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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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