标题： 一类由可换扩张构造的（无限维）余半单Hopf代数 显示英文标题

标题： A class of (infinite-dimensional) cosemisimple Hopf algebras constructed via abelian extensions

摘要： 本文旨在研究某些无限群的 Abel 扩张。 我们证明了对于某些（无限）群 $F$ 和有限群 $G$，通过 $\Bbbk^G$ 对 $\Bbbk F$ 进行阿贝尔扩张而构造出的 Hopf 代数 $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ 是余半单的，并且讨论了当 $\Bbbk$ 是复数域 $\mathbb{C}$ 时它何时允许紧量子群结构。我们还找到了所有简单的 $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-余模，并尝试确定有限维右 $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-余模类别的 Grothendieck 环。 此外，给出了一些新的性质并构造了一些新的实例。 显示英文摘要

摘要： In this paper, we aim to study abelian extensions for some infinite group. We show that the Hopf algebra $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$ 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 $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules and attempt to determine the Grothendieck ring of the category of finite-dimensional right $\Bbbk^G{}^\tau\#_{\sigma}\Bbbk F$-comodules. Moreover, some new properties are given and some new examples are constructed.