An example of a variety of Lie algebras with a fractional exponent
type: Item
instance of: scholarly article
title: An example of a variety of Lie algebras with a fractional exponent
zbMATH Open document ID: 0933.17004
DOI: 10.1007/BF02366352
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-16T00:00:00Z
review text: Let \(V\) be a variety of linear algebras and \(F(V)\) its countable rank relatively free algebra generated by elements \(y_1,y_2,\dots\). Let \(P_n(V)\subset F(V)\) be a subspace of multilinear elements in \(y_1,\dots,y_n\in F(V)\). Denote \(c_n(V)=\text{dim}P_n(V)\), often these numbers are referred to as the (codimension) growth sequence. One defines lower and upper exponents \[ \underline{\text{EXP}}(V)=\liminf_{n\to\infty}\root n\of{c_n(V)}, \quad \overline{\text{EXP}}(V)=\limsup_{n\to\infty}\root n\of{c_n(V)}. \] For varieties of associative algebras the codimension growth is at most exponential (A. Regev). Moreover, by a recent result of M. Zaitsev and A. Giambruno, over fields of zero characteristic these two numbers always coincide and are integers. For Lie algebras the situation is more complicated. It is known that the codimension growth might be overexponential. As a main result the authors construct a solvable variety of Lie algebras \(V\) with nonintegral exponents: \(3<\underline{\text{EXP}}(V)\leq \overline{\text{EXP}}(V)<4\).
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 1315231
zbMATH Keywords: varieties of Lie algebras
zbMATH Keywords: codimension growth
author: Mikhail Zaicev
author: Sergey Mishchenko
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02366352
OpenAlex ID: W2067811967
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description: scientific article; zbMATH DE number 1315231