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

In this paper, we aim to study the (generalized) quantum double $K^{\ast \mathrm{c}\mathrm{o}\mathrm{p}}{\bowtie }_{\sigma }H$ determined by a (skew) pairing between finite-dimensional Hopf algebras $K^{\ast \mathrm{c}\mathrm{o}\mathrm{p}}$ and $H$, especially the tensor category $\mathsf{R}\mathsf{e}\mathsf{p}(K^{\ast \mathrm{c}\mathrm{o}\mathrm{p}}{\bowtie }_{\sigma }H)$ of its finite-dimensional representations. Specifically, we show that $K^{\ast \mathrm{c}\mathrm{o}\mathrm{p}}{\bowtie }_{\sigma }H$ is a left partially dualized (quasi-)Hopf algebra of $K^{\mathrm{o}\mathrm{p}}\otimes H$, and use this formulation to establish tensor equivalences from $\mathsf{R}\mathsf{e}\mathsf{p}(K^{\ast \mathrm{c}\mathrm{o}\mathrm{p}}{\bowtie }_{\sigma }H)$ to the categories ${}_K^K{\mathcal{M}}_H^K$ and ${}_{K^{\ast }}^{K^{\ast }}{\mathcal{M}}_{K^{\ast }}^{H^{\ast }}$ of two-sided two-cosided relative Hopf modules, as well as the category ${}_H{\mathfrak{Y}\mathfrak{D}}^K$ of relative Yetter-Drinfeld modules.