On the structure of the graded Lie algebra associated with a noncontractible filtration
type: Item
instance of: scholarly article
title: On the structure of the graded Lie algebra associated with a noncontractible filtration
zbMATH Open document ID: 0902.17009
author: Serge Skryabin
published in: Journal of Algebra
publication date: 1998-12-07T00:00:00Z
review text: The theory of graded Lie algebras is the main tool of investigating nonclassical simple Lie algebras over an algebraically closed field of characteristic \(p>0\). This is the reason why it is of particular interest to study graded Lie algebras \(G=\oplus_{i\geq -q}G_i\) not having nontrivial ideals in \(G_0+G_+\) and \(G_-\), and in which \(G_{-1}\) is an irreducible \(G_0\)-module. Let \(\widehat G\) be the minimal ideal of \(G\). In contrast to the case of characteristic zero, the subalgebra \(G_0\) of \(G\) may have a noncentral radical. The main result of this paper is the description of the solvable radical \(\text{solv } G_0\) of \(G_0\). It is proved that when \(p>3\) either \(\widehat {G}_0\) or \(\text{solv}G_0\) is abelian. Using this result the author proves that for \(p>3\) a simple Lie algebra with solvable maximal subalgebra is isomorphic either to \(sl(2)\) or to the Zassenhaus algebra \(W(1:m)\). Many speculations are true for infinite-dimensional Lie algebras. In particular, in the paper the structure of simple linear compact Lie algebras with solvable maximal subalgebra of finite codimension is obtained. It should be pointed out that many technical results on graded Lie algebras are of independent interest.
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 1097705
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: noncontractible filtration
zbMATH Keywords: simple modular Lie algebra
Wikidata QID: Q115395945
reviewed by: Michael Kuznetsov
MaRDI profile type: Publication
full work available at URL: jabr.1997.7124
OpenAlex ID: W2052301072
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DOI: 10.1006/JABR.1997.7124
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description: scientific article; zbMATH DE number 1097705