Isomorphism Types of Hopf Algebras in a Class of Abelian Extensions.I

TL;DR

This paper develops a systematic method for classifying isomorphism classes of certain Hopf algebras that are extensions of group algebras by dual group algebras, focusing on abelian extensions involving cyclic groups of prime order.

Contribution

It introduces a general classification procedure for Hopf algebra extensions of the form  C_p by ^G, specifically for finite abelian p-groups, and applies it to count isomorphism classes of low-dimensional cases.

Findings

Classified isomorphism types of Hopf algebra extensions for specific cases.

Calculated the number of isoclasses for commutative extensions of dimension  ^G with dimension  ^G  p^4.

Provided a systematic approach to classify these Hopf algebras.

Abstract

There is no systematic general procedure by which isomorphism classes of Hopf algebras that are extensions of $\k F$ by ${\k}^G$ can be found. We develop the general procedure for classification of isomorphism classes of Hopf algebras which are extensions of the group algebra $\k C_p$ by ${\k}^G$ where $C_p$ is a cyclic group of prime order $p$ and ${\k}^G$ is the Hopf algebra dual of $\k G$, $G$ a finite abelian $p$-group and $\k$ is an algebraically closed field of characteristic $0$. We apply the method to calculate the number of isoclasses of commutative extensions and certain extensions of this kind of dimension $\le p^4$.