标题： Hopf代数的量子偶通过部分对偶化实现及其表示的张量范畴 显示英文标题

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

摘要： 本文旨在研究由有限维Hopf代数 $K^{\ast\mathrm{cop}}$ 和 $H$ 之间的（斜）配对确定的（广义）量子双代数 $K^{\ast\mathrm{cop}}\bowtie_\sigma H$，特别是其有限维表示的张量范畴 $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_\sigma H)$。 具体而言，我们证明了$K^{\ast\mathrm{cop}}\bowtie_\sigma H$是$K^\mathrm{op}\otimes H$的左部分对偶化（拟）Hopf 代数，并利用这一表述建立了从$\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_\sigma H)$到两类双边双侧面相对 Hopf 模范畴${}^K_K\mathcal{M}^K_H$和${}^{K^\ast}_{K^\ast}\mathcal{M}^{H^\ast}_{K^\ast}$以及相对 Yetter-Drinfeld 模范畴${}_H\mathfrak{YD}^K$的张量等价。 显示英文摘要

摘要： 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.