On the weight spaces of Lie algebra modules and their Jordan kernel
type: Item
instance of: scholarly article
title: On the weight spaces of Lie algebra modules and their Jordan kernel
zbMATH Open document ID: 0625.17005
DOI: 10.1016/0021-8693(87)90070-6
author: Alfons I. Ooms
author: Theo Moons
author: Erna Nauwelaerts
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let L be a finite-dimensional Lie algebra over a field k of characteristic zero and let V be an L module. Put \(V_{\lambda}=\{v\in V|\forall x\in L\), \(xv=\lambda (x)v\}\) and \(V^{\lambda}=\{v\in V|\forall x\in L\), \(\exists n\in {\mathbb{N}}\), \((x-\lambda (x))^ nv=0\}\) where \(\lambda\) is a function from L to k. In the case V is finite-dimensional, \textit{M. K. Smith} [Proc. Am. Math. Soc. 95, 524-526 (1985; Zbl 0603.17004)] has proved that \(V_{\lambda}\) must be nonzero if \(V^{\lambda}\) is nonzero and \(V^{\lambda}\) is a submodule of V. In this paper, these results have been generalized to arbitrary L-modules V and a new characterization of these weight spaces \(V^{\lambda}\) which simplifies their actual computation has been given. Particularly, the details for the case V is either the universal enveloping algebra U(L) of L or its division ring of quotients D(L) have been discussed.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 4014951
zbMATH Keywords: Lie algebra modules
zbMATH Keywords: Jordan kernel
zbMATH Keywords: weight spaces
zbMATH Keywords: universal enveloping algebra
reviewed by: Zhonghu Chen
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90070-6
OpenAlex ID: W1980855803
Wikidata QID: Q115366707
cites work: Q3941610
cites work: On the semi-center of a universal enveloping algebra
cites work: Q4065692
cites work: Q3939967
cites work: Idéaux bilatères des algèbres enveloppantes
cites work: Automorhisms of enveloping algebras
cites work: Weight Spaces of Lie Algebra Modules
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description: scientific article; zbMATH DE number 4014951