Polynomial identities for graded tensor products of algebras.
type: Item
instance of: scholarly article
title: Polynomial identities for graded tensor products of algebras.
zbMATH Open document ID: 1172.16013
author: José António Freitas
author: Plamen Koshlukov
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: Due to the pioneer work of \textit{A. R. Kemer} [see Ideals of identities of associative algebras. Transl. Math. Monogr. 87. Providence: AMS (1991; Zbl 0732.16001), for the account], nowadays the study of \(\mathbb{Z}_2\)-graded polynomial identities is an important part of the theory of PI-algebras. The multiplication in the \(\mathbb{Z}_2\)-graded analogue of the tensor product \(A\overline\otimes B\) of two \(\mathbb{Z}_2\)-graded algebras is defined by \((a_1\overline\otimes b_1)(a_2\overline\otimes b_2)=(-1)^{|b_1||a_2|}a_1a_2\overline\otimes b_1b_2\), where \(a_i\in A\) and \(b_i\in B\) are homogeneous elements of degree \(|a_i|\) and \(|b_i|\), respectively. \textit{A. Regev} and \textit{T. Seeman} [J. Algebra 291, No. 1, 274-296 (2005; Zbl 1083.16017)] established an analogue of the \(A\otimes B\)-theorem of Regev for the \(\mathbb{Z}_2\)-graded tensor product of two PI-algebras. They also conjectured that the graded tensor product of two T-prime PI-algebras in characteristic 0 is PI-equivalent to a T-prime algebra, as in the ordinary case. Regev and Seeman confirmed their conjecture in several cases. (The T-prime algebras are the matrix algebras \(M_n(K)\) and \(M_n(E)\) over the base field \(K\) and over the Grassmann algebra \(E\) and the subalgebras \(M_{k,l}\) of \(M_{k+l}(E)\).) The first main result of the paper under review is to prove completely the conjecture. In the remaining open cases, the authors show the PI-equivalences \(M_{k,l}\overline\otimes E\sim M_{k+l}(E)\) and \(M_{p,q}\overline\otimes M_{r,s}\sim M_{pr+qs,ps+qr}\), as in the ordinary case. (As the authors have added in proof, similar results have been independently obtained by \textit{O. M. Di Vincenzo} and \textit{V. Nardozza} [accepted for publication in Isr. J. Math.].) On the other hand, the paper shows that over an infinite field of characteristic \(p>2\) the conjecture of Regev and Seeman does not hold: the equivalence \(M_{1,1}\overline\otimes E\sim M_2(E)\) in characteristic 0…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 5545853
zbMATH Keywords: algebras with polynomial identity
zbMATH Keywords: graded tensor products
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: T-prime algebras
zbMATH Keywords: graded T-ideals
zbMATH Keywords: commutation factors
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.09.031
OpenAlex ID: W1992466707
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DOI: 10.1016/J.JALGEBRA.2008.09.031
description: scientific article; zbMATH DE number 5545853