The graded identities of upper triangular matrices of size two
type: Item
instance of: scholarly article
title: The graded identities of upper triangular matrices of size two
zbMATH Open document ID: 1007.16019
DOI: 10.1016/S0022-4049(01)00169-4
author: R. Smith
published in: Journal of Pure and Applied Algebra
publication date: 2002-08-22T00:00:00Z
review text: Denote by \(UT_2\) the algebra of \(2\times 2\) upper triangular matrices over a field \(F\) of characteristic 0. This algebra is quite important in the PI theory. The paper under review describes all gradings on this algebra by an arbitrary group \(G\). It turns out that there are two possible gradings only. The first is the trivial one, the second is the usual grading by the group of two elements. Then the author describes the graded identities of \(UT_2\). It is shown that the two identities \(y_1y_2-y_2y_1=0\) and \(z_1z_2=0\) generate the graded T-ideal of \(UT_2\). Here \(y_i\) are 0-variables while \(z_i\) are 1-variables. Furthermore the graded cocharacter is computed. As corollaries, various numerical invariants of the graded identities for \(UT_2\) are described such as the graded codimenison and the graded length. Finally, it is shown that the graded codimension is polynomially bounded for any superalgebra whose graded T-ideal contains strictly all polynomial identities of \(UT_2\).
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1787479
zbMATH Keywords: graded identities
zbMATH Keywords: multilinear identities
zbMATH Keywords: growth
zbMATH Keywords: graded codimensions
zbMATH Keywords: algebras of triangular matrices
zbMATH Keywords: graded algebras
zbMATH Keywords: graded cocharacters
zbMATH Keywords: graded T-ideals
zbMATH Keywords: polynomial identities
MaRDI profile type: Publication
cites work: Cocharacters, Codimensions and Hilbert Series of the Polynomial Identities for 2 × 2 Matrices with Involution
cites work: Group Actions and Asymptotic Behavior of Graded Polynomial Identities
cites work: Wreath products and P.I. algebras
cites work: Q3952291
cites work: T-ideals with power growth of the codimensions are Specht
cites work: Complexity of nonmatrix varieties of associative algebras. I
cites work: A star-variety with almost polynomial growth
cites work: Existence of identities in \(A \otimes B\)
cites work: Abelian Gradings on Upper Block Triangular Matrices
full work available at URL: s0022-4049(01)00169-4
OpenAlex ID: W2018465721
description: scientific article; zbMATH DE number 1787479