# Copointed Hopf algebras over $\mathbb{S}_4$

## TL;DR

This paper classifies finite-dimensional copointed Hopf algebras over the symmetric group S4 by analyzing braided vector spaces of rack type as Yetter-Drinfeld modules and computing their liftings.

## Contribution

It provides a complete classification of copointed Hopf algebras over S4 using a novel application of liftings of braided vector spaces as Yetter-Drinfeld modules.

## Key findings

- Classification of all finite-dimensional copointed Hopf algebras over S4
- Explicit descriptions of liftings of braided vector spaces of rack type
- Application of a systematic strategy to compute liftings in this context

## Abstract

We study the realizations of certain braided vector spaces of rack type as Yetter-Drinfeld modules over a cosemisimple Hopf algebra $H$. We apply the strategy developed in arXiv:1212.5279 to compute their liftings and use these results to obtain the classification of finite-dimensional copointed Hopf algebras over $\mathbb{S}_4$.