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Varieties of polynomial growth of Lie algebras over a field of characteristic zero. (English. Russian original) Zbl 0624.17007
Math. Notes 40, 901-905 (1986); translation from Mat. Zametki 40, No. 6, 713-722 (1986).
Let $$N_ cA$$ be the variety of Lie algebras over a field of characteristic 0 whose derived algebra is nilpotent of class at most c. Let $$P_ n$$ be the space of polylinear elements of degree n of the free associative algebra. If M is a Lie algebra variety then let $$T_ n(M)\subset P_ n$$ be the subspace of identities of algebras of M. The sequence $$d_ n=d_ n(M)=\dim P_ n/T_ n(M)$$ is called the codimension sequence of M.
In his previous article [Vestn. Mosk. Univ., Ser. I 1982, No.5, 63-66 (1982; Zbl 0517.17006)] the author proved that $$M\subset N_ cA$$ for some c provided $$d_ n(M)<(\sqrt{2})^{n+2}$$ for $$n>n_ 0$$, in particular if the growth of $$d_ n(M)$$ is subexponential. On the other hand he proved (unpublished) that $$N_ 2A$$ is of exponential growth. In the article under review the author proves that M is of polynomial growth if $$M\subset N_ cA$$ and $$N_ 2A\not\subset M$$. Therefore there is no Lie algebra variety with subexponential and nonpolynomial growth of the codimension sequence.
Reviewer: A.Zalesskij

##### MSC:
 17B99 Lie algebras and Lie superalgebras 08B15 Lattices of varieties 17B30 Solvable, nilpotent (super)algebras 17B65 Infinite-dimensional Lie (super)algebras
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##### References:
 [1] I. I. Benediktovich and A. E. Zalesskii, ?T-ideals of a free Lie algebra with polynomial growth of the sequence of codimensions,? Vestsi Akad. Nauk BSSR, No. 3, 5-10 (1980). · Zbl 0434.17009 [2] S. P. Mishchenko, ?Varieties of Lie algebras with a weak growth of the sequence of codimensions,? Vestn. Mosk. Gos. Univ., Ser. 1, Mat. Mekh., No. 5, 63-66 (1982). [3] S. P. Mishchenko, ?A variety of Lie algebras whose commutator subalgebra is nilpotent of step two,? Vestn. Mosk. Gos. Univ., Ser. 1, Mat., Mekh., No. 5, 94 (1984). [4] Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985). · Zbl 0571.17001 [5] C. Curtis and I. Reiner, Theory of Representations of Finite Groups and Associative Algebras [Russian translation], Nauka, Moscow (1979). [6] I. B. Volichenko, ?On one variety of Lie algebras associated with standard identities, II,? Vestsi Akad. Nauk BSSR, No. 2, 22-29 (1980). · Zbl 0432.17005
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