Simple Lie algebras of small characteristic. II: Exceptional roots
type: Item
instance of: scholarly article
title: Simple Lie algebras of small characteristic. II: Exceptional roots
zbMATH Open document ID: 0939.17016
author: Alexander Premet
author: Helmut Strade
published in: Journal of Algebra
publication date: 2000-03-08T00:00:00Z
review text: Part I, cf. J. Algebra 189, 419-480 (1997; Zbl 0878.17019). This article represents another step in extending to characteristic \(p > 3\) the classification of finite-dimensional simple Lie algebras, which has already been accomplished when the base field \(F\) is algebraically closed of characteristic \(p > 7\). The authors investigate finite-dimensional simple Lie algebras of absolute toral rank 2 where \(F\) has characteristic \(p > 3\) and show that, for such an algebra \(L\) with \(T\) a 2-dimensional torus in the semisimple \(p\)-envelope of \(L\), all roots of \(\Gamma (L,T)\) are nonexceptional unless \(L\) is isomorphic to a Melikyan algebra. It appears that this generalization of a step in the classification by \textit{R. E. Block} and \textit{R. L. Wilson} of the restricted simple Lie algebras for \(p > 7\) [J. Algebra 114, 115-259 (1988; Zbl 0644.17008)] overcomes a major obstacle in achieving the classification in characteristics 5 and 7. To do this the authors prove that for every root \(\alpha \in \Gamma (L,T)\) the subalgebra \(K' (\alpha)\) generated by \(\sum_{i\in {\mathbb F}_p^\ast} K_{i\alpha}\) (where \(K_{i\alpha} = \{ x \in L_{i\alpha} \mid \alpha ([x,L_{-\alpha L} ]) = 0\}\)) acts triangulably on \(L\). Some additional results which may be used in other steps of the classification proof are also obtained.
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 1309257
zbMATH Keywords: prime characteristic
zbMATH Keywords: simple Lie algebras of absolute toral rank 2
zbMATH Keywords: semisimple \(p\)-envelope
zbMATH Keywords: Melikyan algebra
MaRDI profile type: Publication
full work available at URL: jabr.1998.7746
OpenAlex ID: W2013274670
Wikidata QID: Q115395834
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DOI: 10.1006/JABR.1998.7746
description: scientific article; zbMATH DE number 1309257