
University Lecture Series 2002; 158 pp; softcover Volume: 26 ISBN10: 0821832328 ISBN13: 9780821832325 List Price: US$37 Member Price: US$29.60 Order Code: ULECT/26 See also: Finite Dimensional Algebras and Quantum Groups  Bangming Deng, Jie Du, Brian Parshall and Jianpan Wang Geometric Representation Theory and Extended Affine Lie Algebras  Erhard Neher, Alistair Savage and Weiqiang Wang  This book contains most of the nonstandard material necessary to get acquainted with this new rapidly developing area. It can be used as a good entry point into the study of representations of quantum groups. Among several tools used in studying representations of quantum groups (or quantum algebras) are the notions of Kashiwara's crystal bases and Lusztig's canonical bases. Mixing both approaches allows us to use a combinatorial approach to representations of quantum groups and to apply the theory to representations of Hecke algebras. The primary goal of this book is to introduce the representation theory of quantum groups using quantum groups of type \(A_{r1}^{(1)}\) as a main example. The corresponding combinatorics, developed by Misra and Miwa, turns out to be the combinatorics of Young tableaux. The second goal of this book is to explain the proof of the (generalized) LascouxLeclercThibon conjecture. This conjecture, which is now a theorem, is an important breakthrough in the modular representation theory of the Hecke algebras of classical type. The book is suitable for graduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, and related fields. Readership Graduate students and research mathematicians interested in representation theory of algebraic groups and quantum groups, the theory of Hecke algebras, algebraic combinatorics, and related fields. Reviews "The author gives a good introduction to the algebraic aspects of this fastdeveloping field ... Overall, this is a wellwritten and clear exposition of the theory needed to understand the latest advances in the theory of the canonical/global crystal basis and the links with the representation theory of symmetric groups and Hecke algebras. The book finishes with an extensive bibliography of papers, which is well organised into different areas of the theory for easy reference."  Zentralblatt MATH "Well written and covers ground quickly to get to the heart of the theory ... should serve as a solid introduction ... abundant references to the literature are given."  Mathematical Reviews Table of Contents



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