Commutative Hopf structures over a loop

TL;DR

This paper classifies all finite-dimensional commutative Hopf algebras over subcoalgebras of a loop's path coalgebra in characteristic p, and as a result, classifies certain one-dimensional infinitesimal groups.

Contribution

It provides a complete classification of finite-dimensional commutative Hopf algebras over subcoalgebras of a loop's path coalgebra and classifies one-dimensional infinitesimal groups.

Findings

All such Hopf algebras are classified.

All commutative infinitesimal groups with Lie dimension 1 are classified.

The classification is over an algebraically closed field of characteristic p > 0.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the commutative infinitesimal groups $\mathcal{G}$ with dim$_{k}$Lie$(\mathcal{G})=1$ are classified.