Graded identities of block-triangular matrices

Let \(F\) be an infinite field and \(UT(d_1,\dots, d_n)\) be the algebra of upper block-triangular matrices over \(F\). In this paper we describe a basis for the \(G\)-graded polynomial identities of \(UT(d_1,\dots, d_n)\), with an elementary grading induced by an \(n\)-tuple of elements of a group...

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Bibliographic Details
Published in:arXiv.org 2015-09
Main Authors: Diogo Diniz Pereira da Silva e Silva, Thiago Castilho de Mello
Format: Article
Language:English
Subjects:
Algebra
Color
Grading
Mathematical analysis
Polynomials
Tensors
Online Access:Get full text
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Summary:Let \(F\) be an infinite field and \(UT(d_1,\dots, d_n)\) be the algebra of upper block-triangular matrices over \(F\). In this paper we describe a basis for the \(G\)-graded polynomial identities of \(UT(d_1,\dots, d_n)\), with an elementary grading induced by an \(n\)-tuple of elements of a group \(G\) such that the neutral component corresponds to the diagonal of \(UT(d_1,\dots,d_n)\). In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to \(2n-1\). Our results generalize for infinite fields of arbitrary characteristic, previous results in the literature which were obtained for fields of characteristic zero and for particular \(G\)-gradings. In the characteristic zero case we also generalize results for the algebra \(UT(d_1,\dots, d_n)\otimes C\) with a tensor product grading, where \(C\) is a color commutative algebra generating the variety of all color commutative algebras.
ISSN:2331-8422
DOI:10.48550/arxiv.1504.04238