Reflection of Nichols Algebras over Coquasi-Hopf Algebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866908855067213824 |
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| author | Li, Bowen Liu, Gongxiang |
| author_facet | Li, Bowen Liu, Gongxiang |
| contents | This paper extends the foundational reflection theory of Nichols algebras to the setting of some certain coquasi-Hopf algebras. Our primary motivation arises from the classification of pointed finite-dimensional coquasi-Hopf algebras. We develop a reflection theory for tuples of simple Yetter-Drinfeld modules in the category $\GG$, where $G$ is a finite group and $Φ$ is a 3-cocycle on $G$. We prove that such a tuple gives rise to a semi-Cartan graph if admitting all reflections. Consequently, its Weyl groupoid is well-defined. We further establish several criteria for the finite-dimensionality of Nichols algebras in terms of the associated semi-Cartan graph. As an application, we provide a new proof for the infinite-dimensionality of a specific class of Nichols algebras previously studied in \cite{huang2024classification}, bypassing extensive computational arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reflection of Nichols Algebras over Coquasi-Hopf Algebras Li, Bowen Liu, Gongxiang Quantum Algebra This paper extends the foundational reflection theory of Nichols algebras to the setting of some certain coquasi-Hopf algebras. Our primary motivation arises from the classification of pointed finite-dimensional coquasi-Hopf algebras. We develop a reflection theory for tuples of simple Yetter-Drinfeld modules in the category $\GG$, where $G$ is a finite group and $Φ$ is a 3-cocycle on $G$. We prove that such a tuple gives rise to a semi-Cartan graph if admitting all reflections. Consequently, its Weyl groupoid is well-defined. We further establish several criteria for the finite-dimensionality of Nichols algebras in terms of the associated semi-Cartan graph. As an application, we provide a new proof for the infinite-dimensionality of a specific class of Nichols algebras previously studied in \cite{huang2024classification}, bypassing extensive computational arguments. |
| title | Reflection of Nichols Algebras over Coquasi-Hopf Algebras |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2512.04560 |