Efficient Decomposition of Associative Algebras over Finite Fields

We present new, e cient algorithms for some fundamental computations with nite dimensional (but not necessarily commutative) associative algebras over nite elds. For a semisimple algebra A we show how to compute a complete Wedderburn decomposition of A 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 A is given by a generating set of matrices in Fm m then our algorithm requires about O(m3) 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 A. We also show how to compute a complete set of orthogonal primitive idempotents in any associative algebra over a nite eld in this same time.