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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura

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1869-6104
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How to compute the Wedderburn decomposition of a finite-dimensional associative algebra

Murray R. Bremner
Published Online: 2011-01-20 | DOI: https://doi.org/10.1515/gcc.2011.003

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 .

Keywords.: Structure theory of associative algebras; Dickson's Theorem on the radical; Wedderburn–Artin Theorem; Wedderburn–Malcev Theorem; computational algebra; finite transformation semigroups; representation theory of finite semigroups

About the article

Received: 2010-06-17

Published Online: 2011-01-20

Published in Print: 2011-05-01


Citation Information: Groups – Complexity – Cryptology, Volume 3, Issue 1, Pages 47–66, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc.2011.003.

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[1]
Hader A. Elgendy
Communications in Algebra, 2014, Volume 42, Number 4, Page 1785
[2]
Murray Bremner and Sara Madariaga
Journal of Algebra, 2014, Volume 412, Page 51
[3]
Murray R. Bremner
Linear Algebra and its Applications, 2014, Volume 449, Page 381
[4]
Gábor Ivanyos, Lajos Rónyai, and Josef Schicho
Journal of Algebra, 2012, Volume 354, Number 1, Page 211

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