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A criterion for polynomial growth of varieties of Lie superalgebras. (English. Russian original) Zbl 0915.17004
Izv. Math. 62, No. 5, 953-967 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 5, 103-116 (1998).
Suppose that $$V$$ is a variety of Lie superalgebras, i.e. a class of Lie superalgebras that satisfy some set of graded identical relations. Suppose that $$x_1,\dots,x_n$$ are arbitrary (nonhomogeneous) elements in an algebra from $$V$$, one considers the dimension of the spaces of multilinear polynomials in these variables. The supremum of these dimensions is called the codimension growth sequence $$c_n(V)$$. The authors find a criterion for a variety $$V$$ of Lie superalgebras over a field of characteristic zero to have polynomial codimension growth. Namely, $$V$$ has polynomial growth iff the following three conditions hold: 1) $$V$$ has a nilpotent commutator subalgebra, 2) each multilinear polynomial containing at least $$k$$ even and $$k$$ odd variables is an identity for $$V$$, 3) $$V$$ satisfies some additional specific identities.

##### MSC:
 17B01 Identities, free Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras 17B65 Infinite-dimensional Lie (super)algebras 16R10 $$T$$-ideals, identities, varieties of associative rings and algebras
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