On special varieties of Lie algebras
type: Item
instance of: scholarly article
title: On special varieties of Lie algebras
zbMATH Open document ID: 0688.17009
DOI: 10.1007/BF01980605
author: A. Ya. Vajs
published in: Algebra and Logic
publication date: 1989-01-01T00:00:00Z
review text: A variety \({\mathfrak M}\) of Lie algebras over a field F of characteristic 0 is called special if it is generated by an algebra R such that the associative subalgebra \(<ad r |\) \(r\in R>\) of \(End_ FR\) satisfies a polynomial identity. The main result of the paper under review is the following theorem: For every special variety \({\mathfrak M}\) of Lie algebras, there exists a finitely generated F-superalgebra \(A=A_ 0\oplus A_ 1\) such that \({\mathfrak M}=var G(A)\). Here \(G=G_ 0\oplus G_ 1\) is the canonical \({\mathbb{Z}}_ 2\)-grading of the Grassmann (or exterior) algebra and \(G(A)=G_ 0\otimes A_ 0\oplus G_ 1\otimes A_ 1\) is the Grassmann enveloping algebra of A. This result is an analogue of a well known result for varieties of associative algebras [\textit{A. R. Kemer}, Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.5, 1042-1059 (1984; Zbl 0586.16010)]. As a consequence the author establishes the solvability of a special variety of Lie algebras which does not contain the three- dimensional simple algebra \({\mathfrak sl}(2,F)\). As another application, some upper bounds for the class of nilpotency of Engel Lie algebras are obtained.
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 4127467
zbMATH Keywords: Grassmann algebra
zbMATH Keywords: superalgebras
zbMATH Keywords: polynomial identity
zbMATH Keywords: solvability
zbMATH Keywords: special variety of Lie algebras
zbMATH Keywords: class of nilpotency
zbMATH Keywords: Engel Lie algebras
Wikidata QID: Q115393017
MaRDI profile type: Publication
cites work: A 3-local characterization of Held's group
cites work: Q3337615
cites work: Q3996716
cites work: Q4435456
cites work: Existence of polynomial identities in $A \otimes \,_F B$
cites work: Lie superalgebras
cites work: Q3950687
cites work: Q3243334
cites work: Q3659932
cites work: Q5668945
description: scientific article; zbMATH DE number 4127467