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Varieties of Lie algebras with two-step nilpotent commutant. (Russian. English summary) Zbl 0646.17013
The variety $${\mathfrak N}_ 2{\mathfrak A}$$ of Lie algebras over a field $$K$$ of characteristic zero defined by the identity $$(x_ 1x_ 2)(x_ 3x_ 4)(x_ 5x_ 6)=0$$ is considered in the paper. The multilinear polynomials $$P_ n({\mathfrak N}_ 2{\mathfrak A})$$ of degree $$n$$ in the relatively free algebra of rank $$n$$ in the variety $${\mathfrak N}_ 2{\mathfrak A}$$ form a module over the symmetric group $$S_ n$$. A complete description of the module structure of this $$S_ n$$-module is given in Theorem 1 using the language of partitions and Young diagrams. It is a well-known fact that the variety $${\mathfrak N}_ 2{\mathfrak A}$$ has an exponential growth of the sequence of codimensions. Theorem 2 states that the variety $${\mathfrak N}_ 2{\mathfrak A}$$ has an extremal property with respect to the codimensions namely, every subvariety of this variety has polynomial growth of the sequence of codimensions. The proof of this theorem is sketched. A necessary and sufficient condition for the distributivity of the lattices of the subvarieties of $${\mathfrak N}_ 2{\mathfrak A}$$ is given in Theorem 3 (the proof is omitted).

##### MSC:
 17B01 Identities, free Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras