The lattice of ideals of a Lie algebra
type: Item
instance of: scholarly article
title: The lattice of ideals of a Lie algebra
zbMATH Open document ID: 0816.17003
author: M. Pilar Benito Clavijo
published in: Journal of Algebra
publication date: 1995-07-23T00:00:00Z
review text: Let \(L\) be a finite dimensional Lie algebra over a field \(F\) and \({\mathfrak L} (L)\) be the lattice of ideals of \(L\). The author classifies the Lie algebras in which the lattice \({\mathfrak L} (L)\) is complemented, complemented and distributive, or abelian and show that in case \(\dim L/L^ 2 >1\), the derived algebra and the solvable radical of \(L\) (for fields of characteristic 0) can be characterized in terms of the ideal lattice structure of \(L\). If the intersection of all maximal subalgebras of \(L\) is trivial, that is, if the Frattini subalgebra of \(L\) is zero, let us say that \(L\) has property \({\mathfrak P}\). Let \(F\) have characteristic zero, \(M\) be another Lie algebra over \(F\) such that \({\mathfrak L} (L)\) and \({\mathfrak L} (M)\) are isomorphic and suppose that \({\mathfrak P}\) holds in \(L\). Then the author shows that either \({\mathfrak P}\) holds in \(M\) or \(F\) is not algebraically closed and \(L\) and \(M\) both have a severely constrained structure.
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 735304
zbMATH Keywords: lattice of ideals
zbMATH Keywords: lattice isomorphism
zbMATH Keywords: complemented lattice
reviewed by: Melih Boral
MaRDI profile type: Publication
Wikidata QID: Q115396218
full work available at URL: jabr.1995.1015
OpenAlex ID: W2099610363
DOI: 10.1006/JABR.1995.1015
description: scientific article; zbMATH DE number 735304