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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
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