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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
Citations:
Zbl 0955.17002
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