A standard identity in special varieties of Lie algebras (Q1345679)

A variety of Lie algebras over a field of characteristic 0 is of finite basis rank if it can be generated by a finitely generated algebra. The author studies the relation between the Lie standard identity and the finite basis rank. His main result is: If the variety \(V\) of solvable Lie algebras is special (i.e. generated by a Lie algebra \(G\) embedded into an associative PI-algebra) then the following four conditions are equivalent: (1) \(G\) satisfies all the polynomial identities of a Lie algebra of upper triangular matrices; (2) \(G\) satisfies a Lie standard identity; (3) \(G\) satisfies the Capelli identities with fixed number of skew-symmetric variables; (4) The basis rank of \(V\) is finite. On the other hand the author gives an example of a non-solvable special variety of Lie algebras which satisfies the standard identity and has no finite basis rank. Finally he establishes an analogue for finite-dimensional Lie superalgebras \(L= L_ 0\oplus L_ 1\) of the classical result for the nilpotency of the commutator ideal of a finite-dimensional Lie algebra. It turns out that \(L^ 2\) is nilpotent if and only if the ideal of \(L\) generated by \(L_ 1\) is nilpotent or, equivalently, the elements of \(L_ 1\) satisfy the identity \((\text{ad}\, x)^ n =0\).