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Lectures on Algebraic Quantum Groups
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| 概要 | 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 p...resented 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).続きを見る |
| 目次 | Preface I. BACKGROUND AND BEGINNINGS I.1. Beginnings and first examples I.2. Further quantized coordinate rings I.3. The quantized enveloping algebra of sC2(k) I.4. The finite dimensional representations of Uq(5r2(k)) I.5. Primer on semisimple Lie algebras I.6. Structure and representation theory of Uq(g) with q generic I.7. Generic quantized coordinate rings of semisimple groups I.8. 0q(G) is a noetherian domain I.9. Bialgebras and Hopf algebras I.10. R-matrices I.11. The Diamond Lemma I.12. Filtered and graded rings I.13. Polynomial identity algebras I.14. Skew polynomial rings satisfying a polynomial identity I.15. Homological conditions I.16. Links and blocks II. GENERIC QUANTIZED COORDINATE RINGS II.1. The prime spectrum II.2. Stratification II.3. Proof of the Stratification Theorem II.4. Prime ideals in 0q (G) II.5. H-primes in iterated skew polynomial algebras II.6. More on iterated skew polynomial algebras II.7. The primitive spectrum II.8. The Dixmier-Moeglin equivalence II.9. Catenarity II.10. Problems and conjectures III. QUANTIZED ALGEBRAS AT ROOTS OF UNITY III.1. Finite dimensional modules for affine PI algebras 1II.2. The finite dimensional representations of UE(5C2(k)) II1.3. The finite dimensional representations of OE(SL2(k)) III.4. Basic properties of PI Hopf triples III.5. Poisson structures 1II.6. Structure of U, (g) III.7. Structure and representations of 0,(G) III.8. Homological properties and the Azumaya locus II1.9. Müller’s Theorem and blocks III.10. Problems and perspectives.続きを見る |
| 冊子版へのリンク | https://hdl.handle.net/2324/1001401112 |
| 本文を見る | Springer Book Archive - Mathematics: 2002 |
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| 登録日 | 2023.09.29 |
| 更新日 | 2024.01.30 |