A semi-abelian extension of a theorem by Takeuchi

TL;DR

This paper establishes that the category of cocommutative Hopf algebras over a field is semi-abelian, extending previous results and providing new insights into their categorical structure and related concepts.

Contribution

It proves the semi-abelian nature of cocommutative Hopf algebras over a field, generalizing earlier characteristic-zero results and exploring categorical properties like action representability.

Findings

Category of cocommutative Hopf algebras is semi-abelian

Category is action representable

Equivalence of two definitions of crossed modules

Abstract

We prove that the category of cocommutative Hopf algebras over a field is a semi-abelian category. This result extends a previous special case of it, based on the Milnor-Moore theorem, where the field was assumed to have zero characteristic. Takeuchi's theorem asserting that the category of commutative and cocommutative Hopf algebras over a field is abelian immediately follows from this new observation. We also prove that the category of cocommutative Hopf algebras over a field is action representable. We make some new observations concerning the categorical commutator of normal Hopf subalgebras, and this leads to the proof that two definitions of crossed modules of cocommutative Hopf algebras are equivalent in this context.