On the structure of abelian Hopf algebras

TL;DR

This paper provides a detailed classification of graded, connected, commutative and cocommutative Hopf algebras over a perfect field of characteristic p, focusing on their indecomposable and p-torsion structures.

Contribution

It offers a complete classification of such Hopf algebras, including p-torsion, free, cofree, and indecomposable modulo p cases, with explicit descriptions.

Findings

p-torsion objects are uniquely decomposable into indecomposables

classification of free and cofree Hopf algebras over the field

complete classification of indecomposables modulo p

Abstract

We study the structure of the category of graded, connected, countable-dimensional, commutative and cocommutative Hopf algebras over a perfect field $k$ of characteristic $p$. Every $p$-torsion object in this category is uniquely a direct sum of explicitly given indecomposables. This gives rise to a similar classification of not necessarily $p$-torsion objects that are either free as commutative algebras or cofree as cocommutative coalgebras. We also completely classify those objects that are indecomposable modulo $p$.