Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum GroupsThe 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. |
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Contents
II | xi |
III | 11 |
IV | 18 |
V | 26 |
VI | 41 |
VII | 46 |
VIII | 61 |
IX | 66 |
XXXI | 167 |
XXXIV | 169 |
XXXV | 173 |
XXXVI | 175 |
XXXVII | 183 |
XXXVIII | 186 |
XXXIX | 190 |
XL | 192 |
Other editions - View all
Algorithmic Methods in Non-Commutative Algebra J. L. Bueso,Jose Gomez-Torrecillas,A. Verschoren No preview available - 2014 |
Common terms and phrases
admissible order assume automorphism basis G Buchberger's Algorithm canonical co-homogeneous completely prime ideal consider COROLLARY defined DEFINITION deg(f deg(g dega deglex denote Diamond Lemma Division Algorithm division ring domain enveloping algebra euclidean domain EXAMPLE exists exp(f exp(g exp(h Exp(I exp(r filtration finite follows Gelfand-Kirillov dimension graded Gröbner basis hence Hilbert function homomorphism Homr implies induction irreducible isomorphism iterated Ore extension k-algebra k-vectorspace left Gröbner basis left ideal left noetherian left PBW ring left R-module LEMMA Let f Let G lexicographical order Lie algebra matrix minimal module monoid monomials non-commutative non-zero elements notations obtain PBW algebra polynomial positive integer previous result prime ideal PROOF PROPOSITION prove quantum groups quantum plane quasi-derivation quotient R-points reduced Gröbner reduction-unique resp stredo submodule syzygy Theorem two-sided ideal unique vectors verify Weyl algebra yields
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References to this book
A Singular Introduction to Commutative Algebra Gert-Martin Greuel,Gerhard Pfister Limited preview - 2007 |
Computational Commutative and Non-commutative Algebraic Geometry Svetlana Cojocaru,Gerhard Pfister,Victor Ufnarovski No preview available - 2005 |
Bibliographic information
| Title | Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups Volume 17 of Mathematical Modelling: Theory and Applications |
| Authors | J.L. Bueso, José Gómez-Torrecillas, A. Verschoren |
| Edition | illustrated |
| Publisher | Springer Science & Business Media, 2003 |
| ISBN | 1402014023, 9781402014024 |
| Length | 300 pages |
| Subjects | › › Computers / Information Technology Computers / Programming / Algorithms Mathematics / Algebra / Abstract Mathematics / Algebra / General Mathematics / Discrete Mathematics Mathematics / Geometry / Algebraic Mathematics / History & Philosophy Mathematics / Numerical Analysis |
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