Constructing New Braided $T$-Categories via Weak Monoidal Hom-Hopf Algebras

TL;DR

This paper introduces weak monoidal Hom-Hopf algebras, generalizing existing structures, and constructs a braided T-category from these algebras, expanding the framework of Hom-Hopf algebra theory.

Contribution

The paper defines weak monoidal Hom-Hopf algebras and constructs a new braided T-category integrating these structures, which is a novel development in the field.

Findings

Defined weak monoidal Hom-Hopf algebras.

Constructed a braided T-category from these algebras.

Unified various categories into a braided T-category.

Abstract

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. If $H$ is a weak monoidal Hom-Hopf algebra with bijective antipode and let $Aut_{wmHH}(H)$ be the set of all automorphisms of $H$. Then we introduce a category ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut_{wmHH}(H)$ and construct a braided $T$-category $\mathcal{WMHYD}(H)$ that having all the categories ${_{H}\mathcal{WMHYD}^{H}}(\alpha,\beta)$ as components.