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.

Sommario

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.

Riassunto

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.

Dettagli sul prodotto

• Autori: 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
• Con la collaborazione di: José Bueso (Editore), José Gómez-Torrecillas (Editore), Alain Verschoren (Editore)
• Editore: Springer Netherlands
• Lingue: Inglese
• Formato: Tascabile
• Pubblicazione: 07.10.2010
• EAN: 9789048163281
• ISBN: 978-90-481-6328-1
• Pagine: 300
• Peso: 482 g
• Illustrazioni: XI, 300 p.
• Serie: Mathematical Modelling: Theory and Applications; Mathematical Modelling: Theory and Applications
• Categorie: Scienze naturali, medicina, informatica, tecnica > Matematica > Teoria delle probabilità, stocastica, statistica matematica; 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