On a problem of V. G. Kac: the classification of certain simple graded Lie algebras
type: Item
instance of: scholarly article
title: On a problem of V. G. Kac: the classification of certain simple graded Lie algebras
zbMATH Open document ID: 0596.17012
DOI: 10.1016/0021-8693(86)90120-1
author: Olivier Mathieu
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: The author proves the following theorem: Let \(K\) be a field of characteristic zero. Let \(L=\oplus L_n\) be a Lie algebra over \(K\), graded over \(\mathbb{Z}\) such that (1) \(L\) is simple graded and (2) \(\dim L_n\leq 1\) \(\forall n\in \mathbb{Z}\). Then \(L\) is isomorphic to one of the following Lie algebras suitably graded: \[ \mathfrak{sl}(2), \quad A_1^{(1)}, \quad A_2^{(2)}, \quad W_1, \quad W. \] Here \(W\) is the Witt algebra, i.e. it has a basis \(\{u_i\}_{i\in\mathbb{Z}}\) such that \([u_i,u_j] = (j -i)u_{i+j}\) and \(W_1\) is the subalgebra of \(W\) with basis \(\{u_i\}_{i\ge -1}\). The case when \(\dim L_n=1\) \(\forall n\) had been conjectured by V. G. Kac. The following lemma was very useful in the demonstration: Let \(L\) be a \(\mathbb{Z}\)-graded Lie algebra and let \(A\) and \(B\) be two graded vector subspaces such that \(L=A+B\) and \([A,B]\subset B\). Then \(B+[B,B]\) is a graded ideal. [This article is identical to the author's thesis, Univ. Paris VII, 57 p. (1984).]
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 3959724
zbMATH Keywords: simple Lie algebras
zbMATH Keywords: graded Lie algebras
zbMATH Keywords: affine Kac-Moody
zbMATH Keywords: algebras
zbMATH Keywords: Witt algebra
reviewed by: Fernando Levstein
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90120-1
OpenAlex ID: W1990686282
Wikidata QID: Q106364568
cites work: Q4065692
cites work: On defining relations of certain infinite-dimensional Lie algebras
cites work: Introduction to Lie Algebras and Representation Theory
cites work: On simplicity of certain infinite dimensional Lie algebras
cites work: SIMPLE IRREDUCIBLE GRADED LIE ALGEBRAS OF FINITE GROWTH
cites work: Q3693664
cites work: The Virasoro algebra
cites work: Q4200346
Recommended article: Q5457595
Recommended article: On the fine gradings of simple classical Lie algebras
Recommended article: Q4788503
Recommended article: Q5465223
Recommended article: Q4011185
Recommended article: Classification of simple graded Lie algebras of finite growth
Recommended article: Fine gradings on simple classical Lie algebras
Recommended article: Q3824547
Recommended article: CLASSIFICATION OF SIMPLE GRADED LIE ALGEBRAS WITH NONSEMISIMPLE COMPONENT $ L_0$
Recommended article: The classification of the simple modular Lie algebras. IV: Determining the associated graded algebra
description: scientific article; zbMATH DE number 3959724