Lie derived length and involutions in group algebras.
type: Item
instance of: scholarly article
title: Lie derived length and involutions in group algebras.
zbMATH Open document ID: 1263.16041
author: Zsolt Balogh
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C07
zbMATH DE Number: 6094118
zbMATH Keywords: group algebras
zbMATH Keywords: Lie derived lengths
zbMATH Keywords: involutions
zbMATH Keywords: symmetric elements
zbMATH Keywords: skew-symmetric elements
zbMATH Keywords: nonabelian subgroups
reviewed by: Nako Nachev
MaRDI profile type: Publication
full work available at URL: j.jpaa.2011.12.013
OpenAlex ID: W1997777006
Wikidata QID: Q115345503
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cites work: Group algebras whose symmetric and skew elements are Lie solvable
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cites work: Lie Solvable Group Rings
cites work: The derived length of Lie soluble group rings. I
DOI: 10.1016/J.JPAA.2011.12.013
review text: The main result of this article is the following. Theorem 1. Let \(p\) be an odd prime, \(G\) a group in which the set of \(p\)-elements forms a finite nonabelian subgroup and let \(FG\) be its group algebra over a field \(F\) of characteristic \(p\). Then NEWLINE\[NEWLINE\mathrm{dl}(FG_*^+)\geq\left\lceil\log_2(p+1)\right\rceilNEWLINE\]NEWLINE for every involution \(*\) which arises from \(G\).NEWLINENEWLINE The group algebra \(FG\) may be considered as a Lie algebra with the Lie operation defined by \([x,y]=xy-yx\) for all \(x,y\in FG\). For a subset \(S\subseteq FG\) we define the Lie derived series in the following way. Let \(\delta^{[0]}(S)\) be the vector space over \(F\) spanned by \(S\) and let \(\delta^{[n+1]}(S)\) be the vector space spanned by all Lie commutators \([x,y]\) with \(x,y\in\delta^{[n]}(S)\). The subset \(S\) is said to be Lie solvable if there exists \(n\) such that \(\delta^{[n]}(S)=0\), and the smallest such \(n\) is called the Lie derived length of \(S\) which will be denoted by \(\mathrm{dl}(S)\).NEWLINENEWLINE Let \(*\) be an involution on \(FG\). Denote \(FG_*^+\) and \(FG_*^-\) the set symmetric and skew-symmetric elements in \(FG\) under the involution \(*\), respectively, that is \(FG_*^+=\{x\in FG\mid x^*=x\}\) and \(FG_*^-=\{x\in FG\mid x^*=-x\}\). Evidently, they are vector spaces over \(F\). -- If \(x\) is a real number, then \(\left\lceil x\right\rceil\) is the least integer which is greater or equal to \(x\).NEWLINENEWLINE Theorem 1 and Proposition B of \textit{I. B. S. Passi, D. S. Passman} and \textit{S. K. Sehgal} [Can. J. Math. 25, 748-757 (1973; Zbl 0266.16011)] give immediately the following corollary.NEWLINENEWLINE Corollary 2. Let \(F\) be a field of characteristic \(p>2\) and \(G\) a group in which the set of \(p\)-elements forms a finite nonabelian subgroup with central derived subgroup of order \(p\). Then \(\mathrm{dl}(FG_*^+)=\left\lceil\log_2(p+1)\right\rceil\) for every involution \(*\) which arises from \(G\).
description: scientific article; zbMATH DE number 6094118