On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908981357707264 |
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| author | Yang, Yuping |
| author_facet | Yang, Yuping |
| contents | We prove that every pre-Nichols algebra of a nondiagonal object in the twisted Yetter-Drinfeld category ${_{\k G}^{\k G} {\mathcal{YD}^Φ}}$ has infinite Gelfand-Kirillov dimension, where $G$ is a finite abelian group and $Φ$ is a $3$-cocycle on $G$. This leads to a complete characterization of finite GK-dimensional Nichols algebras in this category. Specifically, for any finite-dimensional $V\in {_{\k G}^{\k G} {\mathcal{YD}^Φ}}$, we show that the Nichols algebra $B(V)$ has finite Gelfand-Kirillov dimension if and only if it is of diagonal type and its associated root system is finite, that is, an arithmetic root system. Via bosonization, this result yields a classification of finite GK-dimensional coradically graded pointed coquasi-Hopf algebras over finite abelian groups that are generated by group-like and skew-primitive elements. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_12206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups Yang, Yuping Quantum Algebra Rings and Algebras We prove that every pre-Nichols algebra of a nondiagonal object in the twisted Yetter-Drinfeld category ${_{\k G}^{\k G} {\mathcal{YD}^Φ}}$ has infinite Gelfand-Kirillov dimension, where $G$ is a finite abelian group and $Φ$ is a $3$-cocycle on $G$. This leads to a complete characterization of finite GK-dimensional Nichols algebras in this category. Specifically, for any finite-dimensional $V\in {_{\k G}^{\k G} {\mathcal{YD}^Φ}}$, we show that the Nichols algebra $B(V)$ has finite Gelfand-Kirillov dimension if and only if it is of diagonal type and its associated root system is finite, that is, an arithmetic root system. Via bosonization, this result yields a classification of finite GK-dimensional coradically graded pointed coquasi-Hopf algebras over finite abelian groups that are generated by group-like and skew-primitive elements. |
| title | On the classification of finite GK-dimensional pre-Nichols algebras and quasi-quantum groups |
| topic | Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2504.12206 |