Lectures on Algebraic Quantum Groups

1. Front Matter

   Pages i-ix

   PDF

2. Beginnings and First Examples

   • Ken A. Brown, Ken R. Goodearl

   Pages 1-14

3. Further Quantized Coordinate Rings

   • Ken A. Brown, Ken R. Goodearl

   Pages 15-23

4. The Quantized Enveloping Algebra of sl₂(k)

   • Ken A. Brown, Ken R. Goodearl

   Pages 25-28

5. The Finite Dimensional Representations of U _q (sl₂(k))

   • Ken A. Brown, Ken R. Goodearl

   Pages 29-37

6. Primer on Semisimple Lie Algebras

   • Ken A. Brown, Ken R. Goodearl

   Pages 39-44

7. Structure and Representation Theory of U _q (g)with q Generic

   • Ken A. Brown, Ken R. Goodearl

   Pages 45-58

8. Generic Quantized Coordinate Rings of Semisimple Groups

   • Ken A. Brown, Ken R. Goodearl

   Pages 59-67

9. O _q (G) is a Noetherian Domain

   • Ken A. Brown, Ken R. Goodearl

   Pages 69-80

10. Bialgebras and Hopf Algebras

    • Ken A. Brown, Ken R. Goodearl

    Pages 81-91

11. R-Matrices

    • Ken A. Brown, Ken R. Goodearl

    Pages 93-96

12. The Diamond Lemma

    • Ken A. Brown, Ken R. Goodearl

    Pages 97-101

13. Filtered and Graded Rings

    • Ken A. Brown, Ken R. Goodearl

    Pages 103-111

14. Polynomial Identity Algebras

    • Ken A. Brown, Ken R. Goodearl

    Pages 113-117

15. Skew Polynomial Rings Satisfying a Polynomial Identity

    • Ken A. Brown, Ken R. Goodearl

    Pages 119-123

16. Homological Conditions

    • Ken A. Brown, Ken R. Goodearl

    Pages 125-127

17. Links and Blocks

    • Ken A. Brown, Ken R. Goodearl

    Pages 129-133

18. The Prime Spectrum

    • Ken A. Brown, Ken R. Goodearl

    Pages 135-146

19. Stratification

    • Ken A. Brown, Ken R. Goodearl

    Pages 147-158

20. Proof of the Stratification Theorem

    • Ken A. Brown, Ken R. Goodearl

    Pages 159-164

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).

• algebra
• algebraic group
• quantum groups
• representation theory
• ring theory

"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

• Department of Mathematics, University of Glasgow, Glasgow, UK

  Ken A. Brown

• Department of Mathematics, University of California, Santa Barbara, USA

  Ken R. Goodearl

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