On polynomial identities of Jordan pairs of rectangular matrices
type: Item
instance of: scholarly article
title: On polynomial identities of Jordan pairs of rectangular matrices
zbMATH Open document ID: 0881.17030
DOI: 10.1016/S0024-3795(97)80013-4
published in: Linear Algebra and its Applications
publication date: 1997-09-09T00:00:00Z
review text: If \(k\) is a field of characteristic different from 2 and 3, a pair \(V=(V^+,V^-)\) of \(k\)-vector spaces is called a Jordan pair if there is a trilinear mapping \(V^{\sigma}\times V^{-\sigma}\times V^{\sigma}\to V^{\sigma}\), \(\sigma=\pm\), denoted by \(\{xay\}\), \(x,y\in V^{\sigma}\), \(a\in V^{-\sigma}\), such that \(\{xay\}=\{yax\}\) and \(\{xa\{ybz\}\}-\{yb\{xaz\}\}=\{\{xay\}bz\}-\{y\{axb\}z\}\) for all \(x,y,z\in V^{\sigma}\), \(a,b\in V^{-\sigma}\), \(\sigma=\pm\). This definition has a natural generalization when \(k\) is any associative commutative ring and \(V^+,V^-\) are \(k\)-modules. In the paper under review the author studies the polynomial identities of the Jordan pair \(P(m,n)=(M_{m,n},M_{n,m})\), where \(M_{i,j}\) is the set of all \(i\times j\) matrices over \(k\) and the operation is given by \(\{xyz\}=xyz+zyx\). The first result is that over any field the polynomial identities of the pair \(P(m',n')\) are consequences of these of \(P(m,n)\) if and only if \(m\leq m'\) and \(n\leq n'\). Then the author gives a minimal basis of the identities of the pair \(P(1,n)\), \(n=1,2,\ldots,\infty\), over a field of characteristic 0. As a byproduct of his approach the author establishes that central simple (not necessarily associative) algebras of different dimension can be distinguished by means of Capelli identities.
Mathematics Subject Classification ID: 17C05
Mathematics Subject Classification ID: 17C20
Mathematics Subject Classification ID: 17C50
zbMATH DE Number: 1040497
zbMATH Keywords: Jordan pairs
zbMATH Keywords: polynomial identities of Jordan pairs
zbMATH Keywords: matrix algebras
zbMATH Keywords: bases of polynomial identities
zbMATH Keywords: T-ideals
author: A. V. Il'tyakov
reviewed by: Vesselin Drensky
MaRDI profile type: Publication
cites work: Q5340151
cites work: Distinguishing simple jordan algebras by means of polynomial identities
cites work: The Specht property of the ideals of identities of certain simple nonassociative algebras
cites work: Jordan pairs
cites work: Q3663493
cites work: Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero
cites work: Jordan triple systems of degree at most 2
cites work: Q3202278
cites work: Q3910639
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description: scientific article; zbMATH DE number 1040497