Introduction to Quantum Groups and Crystal Bases
( Graduate Studies in Mathematics )
Publication series ：Graduate Studies in Mathematics
Author： Jin Hong;Seok-Jin Kang
Publisher： American Mathematical Society
Publication year： 2002
E-ISBN: 9781470420932
P-ISBN(Paperback): 9780821828748
Subject： O152.5 Lie group
Keyword： Algebra and Algebraic Geometry
Language： ENG
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Introduction to Quantum Groups and Crystal Bases
Description
The notion of a “quantum group” was introduced by V.G. Dinfeldacute 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.
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