The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations

Abstract: In this paper, we aim to study the (generalized) quantum double $K^{\ast\mathrm{cop}}\bowtie_\sigma H$ determined by a (skew) pairing between finite-dimensional Hopf algebras $K^{\ast\mathrm{cop}}$ and $H$, especially the tensor category $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_\sigma H)$ of its finite-dimensional representations. Specifically, we show that $K^{\ast\mathrm{cop}}\bowtie_\sigma H$ is a left partially dualized (quasi-)Hopf algebra of $K^\mathrm{op}\otimes H$, and use this formulation to establish tensor equivalences from $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_\sigma H)$ to the categories ${}^K_K\mathcal{M}^K_H$ and ${}^{K^\ast}{K^\ast}\mathcal{M}^{H^\ast}{K^\ast}$ of two-sided two-cosided relative Hopf modules, as well as the category ${}_H\mathfrak{YD}^K$ of relative Yetter-Drinfeld modules.