On the codimension growth of finite-dimensional Lie algebras
type: Item
instance of: scholarly article
title: On the codimension growth of finite-dimensional Lie algebras
zbMATH Open document ID: 0938.17004
author: Antonio Giambruno
author: Amitai Regev
author: Mikhail Zaicev
published in: Journal of Algebra
publication date: 2000-05-02T00:00:00Z
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 \(c_n(V)=\roman{dim}P_n(V)\), often these numbers are referred to as the (codimension) growth sequence. For varieties of associative algebras the codimension growth is at most exponential (A. Regev). Moreover, by recent result of M. Zaitsev and A. Giambruno, over fields of zero characteristic there exists a limit \(\lim_{n\to\infty}\root n\of{c_n(V)}=k\), where \(k\) is an integer. For Lie algebras the situation is more complicated. It is known that the codimension growth might be overexponential. If \(L\) is an algebra, then \(c_n(V)\) is the growth of the variety generated by \(L\). As a main result the authors prove: Theorem: Let \(L\) be a finite-dimensional Lie algebra over a field of characteristic zero, whose solvable radical is nilpotent. Then there exists the limit \(\lim_{n\to\infty}\root n\of{c_n(L)}=k\), where \(k\) is an integer.
Mathematics Subject Classification ID: 17B01
zbMATH DE Number: 1384469
zbMATH Keywords: varieties of Lie algebras
zbMATH Keywords: codimension growth
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: jabr.1999.7933
OpenAlex ID: W2065724862
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Wikidata QID: Q115395803
DOI: 10.1006/JABR.1999.7933
description: scientific article; zbMATH DE number 1384469