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How to compute the Wedderburn decomposition of a finite-dimensional associative algebra. (English) Zbl 1250.16018

Let \(A\) be finite-dimensional algebra over either a finite field or an algebraically closed field. The paper under review surveys various algorithms available to do the following:
(i)
compute a basis for the radical \(R\) of \(A\);
(ii)
compute structure constants for the semisimple quotient \(A/R\);
(iii)
compute a basis for the center of \(A/R\) consisting of orthogonal idempotents;
(iv)
compute the identity matrices in simple ideals of \(A/R\);
(v)
compute an isomorphism of each simple ideal of \(A/R\) with a full matrix algebra;
(vi)
compute explicit matrices for irreducible representations of \(A\).
The author illustrates these algorithms on the example of the semigroup algebra of the full partial transformation semigroup \(PT_2\) on a \(2\)-element set.

MSC:

16P10 Finite rings and finite-dimensional associative algebras
16Z05 Computational aspects of associative rings (general theory)
16-04 Software, source code, etc. for problems pertaining to associative rings and algebras
16G10 Representations of associative Artinian rings
16K20 Finite-dimensional division rings
16S50 Endomorphism rings; matrix rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
20M20 Semigroups of transformations, relations, partitions, etc.
20M25 Semigroup rings, multiplicative semigroups of rings
20M30 Representation of semigroups; actions of semigroups on sets
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References:

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