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

TL;DR

This paper establishes the structural classification of semisimple Hopf algebras with dimensions involving prime squares, specifically for dimensions $p^2q^2$, $9p^2$, and $25q^2$, over an algebraically closed field of characteristic zero.

Contribution

It provides new structure theorems for semisimple Hopf algebras of specific composite dimensions involving prime squares, extending previous classifications.

Findings

Classified semisimple Hopf algebras of dimension $p^2q^2$ with $p^2<q$

Derived structure theorems for dimensions $9p^2$ and $25q^2$

Extended understanding of semisimple Hopf algebra structures in low dimensions

Abstract

Let $k$ be an algebraically closed field of characteristic $0$. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $p^2q^2$ over $k$, where $p,q$ are prime numbers with $p^2<q$. As an application, we also obtain the structure theorems for semisimple Hopf algebras of dimension $9p^2$ and $25q^2$ for all primes $3\neq p$ and $5\neq q$.