How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

Abstract This is a survey paper on algorithms that have been developed during the last 25 years for the explicit computation of the structure of an associative algebra of finite dimension over either a finite field or an algebraic number field. This constructive approach was initiated in 1985 by Friedl and Rónyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Küronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PTn on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group Sn .