Commutators and crossed modules of color Hopf algebras

Abstract

In a previous paper we showed that the category of cocommutative color Hopf algebras is semi-abelian in case the group $G$ is abelian and finitely generated and the characteristic of the base field is different from 2 (not needed if $G$ is finite of odd cardinality). Here we describe the commutator of cocommutative color Hopf algebras and we explain the Hall's criterion for nilpotence and the Zassenhaus Lemma. Furthermore, we introduce the category of color Hopf crossed modules and we explicitly show that this is equivalent to the category of internal crossed modules in the category of cocommutative color Hopf algebras and to the category of simplicial cocommutative color Hopf algebras with Moore complex of length 1.