Algorithmic Methods in Non-Commutative Algebra - Applications to Quantum Groups

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.

List of contents

1. Generalities on rings.- 2. Gröbner basis computation algorithms.- 3. Poincaré-Birkhoff-Witt Algebras.- 4. First applications.- 5. Gröbner bases for modules.- 6. Syzygies and applications.- 7. The Gelfand-Kirillov dimension and the Hilbert polynomial.- 8. Primality.- References.

Summary

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.

Product details

• Authors: J Bueso, J L Bueso, J. L. Bueso, J.L. Bueso, José Bueso, Jos Gómez-Torrecillas, José Gómez-Torrecillas, A Verschoren, A. Verschoren, Alain Verschoren
• Assisted by: José Bueso (Editor), José Gómez-Torrecillas (Editor), Alain Verschoren (Editor)
• Publisher: Springer Netherlands
• Languages: English
• Product format: Paperback / Softback
• Released: 07.10.2010
• EAN: 9789048163281
• ISBN: 978-90-481-6328-1
• No. of pages: 300
• Weight: 482 g
• Illustrations: XI, 300 p.
• Series: Mathematical Modelling: Theory and Applications; Mathematical Modelling: Theory and Applications
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Algebra, C, Algorithms, Mathematics and Statistics, Algebraic Geometry, Numerical analysis, Rings (Algebra), Mathematical foundations, Associative rings, Associative Rings and Algebras, Category theory (Mathematics), Category Theory, Homological Algebra, Homological algebra, Numeric Computing