Decomposition of algebras

Gianni, Patrizia; Miller, Victor S.; Trager, Barry M.

doi:10.1007/3-540-51084-2_29

Cited by 12 publications

(15 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The large variance that one sees in the runtimes arises from the fact that the code currently uses generic Magma functions for matrix algebras to compute a WM-decomposition A = J ⊕ S. The efficiency of this step can vary quite dramatically depending on the dimension of the Jacobson radical, J. As alluded to earlier, however, an implementation of the methods of [9,16] would likely improve that aspect of the test significantly, and a more thoughtful exploitation of the fact that our algebras Fig. 3.…”

Section: Implementation and Performancementioning

confidence: 97%

“…The code currently uses generic functions to handle certain computations with matrix algebras, and we expect that implementations of algorithms such as [9,16] would improve its performance significantly. Nevertheless, even on test examples that are built to allow comparisons with standard module isomorphism machinery in Magma, our functions fare reasonably well.…”

Section: Implementation and Performancementioning

confidence: 99%

“…Suppose M and N are both 1-dimensional modules over a common field, say GF (9). Up to isomorphism there is only one such module, so we would expect any test of module isomorphism to confirm that M ∼ = N .…”

Section: Introductionmentioning

confidence: 95%

“…The two modules M and N are determined by the action of A and B, respectively, on an identical 2-dimensional k-vector space: not only are A and B equal, so are M and N as k-spaces. So, M and N are indeed 1-dimensional (unital) vector spaces over GF (9). Why, then, does Magma report that M and N are nonisomorphic?…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The module isomorphism problem reconsidered

Brooksbank¹,

Wilson²

2015

Journal of Algebra

View full text Add to dashboard Cite

Algorithms to decide isomorphism of modules have been honed continually over the last 30 years, and their range of applicability has been extended to include modules over a wide range of rings. Highly efficient computer implementations of these algorithms form the bedrock of systems such as GAP and Magma, at least in regard to computations with groups and algebras. By contrast, the fundamental problem of testing for isomorphism between other types of algebraic structures -such as groups, and almost any type of algebra -seems today as intractable as ever. What explains the vastly different complexity status of the module isomorphism problem? This paper argues that the apparent discrepancy is explained by nomenclature. Current algorithms to solve module isomorphism, while efficient and immensely useful, are actually solving a highly constrained version of the problem. We report that module isomorphism in its general form is as hard as algebra isomorphism and graph isomorphism, both well-studied problems that are widely regarded as difficult. On a more positive note, for cyclic rings we describe a polynomial-time algorithm for the general module isomorphism problem. We also report on a Magma implementation of our algorithm.

show abstract

Section: Implementation and Performancementioning

confidence: 97%

Section: Implementation and Performancementioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The module isomorphism problem reconsidered

Brooksbank¹,

Wilson²

2015

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…While theoretically of great interest, these algorithms are probably not practical. For commutative algebras, Gianni et al (1988) gave an efficient algorithm to decompose an associative algebra over Q as a direct sum of local algebras.…”

Section: Historical Perspectivementioning

confidence: 99%

Efficient Decomposition of Associative Algebras over Finite Fields

Eberly

Giesbrecht

2000

Journal of Symbolic Computation

View full text Add to dashboard Cite

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.

show abstract