Algorithmic Methods in Non-Commutative Algebra - A. Verschoren

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Résumé :
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 Poincar?-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.

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