Representations of finitary Lie algebras
type: Item
instance of: scholarly article
title: Representations of finitary Lie algebras
zbMATH Open document ID: 1010.17002
DOI: 10.1016/S0021-8693(02)00129-1
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: For \(F\) an algebraically closed field of characteristic \(\not= 2, 3\) and \(W\) an infinite-dimensional \(F\)-vector space a Lie subalgebra \(L\) of \(\mathfrak{gl}(W)\) all of whose elements have finite rank is said to be finnitary. Finitary Lie algebras \(L\) with \(\text{nil}_W L = (0)\), where \(\text{nil}_W L\) is the maximal ideal of \(L\) consisting of nilpotent transformations, are considered. In such an algebra every ideal contains a minimal ideal \(S\) of \(L\). It is shown that if \(S\) is infinite-dimensional and not nil, then \(SW\) is the finite direct sum of \(L\)-modules, each of which is a nontrivial irreducible \(S\)-module. A classification theorem is also obtained for irreducible Lie subalgebras of \(\mathfrak{gl}(W)\) which contain a nonzero transformation of finite rank. The existence of a local system of finite-dimensional subalgebras of irreducible finitary Lie algebras with certain restrictive properties is a basic tool in this work.
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1868052
zbMATH Keywords: minimal ideal
zbMATH Keywords: irreducible Lie subalgebras
zbMATH Keywords: finitary Lie algebras
Wikidata QID: Q115340534
author: Helmut Strade
MaRDI profile type: Publication
cites work: Some examples of locally finite simple Lie algebras
cites work: Finitary simple Lie algebras
cites work: Infinite dimensional irreducible Lie algebras containing transformations of finite rank
cites work: Finitary Lie algebras.
cites work: Differentiably simple algebras
cites work: Local systems in simple finitary lie algebras
cites work: Irreducible finitary Lie algebras over fields of characteristic zero
cites work: Q3416332
cites work: Q5583129
cites work: Modular lie algebras of cartan type over algebraically non-closed fields.1
cites work: The classification of simple Lie algebras over fields of positive characteristic: methods and results
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description: scientific article; zbMATH DE number 1868052