Minimal varieties of PI-superalgebras with graded involution
type: Item
instance of: scholarly article
title: Minimal varieties of PI-superalgebras with graded involution
zbMATH Open document ID: 1471.16035
author: Viviane Ribeiro Tomaz da Silva
author: Ernesto Spinelli
author: M. Onofrio Di Vincenzo
published in: Israel Journal of Mathematics
publication date: 2021-04-29T00:00:00Z
review text: Let \(A\) be an associative algebra over a field \(F\) of characteristic 0, it is well known that the polynomial identities of \(A\) are consequences of the multilinear ones. Hence in order to study the identities satisfied by \(A\) it suffices to consider only the multilinear identities of \(A\). Denote by \(P_n\) the vector space of the multilinear polynomials in the first \(n\) free variables in the free associative algebra, and let \(I=Id(A)\) be the ideal of identities of \(A\). Thus one might want to study the intersections \(P_n\cap I\) for every \(n\). It is clear that \(P_n\) is a left module over the symmetric group \(S_n\) which acts by permuting the variables. Since \(I\) is invariant under permutations of the variables it follows that \(P_n\cap I\) is a submodule of \(P_n\). Therefore one may apply the well developed theory of the representations of \(S_n\) to the study of polynomial identities in algebras. In fact this approach is not the best one, as it has a significant drawback. Let us denote by \(P_n(A)=P_n/(P_n/\cap I)\) and by \(c_n(A)=\dim P_n(A)\) the \(n\)th codimension of \(A\). \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)] proved that if \(A\) satisfies an identity of degree \(d\) then \(c_n(A)\le (d-1)^{2n}\) for every \(n\). This implies that for \(n\) large enough, the intersection \(P_n\cap I\) becomes very large. Hence studying the quotient module \(P_n(A)\) is better from computational point of view. The codimension sequence \(c_n(A)\), introduced by Regev in that paper, is one of the main numerical invariants of a T-ideal (that is an ideal of identities of an algebra). It is also one of the most studied ones. Let us mention that in very few occasions it is possible (and plausible) to compute it explicitly. Amitsur conjectured that the limit \(\lim_{n\to\infty} (c_n(A)^{1/n})\), called the PI exponent of \(A\), exists for every PI algebra \(A\), and is, moreover, a nonnegative integer. This conjecture was confir…
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 7341471
zbMATH Keywords: minimal varieties
zbMATH Keywords: superalgebras
zbMATH Keywords: graded involution
zbMATH Keywords: block triangular matrices
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: s11856-021-2119-z
OpenAlex ID: W3135925719
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DOI: 10.1007/S11856-021-2119-Z
description: scientific article; zbMATH DE number 7341471