International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-06-23 , DOI: 10.1142/s0129167x20500585
Dongdong Yan; Shuanhong Wang

Let $H$ be a Hom–Hopf T-coalgebra over a group $π$ (i.e. a crossed Hom–Hopf $π$-coalgebra). First, we introduce and study the left–right $α$-Yetter–Drinfel’d category $𝒴𝒟(H)α$ over $H$, with $α∈G$, and construct a class of new braided T-categories. Then, we prove that a Yetter–Drinfel’d module category $𝒴𝒟(H)$ is a full subcategory of the center $𝒵(Rep(H))$ of the category of representations of $H$. Next, we define the quasi-triangular structure of $H$ and show that the representation crossed category $Rep(H)$ is quasi-braided. Finally, the Drinfel’d construction $D(H)$ of $H$ is constructed, and an equivalent relation between $𝒴𝒟(H)$ and the representation of $D(H)$ is given.

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