Minimal varieties of algebras of exponential growth.
type: Item
instance of: scholarly article
title: Minimal varieties of algebras of exponential growth.
zbMATH Open document ID: 1035.16013
DOI: 10.1016/S0001-8708(02)00047-6
author: Antonio Giambruno
author: Mikhail Zaicev
published in: Advances in Mathematics
publication date: 2003-05-04T00:00:00Z
review text: Recently, the authors proved the remarkable result that the exponent of a variety \(V\) of associative algebras over a field of characteristic zero is always an integer [\textit{A. Giambruno, M. Zaicev}, Adv. Math. 142, No. 2, 221-243 (1999; Zbl 0920.16013)]. Namely, let \(c_n(V)\), \(n=1,2,\dots\), be the codimension sequence of a variety \(V\). Then the limit \(\lim_{n\to\infty}\root n\of{c_n(V)}\) exists and is an integer. Now, the authors classify all minimal varieties of a given exponent and of finite basic rank. As a consequence, they describe the corresponding T-ideals of the free algebra and compute the asymptotics of the relative codimension sequences. They prove that the number of these minimal varieties is finite for any given exponent. Also, some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the respective relatively free algebras are given.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 1903519
zbMATH Keywords: varieties of associative algebras
zbMATH Keywords: codimension growth
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: T-ideals
zbMATH Keywords: exponents
zbMATH Keywords: codimension sequences
zbMATH Keywords: relatively free algebras
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1903519