Graded Identities and Isomorphisms on Algebras of Upper Block-Triangular Matrices

TL;DR

This paper extends the classification of G-gradings on upper block-triangular matrix algebras to all abelian groups and explores whether graded identities uniquely determine such graded algebras up to isomorphism.

Contribution

It generalizes previous results to any abelian group G and links graded identities to algebra isomorphisms in this context.

Findings

G-gradings on upper block-triangular matrices are classified for all abelian groups.

Any such grading decomposes into a tensor product of elementary and division gradings.

Graded identities can determine the algebra up to graded isomorphism in certain cases.

Abstract

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\otimes B$ up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O. M. Di Vincenzo and E. Spinelli.