An algorithm for the construction of matrix representations for finitely presented non-commutative algebras
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.
scientific article; zbMATH DE number 4150431
scientific article