Classification of pointed Hopf algebras of dimension $p^2$ over any algebraically closed field

TL;DR

This paper completes the classification of pointed Hopf algebras of dimension p^2 over algebraically closed fields, covering both characteristic p and not p, revealing 14 types including a unique noncommutative, noncocommutative example.

Contribution

It extends the classification of pointed Hopf algebras of dimension p^2 to fields of characteristic p, identifying 14 distinct types including a unique noncommutative, noncocommutative case.

Findings

Classification matches known results for char k ≠ p

Identifies 14 types of algebras in characteristic p

Includes a unique noncommutative, noncocommutative algebra

Abstract

Let $p$ be a prime. We complete the classification on pointed Hopf algebras of dimension $p^2$ over an algebraically closed field $k$. When $\text{char}k \neq p$, our result is the same as the well-known result for $\text{char}k=0$. When $\text{char}k=p$, we obtain 14 types of pointed Hopf algebras of dimension $p^2$, including a unique noncommutative and noncocommutative type.