Algorithms for Finite Dimensional Algebras Over Finite Fields Using Basic Algebras

Abstract

This dissertation describes algorithms for computing information about finite dimensional associative algebra over a finite field. In particular, we provide algorithms for computing a basis, the lattice of two-sided ideals, the center, and the unit group for a finite dimensional associative algebra over a finite field.The primary strategy employed is to first compute the basic algebra using the techniques described by J. Carlson and G. Matthews in [CM06], then perform computations in the basic algebra where possible. The algorithms described in the dissertation have been implemented in GAP as a package called Basic Algebras from Matrix generators by the author. Timings for this implementation are provided.