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

TL;DR

This survey reviews algorithms developed over 25 years for explicitly computing the structure of finite-dimensional associative algebras over various fields, highlighting their development and applications.

Contribution

It summarizes the evolution of algorithms for Wedderburn decomposition of associative algebras and illustrates their application with the case of the rational semigroup algebra for n=2.

Findings

Algorithms enable explicit Wedderburn decomposition of finite-dimensional associative algebras.

Application to the rational semigroup algebra of PT_2 demonstrates the methods.

The survey consolidates developments from multiple researchers over 25 years.

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 Ronyai and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos, Kuronya and Wales. I illustrate these algorithms with the case n = 2 of the rational semigroup algebra of the partial transformation semigroup PT_n on n elements; this generalizes the full transformation semigroup and the symmetric inverse semigroup, and these generalize the symmetric group S_n.