PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic.
type: Item
instance of: scholarly article
title: PI (non)equivalence and Gelfand-Kirillov dimension in positive characteristic.
zbMATH Open document ID: 1192.16018
author: Sérgio Mota Alves
publication date: 2009-10-26T00:00:00Z
review text: The author shows that the so called Tensor Product Theorem of Kemer for verbally prime varieties over fields of zero characteristic cannot be extended to infinite fields of positive characteristic \(p>2\). The author computes the Gelfand-Kirillov dimensions of several relatively free algebras and thus obtains a new proof of the fact that the algebras \(M_{a,a}(E)\otimes E\) and \(M_{2a}(E)\) are not PI equivalent in characteristic \(p>2\). Also, it is shown that the following algebras are not PI equivalent in positive characteristic: \(M_{a,b}(E)\otimes M_{c,d}(E)\) and \(M_{ac+bd,ad+cb}(E)\); and \(M_{a,b}(E)\otimes M_{c,d}(E)\) and \(M_{e,f}(E)\otimes M_{g,h}(E)\) when \(a\geq b\), \(c\geq d\), \(e\geq f\), \(g\geq h\), \(ac+bd=eg+fh\), \(ad+bc=eh+fg\) and \(ac\neq eg\). Here \(E\) is the infinite dimensional Grassmann algebra with 1, and \(M_{a,b}(E)\) the subalgebra of \(M_{a+b}(E)\) of the block matrices with blocks \(a\times a\) and \(b\times b\) on the main diagonal with entries from \(E_0\), and off-diagonal entries from \(E_1\), where \(E=E_0\otimes E_1\) is the natural grading.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 5620618
zbMATH Keywords: graded identities
zbMATH Keywords: verbally prime algebras
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: verbally prime PI algebras
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: T-prime algebras
zbMATH Keywords: matrix algebras
zbMATH Keywords: polynomial identities
zbMATH Keywords: tensor products
zbMATH Keywords: PI equivalent algebras
zbMATH Keywords: T-ideals
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: s12215-009-0011-5
OpenAlex ID: W2050476110
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DOI: 10.1007/S12215-009-0011-5
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5620618