# Lectures on Quantum Groups

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*Jens Carsten Jantzen*

Since its origin about ten years ago, the theory of quantum groups has become one of the most fascinating topics of modern mathematics, with numerous applications to several sometimes rather disparate areas, including low-dimensional topology and mathematical physics. This book is one of the first expositions that is specifically directed to students who have no previous knowledge of the subject. The only prerequisite, in addition to standard linear algebra, is some acquaintance with the classical theory of complex semisimple Lie algebras. Starting with the quantum analog of \(\mathfrak{sl}_2\), the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara–Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebras. The choice of the topics and the style of exposition make Jantzen's book an excellent textbook for a one-semester course on quantum groups.

#### Reviews & Endorsements

Very useful for … understanding and … research in quantum groups, in particular, the chapters on the braid group action and crystal bases … highly recommend[ed] … to all research mathematicians working in quantum groups … The writing is one of the most pleasant attributes of this book. The flow of the words and ideas is very smooth and very conducive to actually understanding what is going on.

-- Mathematical Reviews

Carefully written … there is an agreeable, sometimes informal, spirit with which [the author] tries to indicate to the reader what is really happening. One pleasant feature is his placing of long computations in appendices at the end of certain chapters.

-- Zentralblatt MATH

The material is very well motivated … Of the various monographs available on quantum groups, this one … seems the most suitable for most mathematicians new to the subject … will also be appreciated by a lot of those with considerably more experience.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## Lectures on Quantum Groups

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction 110 free
- Chapter 0. Gaussian Binomial Coefficients 514 free
- Chapter 1. The Quantized Enveloping Algebra U[sub(q)](sl[sub(2)]) 918
- Chapter 2. Representations of U[sub(q)](sl[sub(2)]) 1726
- Chapter 3. Tensor Products or: U[sub(q)](sl[sub(2)]) as a Hopf Algebra 3140
- Chapter 4. The Quantized Enveloping Algebra U[sub(q)](g) 5160
- Chapter 5. Representations of U[sub(q)](g) 6978
- Chapter 5A. Examples of Representations 8796
- Chapter 6. The Center and Bilinear Forms 105114
- Chapter 7. R-matrices and k[sub(q)][G] 129138
- Chapter 8. Braid Group Actions and PBW Type Basis 141150
- Chapter 8A. Proof of Proposition 8.28 173182
- Chapter 9. Crystal Bases I 187196
- Chapter 10. Crystal Bases II 217226
- Chapter 11. Crystal Bases III 237246
- References 259268
- Notations 263272
- Index 265274
- Errata 267276
- Back Cover Back Cover1282