On pointed Hopf algebras associated with alternating and dihedral groups

TL;DR

This paper classifies finite-dimensional pointed Hopf algebras related to alternating and dihedral groups, revealing conditions under which these algebras are finite or infinite-dimensional, and expanding understanding of their structures.

Contribution

It provides a classification of finite-dimensional pointed Hopf algebras over A_5 and analyzes the infinitesimal braiding in relation to conjugacy classes in alternating and dihedral groups.

Findings

Finite-dimensional pointed Hopf algebras over A_5 are classified.

Pointed Hopf algebras with certain conjugacy classes are infinite-dimensional.

Conditions for finite-dimensionality depend on the order of elements in the groups.

Abstract

We classify finite-dimensional complex pointed Hopf algebra with group of group-like elements isomorphic to A_5. We show that any pointed Hopf algebra with infinitesimal braiding associated with the conjugacy class of $\pi$ \in $A_n$ is infinite-dimensional if the order of $\pi$ is odd except for $\pi=(1 2 3)$ in $A_4$. We also study pointed Hopf algebras over the dihedral groups.