Drinfel’d construction for Hom–Hopf T-coalgebras

Abstract

Let $H$ be a Hom-Hopf T-coalgebra over a group $\pi $ (i.e., crossed Hom-Hopf $\pi $-coalgebra). First, we introduce and study the left-right $\alpha $-Yetter-Drinfel'd category $\mathcal{YD}(H)_{\alpha}$ over $H$ with $\alpha \in G$, and construct a class of new braided T-categories. Then, we prove that Yetter-Drinfel'd module category $\mathcal{YD}(H)$ is a full subcategory of the center $\mathcal{Z}(Rep(H))$ of the category of representations of $H$. Next, we define the quasi-triangular structure of $H$, and obtain its representation crossed category $Rep(H)$ is quasi-braided. Finally, the Drinfel'd construction $D(H)$ of $H$ is constructed, and an equivalent relation between $\mathcal{YD}(H)$ and the representation of $D(H)$ is given.