Quantum Groups and Their Primitive Ideals

1. Front Matter

   Pages I-IX

   PDF

2. Introduction

   • Anthony Joseph

   Pages 1-7

3. Hopf Algebras

   • Anthony Joseph

   Pages 8-35

4. Excerpts from the Classical Theory

   • Anthony Joseph

   Pages 36-61

5. Encoding the Cartan Matrix

   • Anthony Joseph

   Pages 62-95

6. Highest Weight Modules

   • Anthony Joseph

   Pages 96-130

7. The Crystal Basis

   • Anthony Joseph

   Pages 131-160

8. The Global Bases

   • Anthony Joseph

   Pages 161-200

9. Structure Theorems for U _q (g)

   • Anthony Joseph

   Pages 201-232

10. The Primitive Spectrum of U _q (g)

    • Anthony Joseph

    Pages 233-261

11. Structure Theorems for R _q [G]

    • Anthony Joseph

    Pages 262-293

12. The Prime Spectrum of R_q[G]

    • Anthony Joseph

    Pages 294-325

13. Back Matter

    Pages 326-383

    PDF

by a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.

• Algebra
• Kristallbasen
• Lie algebra
• Quantengruppen
• crystal bases
• einhüllende Algebren von Lie Algebren
• enveloping algebras of Lie algrebras
• quantum groups

• Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel

  Anthony Joseph

• Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, Paris Cédex 05, France

  Anthony Joseph

Policies and ethics