On modular\(^*\) subalgebras of a Lie algebra
type: Item
instance of: scholarly article
title: On modular\(^*\) subalgebras of a Lie algebra
zbMATH Open document ID: 0874.17003
published in: Journal of Algebra
publication date: 1997-11-11T00:00:00Z
review text: A subalgebra \(U\) of a Lie algebra \(L\) is called modular\(^*\) if \(\langle U,B \rangle \cap C= \langle B,U \cap C \rangle\) for all subalgebras \(B\subseteq C\) and \(\langle U \cap B,C \rangle= \langle B,C \rangle \cap U\) for all subalgebras \(C\subseteq U\). (This is a dualized version of the modular conditions.) This paper contains several results on the structure of Lie algebras containing a modular\(^*\) subalgebra. It is shown that all modular\(^*\) subalgebras of a nilpotent Lie algebra are inside the center. It is also shown that for Lie algebras \(L\) over a field of characteristic zero every proper modular\(^*\) subalgebra is inside the abelian socle of \(L\) unless \(L\) is three-dimensional nonsplit simple or almost abelian, and, finally, that if \(L\) is a Lie algebra over a field of characteristic different from 2, 3 and containing an almost abelian modular\(^*\) subalgebra, then \(L\) is almost abelian also.
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 1014006
zbMATH Keywords: modular\(^*\) subalgebra
zbMATH Keywords: almost abelian modular\(^*\) subalgebra
zbMATH Keywords: structure of Lie algebras
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: characteristic zero
author: David A. Towers
MaRDI profile type: Publication
full work available at URL: jabr.1996.6912
OpenAlex ID: W4210316804
Wikidata QID: Q115396030
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DOI: 10.1006/JABR.1996.6912
description: scientific article; zbMATH DE number 1014006