Graded identities of block-triangular matrices

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.

24 pages and 39 references. We have added section 5 in the text about tensor products by color commutative superalgebras

References in corpus (2)

• On the Graded Identities for Elementary Gradings in Matrix Algebras over Infinite Fields
• On group gradings on PI-algebras

Cited by in corpus (5)

• Group gradings on upper block triangular matrices
• Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field
• Identities and central polynomials of real graded division algebras
• Gradings on the algebra of triangular matrices as a Lie algebra: revisited
• Graded monomial identities and almost non-degenerate gradings on matrices