Split Extensions, Actions and Crossed Modules in the Categories of Hopf Algebras

In this talk, we will investigate some categorical properties of Hopf algebras. The first part of the talk will be devoted to the category of cocommutative Hopf algebras over a field. The fact that this category is semi-abelian will allow us to give a description of internal crossed modules (as defined by Janelidze). In the second part of the talk, we will study the notion of split extensions of general Hopf algebras. In the category of groups, split extensions have a lot of interesting properties. One of them is the fact that the category of split extensions is equivalent to the category of group actions. Unfortunately, this does not hold in any category, for example, the category of monoids does not have this property. Nevertheless, D. Bourn, A. Montoli, N. Martins-Ferreira and M. Sobral proved that there exists such an equivalence if the split extensions of monoids are « Schreier extensions ». We will answer the question: Which conditions on the split extensions of Hopf algebras do we need to have an equivalence with the category of actions of Hopf algebras?