Lectures on Algebraic Quantum Groups

By: John C Brown, Ken R. Goodearl

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Paperback | 1 April 2002

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In September 2000, at the Centre de Recerca Matematica in Barcelona, we pre- sented a 30-hour Advanced Course on Algebraic Quantum Groups. After the course, we expanded and smoothed out the material presented in the lectures and inte- grated it with the background material that we had prepared for the participants; this volume is the result. As our title implies, our aim in the course and in this text is to treat selected algebraic aspects of the subject of quantum groups. Sev- eral of the words in the previous sentence call for some elaboration. First, we mean to convey several points by the term 'algebraic' - that we are concerned with algebraic objects, the quantized analogues of 'classical' algebraic objects (in contrast, for example, to quantized versions of continuous function algebras on compact groups); that we are interested in algebraic aspects of the structure of these objects and their representations (in contrast, for example, to applications to other areas of mathematics); and that our tools will be drawn primarily from noncommutative algebra, representation theory, and algebraic geometry. Second, the term 'quantum groups' itself. This label is attached to a large and rapidly diversifying field of mathematics and mathematical physics, originally launched by developments around 1980 in theoretical physics and statistical me- chanics. It is a field driven much more by examples than by axioms, and so resists attempts at concise description (but see Chapter 1. 1 and the references therein).

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"The proofs, sketches of proofs, and quotations from the literature are carefully written. Numerous examples and exercises are included, and bibliographical notes conclude each chapter. The second and third parts end with a discussion of open problems and perspectives for further research."

--Mathematical Reviews

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Preface
Background and Beginnings
Beginnings and first examples	p. 1
Further quantized coordinate rings	p. 15
The quantized enveloping algebra of sl[subscript 2](k)	p. 25
The finite dimensional representations of U[subscript q](sl[subscript 2](k))	p. 29
Primer on semisimple Lie algebras	p. 39
Structure and representation theory of U[subscript q](g) with q generic	p. 45
Generic quantized coordinate rings of semisimple groups	p. 59
O[subscript q](G) is a noetherian domain	p. 69
Bialgebras and Hopf algebras	p. 81
R-matrices	p. 93
The Diamond Lemma	p. 97
Filtered and graded rings	p. 103
Polynomial identity algebras	p. 113
Skew polynomial rings satisfying a polynomial identity	p. 119
Homological conditions	p. 125
Links and blocks	p. 129
Generic Qantized Coordinate Rings
The prime spectrum	p. 135
Stratification	p. 147
Proof of the Stratification Theorem	p. 159
Prime ideals in O[subscript q](G)	p. 165
H-primes in iterated skew polynomial algebras	p. 173
More on iterated skew polynomial algebras	p. 185
The primitive spectrum	p. 195
The Dixmier-Moeglin equivalence	p. 205
Catenarity	p. 215
Problems and conjectures	p. 229
Quantized algebras at Roots of Unity
Finite dimensional modules for affine PI algebras	p. 237
The finite dimensional representations of U[subscript [epsilon]](sl[subscript 2](k))	p. 247
The finite dimensional representations of O[subscript [epsilon]](SL[subscript 2](k))	p. 255
Basic properties of PI Hopf triples	p. 259
Poisson structures	p. 273
Structure of U[subscript [epsilon]](g)	p. 281
Structure and representations of O[subscript [epsilon]](G)	p. 289
Homological properties and the Azumaya locus	p. 303
Muller's Theorem and blocks	p. 313
Problems and perspectives	p. 323
Bibliography	p. 331
Index	p. 341
Table of Contents provided by Blackwell. All Rights Reserved.

Lectures on Algebraic Quantum Groups

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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