Superalgebras with graded involution: classifying minimal varieties of quadratic growth
type: Item
instance of: scholarly article
title: Superalgebras with graded involution: classifying minimal varieties of quadratic growth
zbMATH Open document ID: 1471.16036
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 2021-04-24T00:00:00Z
review text: All algebras considered in the paper are over a field of characteristic 0. In the study of the identities satisfied by a given algebra \(A\) a prominent role plays its codimension sequence \(c_n(A)\). Let \(F\langle X\rangle\) be the free associative algebra and denote by \(P_n\) the vector space of all multilinear polynomials of degree \(n\) in the variables \(x_1\), \dots, \(x_n\) in it. Clearly \(\dim P_n=n!\). The symmetric group \(S_n\) acts on the left on \(P_n\) by permuting the variables, thus endowing it with the structure of an \(S_n\)-module. This module is clearly isomorphic to the regular \(S_n\)-module \(FS_n\). If \(I\) is the T-ideal of the algebra \(A\) then \(P_n\cap I\) is a submodule, and one employs the representation theory of the symmetric group in the study of PI algebras. In characteristic 0 the T-ideal \(I\) is generated by all of its multilinear elements, namely by the intersections \(I\cap P_n\). It should be noted though that these intersections grow very fast, and according to a well-known theorem of \textit{A. Regev} [Isr. J. Math. 11, 131--152 (1972; Zbl 0249.16007)], \(I\cap P_n\) tends to become almost the whole of \(P_n\) for large \(n\). That is why one studies the quotient \(P_n/(I\cap P_n\). Its dimension \(c_n(I)\) is the \(n\)-th codimension of \(I\), and of any algebra \(A\) that has \(I\) as its T-ideal. According to Regev's theorem \(c_n(I)\le (d-1)^{2n}\) for every \(n\) where \(d\) is the degree of a nontrivial identity in \(I\). \par One defines in a similar way the codimension sequence of an algebra with involution, and for a superalgebra (that is a 2-graded algebra) with an involution, and with a graded involution. Of particular interest are the varieties of algebras, superalgebras, and so on, where the codimension sequence grows like a polynomial function. A variety \(V\) is said to be of growth \(n^k\) if its codimensions grow like the polynomial function \(n^k\). It is a minimal variety of polynomial growth \(n^k\) …
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16P90
zbMATH DE Number: 7337672
zbMATH Keywords: polynomial identity
zbMATH Keywords: superalgebra
zbMATH Keywords: algebra with involution
zbMATH Keywords: graded involution
zbMATH Keywords: codimension
zbMATH Keywords: growth
reviewed by: Plamen Koshlukov
MaRDI profile type: Publication
full work available at URL: j.laa.2021.03.011
OpenAlex ID: W3134249409
cites work: Minimal Identities for Algebras
cites work: Q4700803
cites work: Characterizations of *-superalgebras of polynomial growth
cites work: Glxgl- representations and ∗-polynomial identities
cites work: Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions
cites work: Classifying the Minimal Varieties of Polynomial Growth
cites work: Identities of *-superalgebras and almost polynomial growth
cites work: On codimension growth of finitely generated associative algebras
cites work: Exponential codimension growth of PI algebras: an exact estimate
cites work: Asymptotics for the standard and the Capelli identities.
cites work: Minimal \(\ast\)-varieties and minimal supervarieties of polynomial growth
cites work: Some results concerning the multiplicities of cocharacters of superalgebras with graded involution
cites work: Polynomial codimension growth of algebras with involutions and superinvolutions
cites work: T-ideals with power growth of the codimensions are Specht
cites work: Varieties of superalgebras of almost polynomial growth.
cites work: Polynomial growth and star-varieties.
cites work: Algebras with involution with linear codimension growth.
cites work: Minimal star-varieties of polynomial growth and bounded colength
cites work: Superalgebras with graded involution and star-graded colength bounded by 3
cites work: Existence of identities in \(A \otimes B\)
cites work: Q2716065
DOI: 10.1016/J.LAA.2021.03.011
Recommended article: Superalgebras and algebras with involution: classifying varieties of quadratic growth
Recommended article: Minimal varieties of PI-superalgebras with graded involution
Recommended article: Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions
Recommended article: Varieties of algebras with superinvolution of almost polynomial growth
Recommended article: Unitary superalgebras with graded involution or superinvolution of polynomial growth
Recommended article: Classifying algebras with graded involutions or superinvolutions with multiplicities of their cocharacter bounded by one
Recommended article: Varieties of superalgebras of almost polynomial growth.
Recommended article: Minimal superalgebras generating minimal supervarieties
Recommended article: Group gradings on superinvolution simple superalgebras.
Recommended article: Codimension growth and minimal superalgebras
description: scientific article; zbMATH DE number 7337672