Varieties of solvable Lie algebras with a distributive lattice of subvarieties (Q1206262)

A very difficult and still open problem about varieties of Lie algebras over a field of characteristic zero is to determine the varieties with distributive lattice of subvarieties. A remarkable achievement of the present author is the proof, given in this paper, of the following result. Any solvable variety over a field of characteristic zero with distributive lattice of subvarieties is either a variety with polynomial growth, or is a so called Volichenko variety, i.e. the least variety with no standard identity satisfied, or an explicitly defined super-variety of the preceding variety. A corollary of interest is that any solvable variety of Lie algebras over a field of characteristic zero has finite basis for its laws provided its lattice of subvarieties is distributive.