On a Regev-Seeman conjecture about \(\mathbb{Z}_2\)-graded tensor products.
type: Item
instance of: scholarly article
title: On a Regev-Seeman conjecture about \(\mathbb{Z}_2\)-graded tensor products.
zbMATH Open document ID: 1200.16030
published in: Israel Journal of Mathematics
publication date: 2010-06-29T00:00:00Z
review text: If \(A,B\) are superalgebras then, besides the ordinary tensor product \(A\otimes B\), a \(\mathbb{Z}_2\)-graded tensor product \(A\overline\otimes B\) arises. The multiplication in \(A\overline\otimes B\) 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 \(A\overline\otimes B\), where \(A\) and \(B\) are PI-superalgebras. Kemer proved that if \(A,B\) are T-prime algebras then \(A\otimes B\) has the same polynomial identities as a suitable explicitly given T-prime algebra \(C\). Regev and Seeman conjectured that, as in the ordinary case, the graded tensor product of two T-prime PI-algebras in characteristic 0 is PI-equivalent to a T-prime algebra and confirmed their conjecture in several cases. In the paper under review the authors establish results which imply the conjecture of Regev and Seeman. Using techniques for group graded polynomial identities developed in their previous papers, the authors establish that over any infinite field of characteristic \(\neq 2\) the \(\mathbb{Z}_2\)-graded tensor product of two T-prime algebras has the same multilinear polynomial identities as a T-prime algebra. They also give rules for the \(\mathbb{Z}_2\)-graded tensor multiplication of T-prime algebras. It is interesting to mention that in the supercase these rules are different from the rules in the ordinary case. For example, for the Grassmann algebra \(E\) the rules are \(E\overline\otimes E\sim E\) (as shown already by Regev and Seeman) and \(E\otimes E\sim M_{1,1}\), respectively. Clearly, this difference implies other differences in the tensor multiplication, e.g. \(M_m(E)\overline\otimes M_n(E)\sim M_{mn}(E)\) and \(M_m(E)\otimes M_n(E…
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: 5728341
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: ideals of identities
zbMATH Keywords: PI-superalgebras
zbMATH Keywords: multilinear identities
zbMATH Keywords: Grassmann algebras
Wikidata QID: Q123029295
author: Vincenzo Nardozza
author: M. Onofrio Di Vincenzo
MaRDI profile type: Publication
full work available at URL: s11856-010-0020-2
OpenAlex ID: W2037703315
cites work: Group gradings on associative algebras
cites work: Graded Polynomial Identities for Tensor Products by the Grassmann Algebra
cites work: GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS
cites work: POLYNOMIAL IDENTITIES ON SUPERALGEBRAS AND ALMOST POLYNOMIAL GROWTH
cites work: Identities of associative algebras
cites work: Polynomial identities of related rings
cites work: \(\mathbb{Z}_2\)-graded tensor products of p.i. algebras.
cites work: The theory of Lie superalgebras. An introduction
cites work: Z-Graded Polynomial Identities of the Full Matrix Algebra
cites work: $\mathbf {Z}_n$-graded polynomial identities of the full matrix algebra of order $n$
DOI: 10.1007/S11856-010-0020-2
description: scientific article; zbMATH DE number 5728341