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Efficient Decomposition of Associative Algebras over Finite Fields

✍ Scribed by W. Eberly; M. Giesbrecht

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 369 KB
Volume: 29
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.1999.0308

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✦ Synopsis

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 ) 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.

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