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arXiv · 1504.04238

Graded identities of block-triangular matrices

Diogo Diniz Pereira da Silva e Silva·Thiago Castilho de Mello
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Abstract

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.

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Diogo Diniz Pereira da Silva e Silva, Thiago Castilho de Mello. 2015-04-16. Graded identities of block-triangular matrices. https://doi.org/10.1016/j.jalgebra.2016.07.005

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