On classification of finite-dimensional semisimple Hopf algebras

TL;DR

This paper classifies certain finite-dimensional semisimple Hopf algebras with abelian group of grouplikes of prime index, providing explicit classifications for those of dimension p^4 over an algebraically closed field of characteristic zero.

Contribution

It develops a classification mechanism for semisimple Hopf algebras with abelian grouplike groups of prime index and explicitly classifies those of dimension p^4 for odd primes.

Findings

Explicit classification for Hopf algebras of dimension p^4

Description of second cohomology groups for specific extensions

Structural insights into semisimple Hopf algebras with prime index abelian groups

Abstract

We develop a mechanism for classication of isomorphism types of non-trivial semisimple Hopf algebras whose group of grouplikes $G(H)$ is abelian of prime index $p$ which is the smallest prime divisor of $|G(H)|$. We describe structure of the second cohomology group of extensions of $\k C_p$ by $\k^G$ where $C_p$ is a cyclic group of order $p$ and $G$ a finite abelian group. We carry out an explicit classification for Hopf algebras of this kind of dimension $p^4$ for any odd prime $p$. The ground field is algebraically closed of characteristic $0$.