Classification of connected Hopf algebras of dimension $p^3$ I

TL;DR

This paper classifies connected Hopf algebras of dimension p^3 over an algebraically closed field of characteristic p, detailing generators, relations, and structures, with special cases for primitive spaces and parameters.

Contribution

It provides a comprehensive classification of such Hopf algebras, including parameterized families, except for a specific primitive space case.

Findings

Classification of connected Hopf algebras of dimension p^3

Description of isomorphism classes via generators and relations

Parameterization of infinite families when primitive space is one-dimensional

Abstract

Let $p$ be a prime, $k$ be an algebraically closed field of characteristic $p$. In this paper, we provide the classification of connected Hopf algebras of dimension $p^3$, except the case when the primitive space of the Hopf algebra is two dimensional and abelian. Each isomorphism class is presented by generators $x, y, z$ with relations and Hopf algebra structures. Let $\mu$ be the multiplicative group of $(p^2+p-1)$-th roots of unity. When the primitive space is one-dimensional and $p$ is odd, there is an infinite family of isomorphism classes, which is naturally parameterized by $A_{k}^1/\mu$.