On special varieties of Lie algebras (Q1263660)

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.