On Hopf algebras of dimension $p^n$ in characteristic $p$

TL;DR

This paper classifies and analyzes the structure of Hopf algebras of dimension p^n over an algebraically closed field of characteristic p, revealing their pointedness, extensions, and unique properties of Radford algebras.

Contribution

It provides a classification of p^2-dimensional Hopf algebras over fields of characteristic p, including characterizations of Radford algebras and extension structures.

Findings

Hopf algebras of dimension p^2 are pointed or basic for p ≤ 5

Radford algebra R(p) is the unique nontrivial extension of certain group algebras

All extensions of p-dimensional Hopf algebras are pointed

Abstract

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$. We study the general structures of $p^n$-dimensional Hopf algebras over $\Bbbk$ with $p^{n-1}$ group-like elements or a primitive element generating a $p^{n-1}$-dimensional Hopf subalgebra. As applications, we have proved that Hopf algebras of dimension $p^2$ over $\Bbbk$ are pointed or basic for $p \le 5$, and provided a list of characterizations of the Radford algebra $R(p)$. In particular, $R(p)$ is the unique nontrivial extension of $\Bbbk[C_p]^*$ by $\Bbbk[C_p]$, where $C_p$ is the cyclic group of order $p$. In addition, we have proved a vanishing theorem for some 2nd Sweedler cohomology group and investigated the extensions of $p$-dimensional Hopf algebras. All these extensions have been identified and shown to be pointed.