Hopf algebras of prime dimension in positive characteristic

TL;DR

This paper classifies Hopf algebras of prime dimension over fields of positive characteristic, showing they are either group algebras or restricted universal enveloping algebras under certain conditions.

Contribution

It proves the structure of prime-dimensional Hopf algebras in positive characteristic and identifies conditions for commutativity and cocommutativity.

Findings

Hopf algebra of prime dimension p in characteristic p is either a group algebra or a restricted universal enveloping algebra.

Hopf algebra of prime dimension p over a field of characteristic q>0 is commutative and cocommutative when q=2 or p<4q.

Open problem remains for q>2 and p/4<p in positive characteristic.

Abstract

We prove that a Hopf algebra of prime dimension $p$ over an algebraically closed field, whose characteristic is equal to $p$, is either a group algebra or a restricted universal enveloping algebra. Moreover, we show that any Hopf algebra of prime dimension $p$ over a field of characteristic $q>0$ is commutative and cocommutative when $q=2$ or $p<4q$. This problem remains open in positive characteristic when $2<q<p/4$.