Hopf formulas for cocommutative Hopf algebras

In recent years, numerous new applications of categorical Galois theory have emerged in various interesting non-abelian algebraic contexts. In particular, within semi-abelian categories, this approach has led to some new calculations of higher fundamental groups in terms of generalized commutators in categories such as that of compact groups, crossed modules, and skew braces. These categories share some structural properties with the categories of groups and of Lie algebras, and also with the category of cocommutative Hopf algebras over a field, which is also semi-abelian.
This raises the natural question of whether similar homological methods can be applied to study cocommutative Hopf algebras as well.

In this talk, after reviewing some fundamental properties of semi-abelian categories and some motivating examples, I will explain that the answer to the above question is affirmative. By using the exactness properties of cocommutative Hopf algebras and the free functor universally associating a Hopf algebra with any coalgebra it is possible to establish some new Hopf-type formulae for the homology of cocommutative Hopf algebras. An important role is played by cleft extensions, namely those surjective morphisms of Hopf algebras that are split as coalgebra morphisms.
With any cleft extension, one can associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory. This new approach can also be applied to investigate the homology of cocommutative Hopf braces, which are interesting structures that naturally occur in the study of solutions to the so-called quantum Yang-Baxter equation. The category of cocommutative Hopf braces turns out to be both semi-abelian and monadic on the category of coalgebras, so that it is possible to investigate it from the perspective of non-abelian homological algebra.

This talk is based on a joint work with Andrea Sciandra.