Efficient decomposition of associative algebras over finite fields

We present new, efficient algorithms for some fundamental computations with finite-dimensional (but not necessarily commutative) associative algebras over finite fields. For a semisimple algebra. U we show how to compute a complete Wedderburn decomposition of U as a direct sum of simple algebras, an isomorphism between each simple component and a full matrix algebra, and a basis for the centre of a. If ill is given by a generating set of matrices in F-mxm, then our algorithm requires about O(m(3)) operations in F, in addition to the cost of factoring a polynomial in F[x] of degree O(m), and the cost of generating a small number of random elements from U. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a finite field in this same time. (C) 2000 Academic Press.