Semi-modular subalgebras of a Lie algebra
type: Item
instance of: scholarly article
title: Semi-modular subalgebras of a Lie algebra
zbMATH Open document ID: 0594.17004
DOI: 10.1016/0021-8693(86)90178-X
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(L\) be a Lie algebra, the subalgebra \(U\) of \(L\) is called upper semimodular if, whenever \(B\) is a subalgebra of \(L\) such that \(U\cap B\) is a maximal subalgebra of \(B\), then \(U\) is a maximal subalgebra of \(<U,B>\) (the subalgebra generated by \(U\) and \(B\)). The subalgebra \(U\) of \(L\) is called lower semimodular if, whenever \(B\) is a subalgebra of \(L\) such that \(U\) is maximal in \(<U,B>\), then \(U\cap B\) is maximal in \(B\). In spite of its appearance these are not dual concepts in lattice theory. The subalgebra \(U\) of \(L\) is called semimodular if it is both upper and lower semimodular. Ideals are obvious examples of these three kinds of subalgebras. The first two conditions are shown to be genuinely weaker than modularity, studied by \textit{R. K. Amayo} and \textit{J. Schwarz} [Hiroshima Math. J. 10, 311--322 (1980; Zbl 0441.17002)], and for the third it is proved that, for a finite dimensional Lie algebra over a field of characteristic zero, a subalgebra is semimodular if and only if it is modular.
reviewed by: Alberto Elduque
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 3955160
zbMATH Keywords: Lie algebra
zbMATH Keywords: upper semimodular
zbMATH Keywords: maximal subalgebra
zbMATH Keywords: lower semimodular
zbMATH Keywords: semimodular
Wikidata QID: Q115366818
author: David A. Towers
MaRDI profile type: Publication
cites work: Quasi-Ideals of Lie Algebras I
cites work: Modularity in Lie algebras
cites work: Semimodular Lie algebras
cites work: On Complemented Lie Algebras
cites work: Lattice isomorphisms of Lie algebras
cites work: Lie algebras whose maximal subalgebras are modular
description: scientific article; zbMATH DE number 3955160