Polynomial identities of algebras in positive characteristic.
type: Item
instance of: scholarly article
title: Polynomial identities of algebras in positive characteristic.
zbMATH Open document ID: 1113.16030
author: Sérgio Mota Alves
author: Plamen Koshlukov
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Let \(E\) (\(E'\)) be the Grassmann algebra with 1 (without 1) over an infinite field \(K\), \(A_{a,b}\) is the subalgebra of \(M_{a+b}(E)\) consisting of all matrices \((a_{ij})\), such that \(a_{ij}\in E\) if either \(1\leq i,j\leq a\) or \((a+1)\leq i,j\leq a+b\) and \(a_{ij}\in E'\) if either \(1\leq i\leq a\), \(a+1\leq j\leq a+b\), or \(1\leq j\leq a\), \(a+1\leq i\leq a+b\). Let also \(M_{a,b}(E)\) be 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 \(E_1\) (\(E=E_0+E_1\) is the natural grading on \(E\)). The authors compute the Gelfand-Kirillov dimensions of the free algebras in the varieties determined by \(E\otimes E\), \(M_{1,1}(E)\otimes E\), \(A_{2,1}\), \(A_{2,2}\) and prove the following main results (\(\text{char\,}K=p>2\)): (i) if \(a+b=c+d\), \(a\leq b\), \(c\leq d\), \(c>a\) then \(M_{a,b}(E)\otimes E\) is not PI equivalent to \(M_{c,d}(E)\otimes E\); (ii) if \(a+b=c+d\), \(a\geq b\), \(c\geq d\), \(a\neq c\) then \(T(A_{a,b})\neq T(A_{c,d})\); (iii) \(M_{1,1}(E)\otimes M_{1,1}(E)\) and \(M_{2,2}(E)\) are not PI equivalent.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 5078325
zbMATH Keywords: graded identities
zbMATH Keywords: verbally prime algebras
zbMATH Keywords: GK-dimension
zbMATH Keywords: polynomial identities
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: tensor products
reviewed by: Yu. N. Maltsev
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.04.009
OpenAlex ID: W2031641749
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DOI: 10.1016/J.JALGEBRA.2006.04.009
description: scientific article; zbMATH DE number 5078325