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Constructing New Braided $T$-categories over Monoidal Hom-Hopf Algebras

Let $ Aut_{mHH}(H)$ denote a set of all automorphisms of a monoidal Hopf algebra $H$ with bijective antipode in the sense of Caenepeel S. and Goyvaerts I. (Commun. Algebra 39, 2216-2240, 2011) and let $G$ be a crossed product group $ Aut_{mHH}(H)\times Aut_{mHH}(H)$. The main aim of this paper is to provide further examples of braided $T$-category in the sense of Turaev (1994, 2008). For this purpose, we first introduce a class of new categories $_{H}\mathcal {MHYD}^{H}(A, B)$ of monoidal Hom $(A, B)$-Yetter-Drinfeld modules with $A, B \in Aut_{mHH}(H)$. Then we show that the category ${\cal MHYD}(H)=\{{}_{H}\mathcal {MHYD}^{H}(A, B)\}_{(A, B)\in G}$ forms a braided $T$-category, generalizing the main constructions construction by Panaite and Staic (Isr J Math 158:349-365, 2007).