An algorithm for the construction of matrix representations for finitely presented non-commutative algebras
type: Item
instance of: scholarly article
title: An algorithm for the construction of matrix representations for finitely presented non-commutative algebras
zbMATH Open document ID: 0702.16011
DOI: 10.1016/S0747-7171(08)80004-1
author: Gilles Labonté
published in: Journal of Symbolic Computation
publication date: 1990-01-01T00:00:00Z
review text: In many applications, it is very useful to have available explicit matrix representations for the elements of associative algebras. The author's initial motivation for this study in fact stems from such an application, made while studying quantum mechanical wave equations which are covariant under the Poincaré group. For a finite presentation of an associative algebra E, with identity, over a field, the author studies an algorithm for the construction, from this presentation, of matrix representations of E. A set of vector constraints which is given as a part of the initial data determines which particular representative of E is produced. This construction problem for the algebra is solved through a reduction of it to the much simpler problem of constructing a Gröbner basis for a left module. Convergence of the algorithm is proved for all cases where the representation to be found is finite dimensional, which is always the case, for example, when E is finite. In addition, some examples are provided, which illustrate the close relationship that exists between this method and the Todd-Coxeter coset-enumeration method for group theory.
reviewed by: Tong Wenting
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 16Z05
Mathematics Subject Classification ID: 16S15
zbMATH DE Number: 4150431
zbMATH Keywords: matrix representations
zbMATH Keywords: finite presentation
zbMATH Keywords: associative algebra
zbMATH Keywords: algorithm
zbMATH Keywords: Gröbner basis
describes a project that uses: Algorithm 628
MaRDI profile type: Publication
cites work: An extension of Buchberger's algorithm and calculations in enveloping fields of Lie algebras
cites work: Computation and simplification in lie fields
cites work: The diamond lemma for ring theory
cites work: Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Translation from the German
cites work: Q3208084
cites work: Q3714165
cites work: Q3772359
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems
cites work: Non-commutative Gröbner bases in algebras of solvable type
cites work: Q5581665
cites work: On the solution of matrix equations. Example: Application to invariant equations
cites work: Report on a program for solving polynomial equations in non-commuting variables
cites work: Q3992832
cites work: Q5727698
cites work: Q3808248
cites work: Q3963209
cites work: A practical method for enumerating cosets of a finite abstract group
cites work: Q3745825
cites work: Algorithm 628
description: scientific article; zbMATH DE number 4150431