PI-algebras with slow codimension growth.
type: Item
instance of: scholarly article
title: PI-algebras with slow codimension growth.
zbMATH Open document ID: 1071.16021
author: Antonio Giambruno
author: Daniela La Mattina
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: Let \(A\) be an associative non-unitary algebra over a field of characteristic zero. The multilinear polynomial identities satisfied by \(A\) determine all the polynomial identities satisfied by \(A\). Denote by \(c_n(A)\), \(n=1,2,\dots\), the codimension of the subspace of multilinear identities of \(A\) in the \(n\)-variable multilinear component of the free associative algebra. It is known that for a PI-algebra \(A\) either the codimension sequence grows exponentially (typical case) or it is polynomially bounded. In the present paper the algebras with linearly bounded codimension sequence are classified up to PI-equivalence. The possible linearly bounded codimension sequences are determined. The authors present a list of five explicitly given finite dimensional algebras such that \(A\) has linearly bounded codimension sequence if and only if none of the given algebras belongs to the variety generated by \(A\). Linear growth of the codimensions turns out to be closely related to the property that the limit of \(l_n(A)\) exists and is bounded by \(2\), where \(l_n(A)\) counts the number of \(S_n\)-irreducibles in the \(n\)-th cocharacter of \(A\).
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 2136979
zbMATH Keywords: multilinear polynomial identities
zbMATH Keywords: T-ideals
zbMATH Keywords: linearly bounded codimension sequences
zbMATH Keywords: characters of symmetric groups
zbMATH Keywords: linear growth
zbMATH Keywords: cocharacters
reviewed by: Matyas Domokos
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.09.003
OpenAlex ID: W2057311619
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DOI: 10.1016/J.JALGEBRA.2004.09.003
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description: scientific article; zbMATH DE number 2136979