Minimal supervarieties with factorable ideal of graded polynomial identities.
type: Item
instance of: scholarly article
title: Minimal supervarieties with factorable ideal of graded polynomial identities.
zbMATH Open document ID: 1347.16019
author: Viviane Ribeiro Tomaz da Silva
author: Ernesto Spinelli
author: M. Onofrio Di Vincenzo
published in: Journal of Pure and Applied Algebra
publication date: 2015-12-08T00:00:00Z
review text: A well known result of Giambruno and Zaicev states that a variety \(\mathcal V\) of PI algebras of finite basic rank (i.e., generated by a finite dimensional algebra) is minimal if, and only if, it is generated by an upper block triangular matrix algebra \(UT(m_1,\ldots,m_n)\). The T-ideal (the ideal of identities) of such algebras factors as the product of the ideals of the main blocks. In the superalgebras setting, one replaces upper block triangular matrix algebras with minimal superalgebras but the factorability property for their \(T_{\mathbb Z_2}\)-ideals (ideals of their \(\mathbb Z_2\)-graded identities) fails. If \(A\) is a minimal superalgebra let \(A=A_{ss}+J\) be its Wedderburn-Malcev decomposition and \(A_{ss}=A_1\oplus\cdots\oplus A_n\) be the decomposition of \(A_{ss}\) as a sum of graded simple algebras. In the paper under review, the authors investigate the question of when the ideal of graded identities of such algebra \(A\) factors as the product of the ideals of the graded simple components of \(A_{ss}\). They provide a positive answer for the case in which the algebras \(A_1,\ldots,A_n\) are non-simple graded simple except for at most one between \(A_1\) and \(A_n\). If more than one of the first components are simple and the other are non-simple (or conversely), they prove that such ideal factorizes in a weaker sense, also as a product of \(T_{\mathbb Z_2}\)-ideals of subalgebras of \(A\). In [\textit{O. M. Di Vincenzo} and \textit{E. Spinelli}, J. Algebra 368, 182-198 (2012; Zbl 1279.16021)] it has been stated that any supervariety of superexponent \(\geq 2\) is generated by one of the so called minimal superalgebras. But the problem of which minimal superalgebras generate a minimal supervariety of a fixed graded exponent is still open and according to the authors of the paper under review, ``its possible solution seems to be more involved than that of the ungraded case''. In the mentioned paper it is proved that if \(A\) is a minimal superalg…
reviewed by: Thiago Castilho de Mello
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16R10
zbMATH DE Number: 6517746
zbMATH Keywords: minimal supervarieties
zbMATH Keywords: minimal superalgebras
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: graded simple components
zbMATH Keywords: superexponents
zbMATH Keywords: supercodimension sequences
zbMATH Keywords: block triangular matrix algebras
zbMATH Keywords: T-ideals
zbMATH Keywords: ideals of identities
zbMATH Keywords: graded simple algebras
MaRDI profile type: Publication
full work available at URL: j.jpaa.2015.09.004
OpenAlex ID: W1776651865
cites work: Multialternating graded polynomials and growth of polynomial identities.
cites work: Graded polynomial identities and exponential growth
cites work: Polynomial identities on superalgebras and exponential growth
cites work: Block-triangular matrix algebras and factorable ideals of graded polynomial identities.
cites work: On some minimal supervarieties of exponential growth.
cites work: Graded polynomial identities and codimensions: computing the exponential growth.
cites work: Wreath products and P.I. algebras
cites work: On codimension growth of finitely generated associative algebras
cites work: Exponential codimension growth of PI algebras: an exact estimate
cites work: Minimal varieties of algebras of exponential growth.
cites work: Codimension growth and minimal superalgebras
cites work: Q5714236
cites work: Identities of associative algebras
cites work: A Matrix Representation for Associative Algebras. I
cites work: Existence of identities in \(A \otimes B\)
DOI: 10.1016/J.JPAA.2015.09.004
description: scientific article; zbMATH DE number 6517746