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Quantum Groups and Their Primitive Ideals

  • Book
  • © 1995

Overview

Authors:
  1. Anthony Joseph

    1. Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel
      Laboratoire de Mathématiques Fondamentales, Université Pierre et Marie Curie, Paris Cédex 05, France

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Table of contents (11 chapters)

  1. Front Matter

    Pages I-IX
    Download chapter 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 Rq[G]

    • Anthony Joseph
    Pages 294-325

  13. Back Matter

    Pages 326-383
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Keywords

About this book

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.

Authors and Affiliations

  • 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

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