The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations
Ji-Wei He, Xiaojie Kong, Kangqiao Li
TL;DR
This work develops a categorical framework for the generalized quantum double $K^{*cop}\bowtie_\sigma H$, showing that it arises as a left partial dual of the tensor product $K^{op}\otimes H$ and that its finite representations are captured by relative Doi–Hopf and Yetter–Drinfeld module categories. Through reconstruction from partial duality, the authors establish tensor equivalences $\mathsf{Rep}(K^{*cop}\bowtie_\sigma H) \simeq {}^{}_{K^{*cop}}\mathfrak{M}^{K^{*cop}\otimes H^{*}}_{K^{*cop}} \simeq {}_H\mathfrak{YD}^K$, and show dual descriptions with $({}_{K^*}\mathfrak{YD}^{H^*})^{rev}$, unifying several relative module theories. A key strand is the realization of $D=K^{*cop}\bowtie_\sigma H$ via a partially admissible mapping system, which yields a reconstruction functor linking representations to relative Hopf-module structures and relative centers; in particular, when $\sigma_r$ is surjective, $\mathcal{Z}_{\mathsf{Rep}(K)}(\mathsf{Rep}(H))$ is tensor-equivalent to $\mathsf{Rep}(K^{*cop}\bowtie_\sigma H)$ and to ${}_H\mathfrak{YD}^K$. The results also provide dual and cotensor descriptions and extend Schauenburg’s equivalence to the relative setting, offering a cohesive categorical picture for quantum doubles, partial duality, and relative center phenomena.
Abstract
In this paper, we aim to study the (generalized) quantum double $K^{\ast\mathrm{cop}}\bowtie_σH$ determined by a (skew) pairing between finite-dimensional Hopf algebras $K^{\ast\mathrm{cop}}$ and $H$, especially the tensor category $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_σH)$ of its finite-dimensional representations. Specifically, we show that $K^{\ast\mathrm{cop}}\bowtie_σH$ is a left partially dualized (quasi-)Hopf algebra of $K^\mathrm{op}\otimes H$, and use this formulation to establish tensor equivalences from $\mathsf{Rep}(K^{\ast\mathrm{cop}}\bowtie_σH)$ 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.