Varieties of Lie algebras with the identity \([[x_ 1,x_ 2,x_ 3],[x_ 4,x_ 5,x_ 6]]=0\) over a field of characteristic zero
type: Item
instance of: scholarly article
title: Varieties of Lie algebras with the identity \([[x_ 1,x_ 2,x_ 3],[x_ 4,x_ 5,x_ 6]]=0\) over a field of characteristic zero
zbMATH Open document ID: 0575.17006
DOI: 10.1007/BF00968976
author: I. B. Volichenko
published in: Siberian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The paper under review contains the complete proofs of the author's results announced earlier [Dokl. Akad. Nauk BSSR 25, 1063-1066 (1981; Zbl 0472.17008)]. Let \({\mathfrak AN}_ 2\) be the variety of Lie algebras over a field of characteristic 0, defined by the polynomial identity \([[x_ 1,x_ 2,x_ 3],[x_ 4,x_ 5,x_ 6]]=0.\) The main result of this interesting paper claims that \({\mathfrak AN}_ 2\) satisfies the Specht property, i.e. every subvariety of \({\mathfrak AN}_ 2\) is defined by a finite system of identities. The proof is based on the technique of representation theory of symmetry groups. In order to achieve the result the author proves the Specht property for the ideal of all weak identities generated by \([x_ 1,x_ 2,x_ 3]\) in the free associative algebra with 1. He establishes a series of extremal properties for \({\mathfrak AN}_ 2\) as well. Note that the Specht problem for \({\mathfrak AN}_ k\), \(k>2\), is still open and over a field of characteristic two there exists an infinitely based subvariety of \({\mathfrak AN}_ 2\) [\textit{M. R. Vaughan-Lee}, J. Lond. Math. Soc., II. Ser. 11, 263-266 (1975; Zbl 0316.17007)].
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 3918607
zbMATH Keywords: weak polynomial identities characteristic zero
zbMATH Keywords: variety of Lie algebras
zbMATH Keywords: Specht property
zbMATH Keywords: finite system of identities
zbMATH Keywords: representation theory of symmetry groups
zbMATH Keywords: free associative algebra
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3918607