A note on split extensions of 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.

Reduced the context to algebraically closed fields

References in corpus (2)

• A torsion theory in the category of cocommutative Hopf algebras
• Two characterisations of groups amongst monoids

Cited by in corpus (4)

• A semi-abelian extension of a theorem by Takeuchi
• Two characterisations of groups amongst monoids
• Split extension classifiers in the category of cocommutative Hopf algebras
• A universal Kaluzhnin--Krasner embedding theorem