The GK dimension of relatively free algebras of PI-algebras
type: Item
instance of: scholarly article
title: The GK dimension of relatively free algebras of PI-algebras
zbMATH Open document ID: 1461.16024
author: Lucio Centrone
published in: Journal of Pure and Applied Algebra
publication date: 2019-03-06T00:00:00Z
review text: Let \(\mathfrak{M}\) be a variety of \(PI\)-algebras, \(A=A_n\in\mathfrak{M}\) be an \(n\)-generated relatively free algebra in \(\mathfrak{M}\), \(V_A(k)\) is dimension of space generated by monomials of length \(n\). It is known that \(V_A(k)\sim k^{GK(A_n)}\), where \(GK(A)\) is the Gelfand-Kirillov dimension of an algebra \(A\). This dimension was calculated in the paper [\textit{A. Ya. Belov}, Sb. Math. 195, No. 12, 1703--1726 (2004; Zbl 1090.16007); translation from Mat. Sb. 195, No. 12, 3--26 (2004)]. The Hilbert series \(H_A(t)=\sum_n V_A(n)t^n\) are found to be rational in [\textit{A. Ya. Belov}, Russ. Math. Surv. 52, No. 2, 394--395 (1997; Zbl 0919.16018); translation from Usp. Mat. Nauk 52, No. 2, 153--154 (1997)]. For a graded version of this result see [\textit{E. Aljadeff} and the reviewer, Bull. Lond. Math. Soc. 44, No. 3, 520--532 (2012; Zbl 1259.16023)]. The notion of Hilbert series can be designed to the infinitely generated case for abstract varieties. Consider the relatively free algebra \(A_\infty \in\mathfrak{M}\) with countable number of generators \(x_1,\dots,x_n\dots\). Let \(c_n(\mathfrak{M})\) be the dimension of a space generated by multilinear words of length \(n\) over \(x_1,\dots,x_n\) The codimension series \(\sum c_n(\mathfrak{M}) t^n\) were introduced by A. Regev; it is well known that \(c_n(\mathfrak{M})\sim \lambda^n\), and \(\lambda=\mathrm{Exp}(\mathfrak{M})\) is called the \textit{exponent} of \(\mathfrak{M}\). There is a celebrated series of papers of Giambruno, Zaicev, Mishcenko calculating the exponent in terms of underlying algebras. The rationality result can be generalized somehow to codimension sequences [\textit{A. Berele}, J. Algebra 298, No. 1, 208--214 (2006; Zbl 1106.16024)]. The paper develops this deep relation and provides graded version of it. The author proves a strict relation between the Gelfand-Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a con…
reviewed by: A. Ya Belov
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 7032793
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: generic matrix algebras
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.10.005
OpenAlex ID: W2895159773
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DOI: 10.1016/J.JPAA.2018.10.005
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description: scientific article; zbMATH DE number 7032793