Conjugation in semigroups and finite dimensional algebras
type: Item
instance of: scholarly article
title: Conjugation in semigroups and finite dimensional algebras
zbMATH Open document ID: 0541.20040
author: Hubert Grassmann
published in: Beiträge zur Algebra und Geometrie
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138286
review text: Let \(S\) be a finite semigroup. Then, for \(s\in S\), the idempotent power of \(s\) is denoted by \(e_ s\) and \(e_ ss\) is denoted by \(s^*\). Two elements \(s\) and \(t\) in \(S\) are said to be conjugate if there exist \(p,q\in S\) such that \(s^*=pq\), \(t^*=qp\). It is shown that \(s\) and \(t\) are conjugate if and only if there are mutually inverse elements \(x,y\in S\) such that \(s^*=xt^*y\), \(t^*=ys^*x\) and that the number of conjugacy classes of \(S\) is the same as the number of irreducible representations of \(S\) over the complex numbers, and that \(s\) and \(t\) are conjugate if and only if \(h(s)=h(t)\) for all irreducible characters \(h\) of \(S\). These ideas are generalized to finite dimensional complex algebras \(A\). Let \(T(A)\) be the subspace generated by all commutators \(ab-ba\), \(a,b\in A,\) and let \(J(A)\) denote the Jacobson radical of \(A\). Then it is shown that \(h(a)=h(b)\) for all irreducible characters of \(A\) if and only if \(a-b\in T(A)+J(A)\). Thus the number of nonisomorphic simple \(A\)-modules is \(\dim A/(T(A)+J(A))\).
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 3859411
zbMATH Keywords: finite semigroup
zbMATH Keywords: number of conjugacy classes
zbMATH Keywords: irreducible representations
zbMATH Keywords: irreducible characters
zbMATH Keywords: finite dimensional complex algebras
zbMATH Keywords: Jacobson radical
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3859411