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# On Semisimple Hopf Algebras of Dimension $pq^n$

Published:2014-02-10
Printed: Jun 2014
• Li Dai,
College of Engineering, Nanjing Agricultural University, Nanjing 210031, Jiangsu, China
• Jingcheng Dong,
College of Engineering, Nanjing Agricultural University, Nanjing 210031, Jiangsu, China
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## Abstract

Let $p,q$ be prime numbers with $p^2\lt q$, $n\in \mathbb{N}$, and $H$ a semisimple Hopf algebra of dimension $pq^n$ over an algebraically closed field of characteristic $0$. This paper proves that $H$ must possess one of the following structures: (1) $H$ is semisolvable; (2) $H$ is a Radford biproduct $R\# kG$, where $kG$ is the group algebra of group $G$ of order $p$, and $R$ is a semisimple Yetter--Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^n$.
 Keywords: semisimple Hopf algebra, semisolvability, Radford biproduct, Drinfeld double
 MSC Classifications: 16W30 - Coalgebras, bialgebras, Hopf algebras (See also 16S40, 57T05); rings, modules, etc. on which these act

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