Lie algebras with anisotropic Engel subalgebras
type: Item
instance of: scholarly article
title: Lie algebras with anisotropic Engel subalgebras
zbMATH Open document ID: 0666.17003
DOI: 10.1016/0021-8693(89)90086-0
author: Vicente R. Varea
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let L be a finite dimensional Lie algebra over an arbitrary field. There have been a number of investigations concerning the structure of L when conditions are placed on ad x for all \(x\in L\), the Engel Theorem being a prime example. There are the c.n. algebras of Benkart and Osborn, the E.t. algebras of Winter, the ad-semisimple algebras of Farnsteiner and the P.E 1-algebras of the author. The present paper considers a similar condition, when each proper Engel subalgebra contains no non-zero ad-nilpotent elements. The author calls these E.a. Lie algebras. He considers how the class compares to the classes just mentioned. We are led to another class, called X-algebras, in which ad x is either nilpotent or semisimple for each \(x\in L\). He finds the real simple X-algebras and finds that over algebraically closed fields of prime characteristic, every X-algebra is c.n. Further considerations over the latter fields show that \({\mathfrak sl}(2,F)\), \(W_ p(F)\) and \({\mathfrak sl}(3,F)/F\cdot 1\) are the only simple Lie algebras whose inner derivations are either nilpotent or semisimple and these are the only simple E.a. algebras which are either restricted or of a total rank one.
reviewed by: Ernest L. Stitzinger
Mathematics Subject Classification ID: 17B05
zbMATH DE Number: 4089759
zbMATH Keywords: proper Engel subalgebra
zbMATH Keywords: ad-nilpotent elements
zbMATH Keywords: inner derivations
Wikidata QID: Q115366211
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90086-0
OpenAlex ID: W2064837639
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description: scientific article; zbMATH DE number 4089759