On the *-polynomial identities of \(M_{1,1}(E)\).
type: Item
instance of: scholarly article
title: On the *-polynomial identities of \(M_{1,1}(E)\).
zbMATH Open document ID: 1215.16015
author: Plamen Koshlukov
author: M. Onofrio Di Vincenzo
published in: Journal of Pure and Applied Algebra
publication date: 2011-01-07T00:00:00Z
review text: Let \(F\) be a field of characteristic 0 and let \(E\) be the Grassmann algebra over an infinite dimensional vector space \(L\). If \(\alpha\) and \(\beta\) are, respectively, involutions of the matrix algebra \(M_n(F)\) and on \(E\), then they induce an involution \(\gamma=\gamma_{\alpha,\beta}\) on the algebra \(M_n(E)\). Let \(F\langle Y,Z\rangle\) be the free associative algebra with involution, where \(Y\) and \(Z\) are the free generators which are, respectively, symmetric and skew-symmetric with respect to the involution. The first result of the paper under review is that if \(\beta(L)=L\), then every *-polynomial identity of \(M_n(E)\) is obtained from an ordinary polynomial identity: If \(f(y_1,\dots,y_l,z_1,\dots,z_m)=0\) is a *-polynomial identity for \(M_n(E)\), then \(f(x_1,\dots,x_l,x_{l+1},\dots,x_{l+m})=0\) is an ordinary polynomial identity for \(M_n(E)\). A similar result holds for the algebra \(M_{k,l}(E)\) assuming that \(\alpha\) keeps invariant the \(\mathbb Z_2\)-graded components of \(M_{k+l}(F)\) in the definition of \(M_{k,l}(E)\) and \(\beta(L)=L\). Further, the authors study the *-polynomial identities of the algebra \(M_{1,1}(E)\) equipped with the involution \((ae_{11}+be_{12}+ce_{21}+de_{22})^*=(de_{11}+be_{12}-ce_{21}+ae_{22})\), \(a,d\in E_0\), \(b,c\in E_1\). This involution is induced by the transposition superinvolution of the superalgebra \(M_{1,1}(F)\). In the main result of the paper the authors present an explicit set of 10 polynomials in \(F\langle Y,Z\rangle\) which forms a basis of the *-polynomial identities of \(M_{1,1}(E)\). The basis consists of the polynomials \([y_1,y_2]\), \(z_1z_2z_3-z_3z_2z_1\), \([z_1,z_2][z_3,z_4]\) and 7 more polynomials of degree 4, 5 and 6.
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W55
zbMATH DE Number: 5833471
zbMATH Keywords: algebras with polynomial identity
zbMATH Keywords: algebras with involution
zbMATH Keywords: *-polynomial identities
zbMATH Keywords: T-prime algebras
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: superinvolutions
zbMATH Keywords: superalgebras
zbMATH Keywords: bases of identities
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.04.018
OpenAlex ID: W1558858808
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DOI: 10.1016/J.JPAA.2010.04.018
description: scientific article; zbMATH DE number 5833471