Fast randomized algorithms for the structure of matrix algebras over finite fields (extended abstract)

We discuss randomized algorithms which compute algebra generators of a Wedderburn complement as well as ideal generators of the radical of a matrix algebra over a finite field given by algebra generators. The cost of the algorithms is comparable to that of a poly-logarithmic number of matrix multiplications.

[1]  Derek F. Holt,et al.  Testing modules for irreducibility , 1994, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[2]  W. Eberly,et al.  E cient Decomposition of Associative Algebras over Finite Fields , 1999 .

[3]  Gábor Ivanyos,et al.  Finding the radical of an algebra of linear transformations , 1997 .

[4]  Mark Giesbrecht,et al.  Nearly Optimal Algorithms for Canonical Matrix Forms , 1995, SIAM J. Comput..

[5]  Gábor Ivanyos.  Finding the radical of matrix algebras using Fitting decompositions , 1999 .

[6]  Lajos Rónyai,et al.  Computing the Structure of Finite Algebras , 1990, J. Symb. Comput..

[7]  A. Kertész,et al.  Lectures on Artinian rings , 1987 .

[8]  J. Gathen,et al.  Computations for algebras and group representations , 1989 .

[9]  Mark Giesbrecht,et al.  Factoring in Skew-Polynomial Rings , 1992, LATIN.

[10]  L. Rónyai,et al.  Computations in Associative and Lie Algebras , 1999 .

[11]  Lajos Rónyai,et al.  Polynomial time solutions of some problems of computational algebra , 1985, STOC '85.

[12]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[13]  Lajos Rónyai.  Computations in Associative Algebras , 1991, Groups And Computation.

[14]  Lajos Rónyai,et al.  Computing Levi Decompositions in Lie algebras , 1997, Applicable Algebra in Engineering, Communication and Computing.

[15]  Mark Giesbrecht,et al.  Efficient Decomposition of Associative Algebras over Finite Fields , 2000, J. Symb. Comput..

[16]  Gábor Ivanyos,et al.  Treating the Exceptional Cases of the MeatAxe , 2000, Exp. Math..