The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations
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| Format: | Preprint |
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2025
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| _version_ | 1866911432760623104 |
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| author | He, Ji-Wei Kong, Xiaojie Li, Kangqiao |
| author_facet | He, Ji-Wei Kong, Xiaojie Li, Kangqiao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_06066 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations He, Ji-Wei Kong, Xiaojie Li, Kangqiao Quantum Algebra Category Theory Rings and Algebras 16T05, 18M05 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. |
| title | The Quantum Double of Hopf Algebras Realized via Partial Dualization and the Tensor Category of Its Representations |
| topic | Quantum Algebra Category Theory Rings and Algebras 16T05, 18M05 |
| url | https://arxiv.org/abs/2504.06066 |