Cohomological dimension of braided Hopf algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917812549713920 |
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| author | Bichon, Julien Nguyen, Thi Hoa Emilie |
| author_facet | Bichon, Julien Nguyen, Thi Hoa Emilie |
| contents | We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the trivial left and right module all coincide. We also provide convenient criteria for smoothness and the twisted Calabi-Yau property for such braided Hopf algebras (without the cosemisimplicity assumption on $H$), in terms of properties of the trivial module.These generalize a well-known result in the case of ordinary Hopf algebras. As an illustration, we study the case of the coordinate algebra on the two-parameter braided quantum group $\textrm{SL}_{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomological dimension of braided Hopf algebras Bichon, Julien Nguyen, Thi Hoa Emilie K-Theory and Homology Quantum Algebra Rings and Algebras We show that for a braided Hopf algebra in the category of comodules over a cosemisimple coquasitriangular Hopf algebra, the Hochschild cohomological dimension, the left and right global dimensions and the projective dimensions of the trivial left and right module all coincide. We also provide convenient criteria for smoothness and the twisted Calabi-Yau property for such braided Hopf algebras (without the cosemisimplicity assumption on $H$), in terms of properties of the trivial module.These generalize a well-known result in the case of ordinary Hopf algebras. As an illustration, we study the case of the coordinate algebra on the two-parameter braided quantum group $\textrm{SL}_{2}$. |
| title | Cohomological dimension of braided Hopf algebras |
| topic | K-Theory and Homology Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2410.16768 |