Lectures on Algebraic Quantum Groups

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This book consists of an expanded set of lectures on algebraic aspects of quantum groups, concentrating particularly on quantized coordinate rings of algebraic groups and spaces and on quantized enveloping algebras of semisimple Lie algebras. The approach, a mixture of introductory textbook, lecture notes, and overview survey, is designed to allow access by graduate students and by researchers new to the areas, as well as by experts, and to provide a basis for further study of the subject. Thus, large parts of the material are developed in full textbook style, with many examples and numerous exercises; other portions are discussed with sketches of proofs, while still other material is quoted without proof. Much associated background material is outlined in a series of appendices. Among the topics covered for the first time in book form are a discussion of the nature of the prime spectrum of a "generic" quantum algebra, and details of how the Hopf algebra structure of the algebra and the Poisson algebra structure of the centre carry important consequences for quantized algebras when the quantum parameter is a root of unity. The book is structured in three parts: one introductory part with many examples plus background material, one concentrating on generic quantized coordinate rings, and one dealing with quantized algebras at roots of unity.

"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

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).

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.

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Lectures on Algebraic Quantum Groups
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