Matrices as graded BiHom-algebras and decompositions

TL;DR

This paper explores matrices as graded BiHom-algebras, introducing a connection-based decomposition method that yields simple ideals, and extends these results to general graded BiHom-algebras with applications to classical matrix gradings.

Contribution

It introduces a new framework for decomposing matrices as graded BiHom-algebras using connections, and extends the theory to broader classes of graded BiHom-algebras.

Findings

Decomposition into graded simple ideals under certain assumptions.

Construction of canonical graded ideals via connection in the support.

Reinterpretation of classical matrix gradings within the new framework.

Abstract

We present matrices as graded BiHom-algebras and consider various characteristics of their decompositions. Specifically, we introduce a notion of connection in the support of the grading and use it to construct a family of canonical graded ideals. We show that, under suitable assumptions, such as $\Sigma$-multiplicativity, maximal length, and centre triviality, the matrix BiHom-algebra decomposes into a direct sum of graded simple ideals. We further extend our results to general graded BiHom-algebras over arbitrary base fields. As applications, we reinterpret classical gradings on matrix algebras such as those induced by Pauli matrices and the $\mathbb{Z}_n \times \mathbb{Z}_n $-grading in terms of our setting.