A Note on Braided $T$-categories over Monoidal Hom-Hopf Algebras

TL;DR

This paper constructs new examples of braided T-categories using monoidal Hom-Hopf algebras, demonstrating their properties and establishing isomorphisms with representation categories, thus expanding the framework of braided T-categories.

Contribution

It introduces a monoidal Hom-Hopf T-coalgebra and shows its representation category is isomorphic to a Hom-Yetter-Drinfeld category, also constructing a new braided T-category over integers.

Findings

Construction of monoidal Hom-Hopf T-coalgebra $ ext{MHD}(H)$

Isomorphism between $Rep( ext{MHD}(H))$ and $ ext{MHYD}(H)$ as braided T-categories

Development of a new braided T-category $ ext{ZMHYD}(H)$ over $ ext{Z}$

Abstract

Let $ Aut_{mHH}(H)$ denote the set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel and Goyvaerts \cite{CG2011}. The main aim of this paper is to provide new examples of braided $T$-category in the sense of Turaev \cite{T2008}. For this, first we construct a monoidal Hom-Hopf $T$-coalgebra $\mathcal{MHD}(H)$ and prove that the $T$-category $Rep(\mathcal{MHD}(H))$ of representation of $\mathcal{MHD}(H)$ is isomorphic to $\mathcal {MHYD}(H)$ as braided $T$-categories, if $H$ is finite-dimensional. Then we construct a new braided $T$-category $\mathcal{ZMHYD}(H)$ over $\mathbb{Z},$ generalizing the main construction by Staic \cite{S2007}.