Matrices as graded BiHom-algebras and decompositions
Jiacheng Sun, Shuanhong Wang, Haoran Zhu
TL;DR
This work develops a structural theory for matrix BiHom-algebras graded by a regular BiHom-group, introducing a connectivity notion on the grading support to build canonical graded ideals. Under $Σ$-multiplicativity, maximal length, and centre triviality, the matrix BiHom-algebra decomposes as a direct sum of graded simple ideals, with each summand indexed by a connected component of the support. The results extend to general graded BiHom-algebras over arbitrary fields and recover classical gradings, such as Pauli-based and $Z_n × Z_n$-gradings, within this BiHom framework. The paper provides a unified decomposition mechanism that links grading structure to BiHom-deformations and offers new perspectives for applications in graded algebra and related areas in mathematical physics.
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 $Σ$-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.