Lectures on Quantum Groups

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American Mathematical Soc., 1996 - Mathematics - 266 pages
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The Graduate Studies in Mathematics series is made up of books useful as graduate-level course texts. This book is an introduction to the theory of quantum groups. The main topic is the quantized enveloping algebras introduced independently by Drinfeld and Jimbo. Jantzen considers the crystal (or canonical) bases discovered independently by Lusztig and Kashiwara and looks at the quantum analogue of the Lie algebar SL and then at the quantum analogue of arbitrary finite dimensional complex Lie algebras.
 

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Contents

Gaussian Binomial Coefficients
5
The Quantized Enveloping Algebra Uqsl2
9
Representations of Uqsl2
17
Tensor Products or Uqsl2 as a Hopf Algebra
31
The Quantized Enveloping Algebra Uqg
51
Representations of Uqg
69
Examples of Representations
87
The Center and Bilinear Forms
105
Rmatrices and kqG
129
Braid Group Actions and PBW Type Basis
141
Proof of Proposition 828
173
Crystal Bases I
187
Crystal Bases II
217
Crystal Bases III
237
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Page 18 - An endomorphism of a finite dimensional vector space is diagonalizable if and only if its minimal polynomial splits into linear factors, each occurring with multiplicity 1.
Page 18 - Suppose that q is not a root of unity and that char(fc) ^ 2. Let M be a finite dimensional U-module. Then M is the direct sum of its weight spaces.

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