Primitively generated Hopf orders in characteristic $p$

TL;DR

This paper classifies all Hopf orders over a characteristic p discrete valuation ring for certain commutative, cocommutative Hopf algebras generated by primitive elements, providing explicit examples in specific cases.

Contribution

It constructs all Hopf orders in a given class of Hopf algebras over characteristic p rings, linking them to solutions of a matrix equation.

Findings

All Hopf orders correspond to solutions of a single matrix equation.

Explicit examples of Hopf orders are provided for rank p^2 Hopf algebras.

Complete characterization of Hopf orders in the specified setting.

Abstract

Let $R$ be a characteristic $p$ discrete valuation ring with field of fractions $K$. Let $H$ be a commutative, cocommutative $K$-Hopf algebra of $p$-power rank which is generated as a $K$-algebra by primitive elements. We construct all of the $R$-Hopf orders of $H$ in $K$; each Hopf order corresponds to a solution to a single matrix equation. For $R$ complete, we give explicit examples of Hopf orders in some rank $p^2$ $K$-Hopf algebras.