Another construction of the braided $T$-category

TL;DR

This paper constructs a new braided T-category using group-cograded monoidal Hom-Hopf algebras, expanding the algebraic structures within the Turaev category and introducing a novel p-Yetter-Drinfeld category.

Contribution

It introduces group-cograded monoidal Hom-Hopf algebras and constructs a new braided T-category based on these structures, extending previous algebraic frameworks.

Findings

Established that group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category

Defined the p-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra

Constructed a new kind of braided T-category

Abstract

This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided $T$-categories.