On Lie algebras with modular lattices of subalgebras
type: Item
instance of: scholarly article
title: On Lie algebras with modular lattices of subalgebras
zbMATH Open document ID: 0589.17009
DOI: 10.1016/0021-8693(86)90055-4
author: Alexander Lashkhi
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: An attempt is made to describe Lie algebras over an arbitrary field with a modular lattice of subalgebras. It is clear that an algebra all of whose proper subalgebras are one-dimensional has a modular lattice of subalgebras. A full classification of the simple algebras having this property does not exist. Thus all results were made in supposition that a finite-dimensional simple algebra all of whose proper subalgebras are one-dimensional is three-dimensional (this assertion has been proved by \textit{A. Premet} for algebras over perfect fields of characteristic \(>5\) [Mat. Sb., Nov. Ser. 129, No. 1, 140--153 (1986; Zbl 0595.17007)]). The main result is: Theorem. A locally finite-dimensional Lie algebra has a modular subalgebra lattice if and only if it is abelian, almost abelian or a three-dimensional simple nonsplit algebra. Remark. It is said in the paper that Proposition 4 and the above- mentioned Theorem can be proved in the same way for algebras with an upper semi-modular lattice of subalgebras. There are examples showing that this is not true.
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 3946408
zbMATH Keywords: modular lattice of subalgebras
zbMATH Keywords: locally finite-dimensional Lie algebra
zbMATH Keywords: abelian
zbMATH Keywords: almost abelian
zbMATH Keywords: three-dimensional simple nonsplit algebra
Wikidata QID: Q115366849
reviewed by: A. G. Gein
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90055-4
OpenAlex ID: W2154493166
cites work: Q5510907
cites work: Q5550461
cites work: Lattices of subalgebras of real Lie algebras
cites work: Lattic isomorphisms of lie algebras
cites work: On Lie algebras in which modular pairs of subalgebras are permutable
cites work: Quasi-Ideals of Lie Algebras I
cites work: Modularity in Lie algebras
cites work: Semimodular Lie algebras
cites work: Q4102015
cites work: On Complemented Lie Algebras
cites work: Quasi-Ideals of Lie Algebras II
description: scientific article; zbMATH DE number 3946408