Split extension classifiers in the category of cocommutative Hopf algebras

TL;DR

This paper characterizes split extension classifiers and explores the notions of centralizer and center within the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero, linking categorical and algebraic concepts.

Contribution

It provides a detailed description of split extension classifiers and establishes the equivalence of categorical and algebraic notions of center in cocommutative Hopf algebras.

Findings

Categorical center coincides with the algebraic center in cocommutative Hopf algebras.

Explicit description of split extension classifiers in the semi-abelian category.

Analysis of centralizer and center notions within the categorical framework.

Abstract

We describe the split extension classifiers in the semi-abelian category of cocommutative Hopf algebras over an algebraically closed field of characteristic zero. The categorical notions of centralizer and of center in the category of cocommutative Hopf algebras is then explored. We show that the categorical notion of center coincides with the one that is considered in the theory of general Hopf algebras.