On \(\mathbb Z_2\)-graded identities of the super tensor product of \(UT_2(F)\) by the Grassmann algebra.
type: Item
instance of: scholarly article
title: On \(\mathbb Z_2\)-graded identities of the super tensor product of \(UT_2(F)\) by the Grassmann algebra.
zbMATH Open document ID: 1273.16019
author: Viviane Ribeiro Tomaz da Silva
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: Let \(F\) be a field of characteristic zero and \(E\) the unitary Grassmann algebra generated over an infinite-dimensional vector space \(L\). The author considers arbitrary \(\mathbb Z_2\)-gradings of \(E\): \(\mathcal E=\mathcal E^{(0)}\oplus\mathcal E^{(1)}\) such that the space \(L\) is homogeneous. Let \(\mathcal B=\{e_1,e_2,\dots\}\) be a linear homogeneous basis of \(L\). Then arbitrary gradings are given by a map \(\|*\|\colon\mathcal B\to\mathbb Z_2\). There exist three important possibilities: 1) \(\|e_i\|=0\), \(i=1,\dots,k\), \(\|e_i\|=1\) otherwise, for some \(k\geq 1\); 2) \(\|e_i\|=1\), \(i=1,\dots,k\), \(\|e_i\|=0\) otherwise, for some \(k\geq 1\); 3) \(\|e_i\|=0\) for \(i\) even and \(\|e_i\|=1\) for \(i\) odd. Given an associative superalgebra \(A=A^{(0)}\oplus A^{(1)}\), one defines the superalgebra \(A\widehat\otimes \mathcal E=(A^{(0)}\otimes\mathcal E^{(0)})\oplus (A^{(1)}\otimes\mathcal E^{(1)})\). In case of the canonical grading of \(E\) the algebra \(A\widehat\otimes E\) is the ordinary Grassmann envelope of \(A\). The \(\mathbb Z_2\)-codimensions for the Grassmann algebras with such noncanonical gradings were studied by the author earlier: [\textit{O. M. Di Vincenzo} and \textit{V. Ribeiro Tomaz da Silva}, Linear Algebra Appl. 431, No. 1-2, 56-72 (2009; Zbl 1225.16009); \textit{V. Ribeiro Tomaz da Silva}, Commun. Algebra 37, No. 9, 3342-3359 (2009; Zbl 1192.16021)]. Now the author studies the superalgebras \(UT_2(F)\widehat\otimes \mathcal E\), when the algebra \(UT_2(F)\) of \(2\times 2\) upper triangular matrices over \(F\) is endowed with its canonical grading. The author describes the generators of their graded T-ideals as well as linear bases of the corresponding relatively free algebras. Furthermore, the author describes the \(Y\)-proper codimension sequences and finally gives a complete description of the space of multilinear \(Y\)-proper identities in the language of Young diagrams. As a result, the author finds bases of \(\mathbb …
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16W55
zbMATH DE Number: 6105458
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: PI-algebras
zbMATH Keywords: graded algebras
zbMATH Keywords: codimension growth
zbMATH Keywords: cocharacters
zbMATH Keywords: graded codimensions
zbMATH Keywords: codimension sequences
zbMATH Keywords: graded polynomial identities
zbMATH Keywords: cocharacter sequences
zbMATH Keywords: graded T-ideals
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
full work available at URL: s11856-011-0127-0
OpenAlex ID: W1992151731
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DOI: 10.1007/S11856-011-0127-0
description: scientific article; zbMATH DE number 6105458