An Algorithm for the Construction of Matrix Representations for Finitely Presented Non-Commutative Algebras

Let a finite presentation be given for an associative, in general non-commulative algebra E, with identity, over a field. We study an algorithm for the construction, from this presentation, of linear, i.e, matrix, representations of this algebra. A set of vector constraints which is given as part of the initial data determines which particular representation 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 Gr6bner basis for a left module. The price paid for this simplification is that the latter is then infinitely presented. Convergence of the algorithm is proven for all cases where the representation to be found is finite dimensional; which is always the case, ['or example, when E is finite. Examples are provided, some of which illustrate the close relationship that exists between this method and the Todd-Coxetcr coset-enumeration method for group theory.