# Pointed Hopf algebras of dimension $p^2q$ in characteristic $p$

## TL;DR

This paper classifies all pointed Hopf algebras over an algebraically closed field of characteristic p with dimension p^2q, revealing finitely many classes including new examples not generated by group-like or skew-primitive elements.

## Contribution

It provides a complete classification of pointed Hopf algebras of dimension p^2q in characteristic p, identifying new examples and the structure of isomorphism classes.

## Key findings

- Finitely many isomorphism classes of such Hopf algebras.
- 10 classes are not generated by group-like or skew-primitive elements.
- Discovery of many new finite-dimensional pointed Hopf algebras.

## Abstract

Let $\mathds{k}$ be an algebraically closed field of characteristic $p$. We give the complete classification of pointed Hopf algebras over $\mathds{k}$ of dimension $p^2q$ for a prime number $q$. The result shows that there are finitely many isomorphism classes, including 10 classes that are not generated by group-like elements and skew-primitive elements. In particular, there are many new examples of finite-dimensional pointed Hopf algebras.