Modules for certain Lie algebras of maximal class
type: Item
instance of: scholarly article
title: Modules for certain Lie algebras of maximal class
zbMATH Open document ID: 0845.17011
DOI: 10.1016/0022-4049(94)00002-Z
author: Dietrich Burde
author: Fritz Grunewald
published in: Journal of Pure and Applied Algebra
publication date: 1996-09-11T00:00:00Z
review text: Let \(L\) be a finite-dimensional Lie algebra over a field \(F\) of characteristic zero and let \(\mu (L)= \min \{\dim M\), \(M\) is a faithful \(L\)-module\}. For several classes of Lie algebras, \(\mu (L)\leq 1+\dim L\). Let \(L= a(r, s, t)\) be the algebra with generators \(e_1, e_2, \dots\) and relations \([e_1, e_i ]= e_{i+1}\), \([e_2, e_3 ]= e_5\) and \([e_2, e_5 ]= re_7+ se_8+ te_9\) where \(r, s, t\in F\) and \(i= 1, 2, \dots\). If \(r\neq 0, {9\over 10}, 1, 2, 3\), then there exists a faithful \(L\)-module \(M\) of dimension 22 and \(M\) has no faithful submodule or quotient. Also, for \(s\neq 0\) and certain restrictions on \(r\), there exists no 12-dimensional faithful \(L\)-module.
reviewed by: Ernest L. Stitzinger
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 838656
zbMATH Keywords: Lie algebras of maximal class
zbMATH Keywords: faithful module
zbMATH Keywords: generators
zbMATH Keywords: relations
Wikidata QID: Q115364407
MaRDI profile type: Publication
cites work: Simply Transitive Groups of Affine Motions
cites work: Q4032126
cites work: Q5580319
cites work: Sur les structures affines homotopes à zéro des groupes de Lie. (On nullhomotopic affine structures on Lie groups)
cites work: Left-symmetric structures on simple modular Lie algebras
cites work: Singular fourier integral operators and representations of nilpotent lie groups
cites work: Three-dimensional affine crystallographic groups
cites work: Radical d'une algèbre symétrique à gauche
cites work: Complete left-invariant affine structures on nilpotent Lie groups
cites work: On fundamental groups of complete affinely flat manifolds
cites work: Q3473840
cites work: Examples of compact locally affine spaces
cites work: The structure of complete left-symmetric algebras
cites work: DERIVATIONS AND AFFINE STRUCTURES OF SOME NILPOTENT LIE ALGEBRAS
full work available at URL: 0022-4049(94)00002-z
OpenAlex ID: W2069438714
description: scientific article; zbMATH DE number 838656