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

Let $Au{t}_{mHH}(H)$ 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 $Au{t}_{mHH}(H)\times Au{t}_{mHH}(H)$. 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 ${H}\mathcal {MHYD}^{H}(A, B)$ofmonoidalHom$(A, B)$-Yetter-Drinfeldmoduleswith$A, B \in Aut{mHH}(H)$\mathrm{.}Thenweshowthatthecategory${\cal MHYD}(H)={{}{H}\mathcal {MHYD}^{H}(A, B)}{(A, B)\in G}$formsabraided$T$-category, generalizing the main constructions construction by Panaite and Staic (Isr J Math 158:349-365, 2007).