Hopf formulae for cocommutative Hopf algebras

In recent years, many new applications of categorical Galois theory have emerged in various interesting non-abelian algebraic contexts. In particular, in the realm of semi-abelian categories [1], 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 [2], among many others. These categories share many structural properties with the categories of groups and of Lie algebras. The category of cocommutative Hopf algebras over a field is also semi-abelian [3], raising the natural question of whether some similar homological methods can be applied to study such structures as well.

In this talk, after reviewing some fundamental properties of semi-abelian categories and some motivating examples, I’ll then explain that the answer to the above question is affirmative. Indeed, the exactness properties of cocommutative Hopf algebras together with the existence of the celebrated Takeuchi’s free functor described in [4] - universally associating a Hopf algebra with any coalgebra - make it possible to establish some new Hopf-type formulae for the homology of cocommutative Hopf algebras [5]. Note that in this work an important role is actually played by the cleft extensions, namely by those surjective morphisms of Hopf algebras that are split as coalgebra morphisms. Cleft extensions satisfy all the needed properties in order to build a weak universal central extension of any given cocommutative Hopf algebra. Moreover, 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 [6], which are interesting structures that naturally occur in the study of the solutions of the so-called quantum Yang-Baxter equation. The category of cocommutative Hopf braces turns out to be both semi-abelian and strongly protomodular [7]. It is also monadic on the category of coalgebras [8], so that it is possible to investigate it from the perspective of non-abelian homological algebra.