**Mathematical Surveys and Monographs**

Volume: 56;
1998;
150 pp;
Hardcover

MSC: Primary 17;
Secondary 16; 81

Print ISBN: 978-0-8218-0336-3

Product Code: SURV/56

List Price: $63.00

AMS Member Price: $50.40

MAA Member Price: $56.70

**Electronic ISBN: 978-1-4704-1284-5
Product Code: SURV/56.E**

List Price: $63.00

AMS Member Price: $50.40

MAA Member Price: $56.70

# Algebras of Functions on Quantum Groups: Part I

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*Leonid I. Korogodski; Yan S. Soibelman*

The book is devoted to the study of algebras of functions on quantum groups. The authors' approach to the subject is based on the parallels with symplectic geometry, allowing the reader to use geometric intuition in the theory of quantum groups. The book includes the theory of Poisson Lie groups (quasi-classical version of algebras of functions on quantum groups), a description of representations of algebras of functions, and the theory of quantum Weyl groups. This book can serve as a text for an introduction to the theory of quantum groups.

#### Readership

Graduate students and research mathematicians working in algebra, representation theory, and mathematical physics.

#### Reviews & Endorsements

The book is written carefully and clearly. Every chapter begins with a short review of its content and ends with some historical remarks. All definitions are explained by several illustrative examples, remarks and comments. Proofs are given in detail. The book contains many interesting exercises. Hence we are certain that the book should be interesting for graduate students … it should also be useful for mathematicians working in the area of abstract algebras or representation theory and for anyone who is interested in the study of quantization procedures and symplectic geometry.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Algebras of Functions on Quantum Groups: Part I

- Contents vii8 free
- Chapter 0. Introduction 112 free
- Chapter 1. Poisson Lie Groups 516 free
- Chapter 2. Quantized Universal Enveloping Algebras 5768
- 1. Quantization of Lie bialgebras 5768
- 2. QUE-algebras and R-matrices 6172
- 3. Center of quasi-triangular Hopf algebras 7283
- 4. Center of U[sub(h)g and quantum Harish-Chandra homomorphism 7889
- 5. Finite-dimensional U[sub(h)]g-modules 8293
- 6. Tensor products of U[sub(h)]g-modules and tensor categories 8697
- 7. Fixed quantization parameter 89100
- 8. Historical remarks 94105

- Chapter 3. Quantized Algebras of Functions 95106
- 1. Main definitions 95106
- 2. Properties of the quantized algebras of functions 97108
- 3. Examples: C[SL[sub(2)](C)][sub(q)] and C[SU(2)][sub(q)] 101112
- 4. Representation theory of C[SU(2)][sub(q)] 104115
- 5. Representation theory of C[K][sub(q)] 109120
- 6. Representations of C[K][sub(q)] and symplectic leaves 116127
- 7. Representation theory of the twisted algebras of functions 122133
- 8. Representations of formal quantized algebras of functions 130141
- 9. Historical remarks 131142

- Chapter 4. Quantum Weyl Group and the Universal Quantum R-Matrix 133144
- Bibliography 149160