263282
- Bibliography 297316
- Index of symbols 301320 free
- Index 305324
- Back Cover Back Cover1327