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Algorithmic Methods in Non-Commutative Algebra

Applications to Quantum Groups

  • Book
  • © 2003

Overview

Authors:
  1. José Bueso

    1. University of Granada, Granada, Spain

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  2. José Gómez-Torrecillas

    1. University of Granada, Granada, Spain

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  3. Alain Verschoren

    1. University of Antwerp, Antwerp, Belgium

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Part of the book series: Mathematical Modelling: Theory and Applications (MMTA, volume 17)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xi
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  2. Generalities on rings

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 1-61

  3. Gröbner basis computation algorithms

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 63-108

  4. Poincaré-Birkhoff-Witt Algebras

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 109-135

  5. First applications

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 137-168

  6. Gröbner bases for modules

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 169-202

  7. Syzygies and applications

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 203-237

  8. The Gelfand-Kirillov dimension and the Hilbert polynomial

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 239-261

  9. Primality

    • José Bueso, José Gómez-Torrecillas, Alain Verschoren
    Pages 263-287

  10. Back Matter

    Pages 289-300
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Keywords

About this book

The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc.

Authors and Affiliations

  • University of Granada, Granada, Spain

    José Bueso, José Gómez-Torrecillas

  • University of Antwerp, Antwerp, Belgium

    Alain Verschoren

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