New Braided $T$-Categories over Hopf (co)quasigroups

Abstract: Let $H$ be a Hopf quasigroup with bijective antipode and let $Aut_{HQG}(H)$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a category ${{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ with $\alpha,\beta\in Aut{HQG}(H)$ and construct a braided $T$-category $\mathcal{YDQ}(H)$ having all the categories ${_{H}\mathcal{YDQ}^{H}}(\alpha,\beta)$ as components.