An algorithm for the construction of matrix representationsfor finitely presented non-commutative algebras

Let a finite presentation be given for an associative, in general non-commutative 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 muchsimpler problem of constructing a Gröbner 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, for example, when E is finite. Examples are provided, some of which illustrate the close relationship that exists between this method and the Todd-Coxeter coset-enumeration method for group theory