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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.02427 |
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| _version_ | 1866909373469556736 |
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| author | Garcia, Gaston Andres Mastnak, Mitja |
| author_facet | Garcia, Gaston Andres Mastnak, Mitja |
| contents | This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_ξ$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_ξ \simeq B_{ξ^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_ξ $ as algebra for $ξ\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_02427 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Hopf algebras whose coradical is a cocentral abelian cleft extension Garcia, Gaston Andres Mastnak, Mitja Quantum Algebra 16T05 This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra K_{n}, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K_{3}. This includes a family of 12-dimensional Nichols algebras $B_ξ$ depending on 3rd roots of unity. Here, $B_{1}$ is isomorphic to the well-known Fomin-Kirillov algebra, and $B_ξ \simeq B_{ξ^{2}} $ as graded algebras but $B_1$ is not isomorphic to $B_ξ $ as algebra for $ξ\neq 1$. As a byproduct we obtain new Hopf algebras of dimension 216. |
| title | On Hopf algebras whose coradical is a cocentral abelian cleft extension |
| topic | Quantum Algebra 16T05 |
| url | https://arxiv.org/abs/2304.02427 |