Unitary units and skew elements in group algebras.
type: Item
instance of: scholarly article
title: Unitary units and skew elements in group algebras.
zbMATH Open document ID: 1049.16020
published in: Manuscripta Mathematica
publication date: 2003-08-17T00:00:00Z
review text: Let \(FG\) be the group algebra of a group \(G\) over a field \(F\) and let * denote the canonical involution of \(FG\) induced by the map \(g\mapsto g^{-1}\), \(g\in G\). The unitary units in \(FG\), i.e., the elements \(u\in FG\) for which \(uu^*=1\), form a subgroup \(\text{Un}(FG)\) of the group of units of \(FG\). This paper studies the property of \(\text{Un}(FG)\) being such that it satisfies a group identity. A complete classification is given for the case when \(\text{char\,}F=0\), \(G\) is a torsion group and the group identity is such that it does not become trivial when evaluated at elements of order \(2\). Let \(FG^-\) be the Lie algebra consisting of the skew elements, i.e., the elements \(x\in FG\) for which \(x^*=-x\). The main result on skew elements is a characterization for \(FG^-\) to be Lie nilpotent, provided that \(\text{char\,}F\neq 2\) and \(G\) is such that \(FG\) is semiprime.
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16R50
zbMATH DE Number: 1965483
zbMATH Keywords: group algebras
zbMATH Keywords: unitary units
zbMATH Keywords: skew elements
zbMATH Keywords: Lie nilpotence
zbMATH Keywords: group identities
zbMATH Keywords: groups of units
author: Antonio Giambruno
author: C. Polcino Milies
reviewed by: Inder Bir S. Passi
MaRDI profile type: Publication
full work available at URL: s00229-003-0365-5
OpenAlex ID: W1968312175
DOI: 10.1007/S00229-003-0365-5
description: scientific article; zbMATH DE number 1965483