Algorithmic Methods in Non-Commutative Algebra

Applications to Quantum Groups

By: J.L. Bueso, JosÃ© GÃ³mez-Torrecillas, A. Verschoren

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Hardcover | 31 July 2003

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316 Pages

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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.

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You Can Find This Book In

Non-Fiction Mathematics Groups & Group Theory Algebra Applied Mathematics Mathematical Modelling Medicine Medicine in General Computing & I.T.Computer Programming & Software Development Algorithms & Data Structures Geometry

Algebraic Geometry Computer Science Mathematical Theory of Computation

Introduction	p. vii
Generalities on rings	p. 1
Rings and ideals	p. 2
Modules and chain conditions	p. 13
Ore extensions	p. 20
Factorization	p. 28
Other examples	p. 43
Quantum groups	p. 48
Grobner basis computation algorithms	p. 63
Admissible orders	p. 63
Left Poincare-Birkhoff-Witt Rings	p. 68
Examples	p. 74
The Division Algorithm	p. 77
Grobner bases for left ideals	p. 80
Buchberger's Algorithm	p. 84
Reduced Grobner Bases	p. 94
Poincare-Birkhoff-Witt rings	p. 97
Effective computations for two-sided ideals	p. 101
Poincare-Birkhoff-Witt Algebras	p. 109
Bounding quantum relations	p. 109
Misordering	p. 114
The Diamond Lemma	p. 116
Poincare-Birkhoff-Witt Theorems	p. 123
Examples	p. 127
Iterated Ore Extensions	p. 130
First applications	p. 137
Applications to left ideals	p. 137
Cyclic finite-dimensional modules	p. 143
Elimination	p. 145
Graded and filtered algebras	p. 150
The [omega]-filtration of a PBW algebra	p. 153
Homogeneous Grobner bases	p. 155
Homogenization	p. 162
Grobner bases for modules	p. 169
Grobner bases and syzygies	p. 169
Computation of the syzygy module	p. 171
Admissible orders in stable subsets	p. 175
Grobner bases for modules	p. 177
Grobner bases for subbimodules	p. 185
Elementary applications of Grobner bases for modules	p. 188
Graded and filtered modules	p. 192
The [omega]-filtration of a module	p. 194
Homogeneous Grobner bases	p. 196
Homogenization	p. 197
Syzygies and applications	p. 203
Syzygies for modules	p. 203
Intersections	p. 209
Applications to finitely presented modules	p. 214
Schreyer's order	p. 217
Free resolutions	p. 219
Computation of Hom and Ext.	p. 223
The Gelfand-Kirillov dimension and the Hilbert polynomial	p. 239
The Gelfand-Kirillov dimension	p. 239
The Hilbert function of a stable subset	p. 246
The Hilbert function of a module over a PBW algebra	p. 253
The Gelfand-Kirillov dimension of PBW algebras	p. 255
Primality	p. 263
Localization	p. 263
The Ore condition and syzygies	p. 275
A primality test	p. 276
The primality test in iterated differential operator rings	p. 282
The primality test in coordinate rings of quantum spaces	p. 283
Index	p. 289
References	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

Algorithmic Methods in Non-Commutative Algebra

Applications to Quantum Groups

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Hardcover

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