Graded tensor products.
type: Item
instance of: scholarly article
title: Graded tensor products.
zbMATH Open document ID: 1186.16013
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-04T00:00:00Z
review text: Let \(A=A_1\oplus\cdots\oplus A_r\) be a decomposition of the associative algebra \(A\) as a direct sum of its vector subspaces \(A_i\). This decomposition is regular if for any choice of the indices \(i_j\) one has \(A_{i_1}\cdots A_{i_n}\neq 0\), and furthermore for every \(i\) and \(j\) and every \(x_i\in A_i\), \(x_j\in A_j\) one has \(x_ix_j=\theta_{ij}x_jx_i\) where the scalars \(\theta_{ij}\) are nonzero and depend only on \(i\) and \(j\) (and of course on \(A\)). The matrix \(M^A =(\theta_{ij})\) of order \(r\) is the matrix of commutation relations for the given regular decomposition of \(A\). Regular decompositions were introduced by \textit{A. Regev} and \textit{T. Seeman} [J. Algebra 291, No. 1, 274-296 (2005; Zbl 1083.16017)]. In that paper these decompositions were used in order to study graded tensor products of PI algebras and the corresponding graded identities. In the present paper the authors continue the study and applications of the regular decompositions. They introduce the so-called minimal decomposition of \(A\). They prove (see Proposition 2.1 in the paper) that if \(A\) and \(B\) admit regular decompositions then their tensor product \(A\otimes B\) also does, and if the decompositions of \(A\) and \(B\) are minimal then so is the induced decomposition on \(A\otimes B\). As a corollary the authors obtain that the algebra \(M_n(G)\) is regular (here \(G\) is the infinite dimensional Grassmann algebra). The authors give the interesting Conjecture 2.5: Is it true that a regular decomposition is minimal if and only if the determinant of the corresponding matrix is nonzero? Also, is the number of summands in a minimal decomposition an invariant of the given algebra? Further on the authors study twisted tensor products of graded algebras where the twist is given by a bicharacter on the grading group. They generalize one of the key results of the above cited paper, as well as of the paper by \textit{J. A. Freitas} and the reviewer, [J. Algebra 321,…
reviewed by: Plamen Koshlukov
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16T10
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 15A75
zbMATH DE Number: 5561719
zbMATH Keywords: regular decompositions
zbMATH Keywords: graded tensor products
zbMATH Keywords: graded identities
zbMATH Keywords: twisted tensor products
zbMATH Keywords: bicharacters
zbMATH Keywords: Hopf algebras
zbMATH Keywords: commutation relations
zbMATH Keywords: PI algebras
author: Yuri Bahturin
author: Amitai Regev
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.12.010
OpenAlex ID: W2121146624
cites work: \(\mathbb{Z}_2\)-graded tensor products of p.i. algebras.
cites work: Commutation relations and Vandermonde determinants
cites work: VARIETIES AND $ Z_2$-GRADED ALGEBRAS
cites work: Existence of identities in \(A \otimes B\)
cites work: Bicharacters, twistings, and Scheunert's theorem for Hopf algebras
DOI: 10.1016/J.JPAA.2008.12.010
description: scientific article; zbMATH DE number 5561719