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On the numerical characteristics of subvarieties for three varieties of Lie algebras. (English. Russian original) Zbl 0955.17003
Sb. Math. 190, No. 6, 887-902 (1999); translation from Mat. Sb. 190, No. 6, 111-126 (1999).
Let $$\mathbf V$$ be a variety of Lie algebras over an arbitrary field, $$F({\mathbf V})$$ a $$\mathbf V$$-free algebra over a countable set $$\{x_1,\dots,x_n,\dots\}$$ and $$P_n({\mathbf V})$$ the subspace of all multilinear elements in $$\{x_1,\dots,x_n,\dots\}$$. Let us denote by $$c_n({\mathbf V})$$ the dimension of this subspace. The sequence $$c_n({\mathbf V})$$, $$n=1,2,\dots$$ is called codimension sequence. Put $$\underline\exp({\mathbf V})=\underline\lim_{n\to\infty}\root n\of{c_n({\mathbf V})}$$ and $$\overline\exp({\mathbf V})=\overline\lim_{n\to\infty}\root n\of{c_n({\mathbf V})}$$. If $$\underline\exp({\mathbf V})=\overline\exp({\mathbf V})$$ then this number is called exponent $$\exp({\mathbf V})$$.
The authors study codimension sequences for subvarieties of certain varieties of Lie algebras. The main results of the article are the following three theorems.
Theorem 2.1. Let $$\mathbf V$$ be a subvariety of $${\mathbf N}_s{\mathbf A}$$. Then $$\underline\exp({\mathbf V})=\overline\exp({\mathbf V})$$, the exponent of $$\mathbf V$$ is an integer, $$\exp({\mathbf V})\in\{1,\dots,n\}$$ and if $$\exp({\mathbf V})=1$$ then $$c_n({\mathbf V})$$ is bounded by a polynomial. This fact was proved by the author and S. P. Mishchenko [Commun. Algebra 27, 2223–2230 (1999; Zbl 0955.17002)] under the additional assumption that the ground field is of characteristic 0.
Theorem 4.1. Let $$\mathbf V$$ be a subvariety of $$\mathbf{AN}_2$$ and suppose that for some $$\varepsilon>0$$ there is a subsequence $$c_{n_i}({\mathbf V})$$ of the codimension sequence $$c_n({\mathbf V})$$ such that $$c_{n_i}({\mathbf V})\leq (n_i!)^{1/2-\varepsilon}$$. Then the sequence $$c_n({\mathbf V})$$ is bounded by an exponential function.
Theorem 5.1. Let $$\mathbf V$$ be a subvariety of $${\mathbf A}^3$$ and suppose that for some $$\varepsilon>0$$ there is a subsequence $$c_{n_i}({\mathbf V})$$ of the codimension sequence $$c_n({\mathbf V})$$ such that $$c_{n_i}({\mathbf V})\leq[n_i!\cdot(1-\varepsilon)^{n_i}]/(\ln n_i)^{n_i}$$. Then $$c_n({\mathbf V})\leq(n!)^{1-1/q}$$ for some $$q$$.

MSC:
 17B01 Identities, free Lie (super)algebras 17B30 Solvable, nilpotent (super)algebras
Zbl 0955.17002
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