Structure of semisimple Hopf algebras of dimension $p^2q^2$

TL;DR

This paper classifies semisimple Hopf algebras of dimension p^2q^2, showing they are constructed from group algebras, their duals, or Radford biproducts involving Yetter-Drinfeld Hopf algebras, with specific results for dimension 4q^2.

Contribution

It provides a complete structural classification of semisimple Hopf algebras of dimension p^2q^2 under certain conditions, including explicit construction methods.

Findings

Classification of semisimple Hopf algebras of dimension p^2q^2

Construction from group algebras, duals, or Radford biproducts

Special case analysis for dimension 4q^2

Abstract

Let $p,q$ be prime numbers with $p^4<q$, and $k$ an algebraically closed field of characteristic 0. We show that semisimple Hopf algebras of dimension $p^2q^2$ can be constructed either from group algebras and their duals by means of extensions, or from Radford biproduct $R#kG$, where $kG$ is the group algebra of group $G$ of order $p^2$, $R$ is a semisimple Yetter-Drinfeld Hopf algebra in ${}^{kG}_{kG}\mathcal{YD}$ of dimension $q^2$. As an application, the special case that the structure of semisimple Hopf algebras of dimension $4q^2$ is given.