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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2508.00597 |
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| _version_ | 1866918110025482240 |
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| author | Bulacu, Daniel Misurati, Matteo |
| author_facet | Bulacu, Daniel Misurati, Matteo |
| contents | Inspired by the work of Radford, for $H$ an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension $2$ within the braided category of left Yetter-Drinfeld modules over $H$ and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking $H$ as the Hopf algebra of functions on a group $G$, endowed with the quasi-Hopf algebra structure provided by a non-trivial $3$-cocycle on $G$ (especially when $G$ is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Biproduct Quasi-Hopf Algebras of Rank 2 Bulacu, Daniel Misurati, Matteo Quantum Algebra Rings and Algebras 16W30 Inspired by the work of Radford, for $H$ an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension $2$ within the braided category of left Yetter-Drinfeld modules over $H$ and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking $H$ as the Hopf algebra of functions on a group $G$, endowed with the quasi-Hopf algebra structure provided by a non-trivial $3$-cocycle on $G$ (especially when $G$ is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories. |
| title | Biproduct Quasi-Hopf Algebras of Rank 2 |
| topic | Quantum Algebra Rings and Algebras 16W30 |
| url | https://arxiv.org/abs/2508.00597 |