A note on split extensions of bialgebras

TL;DR

This paper characterizes Hopf algebras within cocommutative bialgebras by a universal property related to split extensions and join decompositions, highlighting limitations in non-cocommutative cases.

Contribution

It provides a universal characterization of Hopf algebras among cocommutative bialgebras based on split extension properties.

Findings

Hopf algebras are characterized by split extension join decompositions

The characterization is specific to cocommutative bialgebras

The result does not extend to non-cocommutative bialgebras

Abstract

We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.