Efficient Decomposition of Associative Algebras over Finite Fields

W. Eberly, M. Giesbrecht
2000 Journal of symbolic computation  
We present new, efficient algorithms for some fundamental computations with finitedimensional (but not necessarily commutative) associative algebras over finite fields. 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 F m×m , then our algorithm requires about O(m 3 )
more » ... 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 finite field in this same time.
doi:10.1006/jsco.1999.0308 fatcat:myr6prcksrge3fufgqbxa6rlsa