A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909636913790976 |
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| author | Yu, Jing Liu, Gongxiang Zhou, Kun Zhen, Xiangjun |
| author_facet | Yu, Jing Liu, Gongxiang Zhou, Kun Zhen, Xiangjun |
| contents | In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^τ\#_σ\Bbbk F$ constructed through abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for some (infinite) group $F$ and finite group $G$ is cosemisimple, and discuss when it admits a compact quantum group structure if $\Bbbk$ is the field of complex numbers $\mathbb{C}.$ We also find all the simple $\Bbbk^G{}^τ\#_σ\Bbbk F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right $\Bbbk^G{}^τ\#_σ\Bbbk F$-comodules. Moreover, some new properties are given and some new examples are constructed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04008 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions Yu, Jing Liu, Gongxiang Zhou, Kun Zhen, Xiangjun Quantum Algebra Representation Theory 16T05, 16T15 In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^τ\#_σ\Bbbk F$ constructed through abelian extensions of $\Bbbk F$ by $\Bbbk^G$ for some (infinite) group $F$ and finite group $G$ is cosemisimple, and discuss when it admits a compact quantum group structure if $\Bbbk$ is the field of complex numbers $\mathbb{C}.$ We also find all the simple $\Bbbk^G{}^τ\#_σ\Bbbk F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right $\Bbbk^G{}^τ\#_σ\Bbbk F$-comodules. Moreover, some new properties are given and some new examples are constructed. |
| title | A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions |
| topic | Quantum Algebra Representation Theory 16T05, 16T15 |
| url | https://arxiv.org/abs/2506.04008 |