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
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)].
weak polynomial identities characteristic zero
representation theory of symmetry groups
scientific article; zbMATH DE number 3918607