State of the art¶

Both categorical Galois theory and Hopf algebras are relatively new disciplines of mathematics. Categorical approach towards Galois theory was initiated by Alexander Grothendieck: he formulated a generalisation of classical Galois theory of field extensions. Grothendieck showed more than merely a bijection between posets of subextensions and subgroups, but rather an equivalence of categories between split algebras and the category of G-sets over the Galois group. In this form it resembles the theory of covering spaces: a categorical equivalence between the category of covering spaces of a suitable topological space and the category of G-sets, where the place of Galois group is played by the fundamental group. Deep similarities between the proofs of both theories let to further developments and new ideas. Later in the 70’s A.R.Magid constructed another important example of a categorical Galois theory, this time for commutative rings. It is based on Stone duality and the Pierce spectrum functor (the Stone space of the boolean algebra of idempotent elements of a commutative ring). In this case the corresponding Galois object is a groupoid. The right hand side category of the equivalence generalises G-sets to (internal) presheaves over this groupoid. It is worth mentioning that in the case of field extension the Galois-groupoid boils down to the Galois group and the equivalence reduces to the Grothendieck–Galois case. In the beginning of 90’s George Janelidze came up with a purely categorical theory which embraces all of the above examples. But also includes new ones: for example the equivalence between the category of central extension of a commutative group B and the comma category over the second homology group H₂(B,Z) (when B is perfect, i.e. B=[B,B]). In this research plan we want to find new examples of the categorical Galois theory coming from different origin: Hopf algebras. We believe that the new example will be of a great importance for both Category Theory and the theory of Hopf Algebras.

The other direction of study that I want to propose, follows the path I have started in my PhD thesis, where I constructed a Galois connection between subalgebras of a comodule algebra over a Hopf algebra H and generalised quotients of H. The Hopf algebra community seemed to be seeking this result for last 20 years, though the roots of Galois theory within Hopf algebra goes back to 60’s and the work of Chase Sweedler. In the beginning of 90’s in the work of Zhang and Oystaeyen the Galois correspondence for the commutative case emerged, then Peter Schauenburg pursued the subject and generalised the Oystaeyen–Zhang construction to non commutative comodule algebras whose coinvariants subalgebra is equal to the commutative base ring. Let us note that both Oystaeyen–Zhang and Schauenburg constructed additional Hopf algebra, closely related to the H-comodule algebra and where considering Galois correspondence between subalgebras of the comodule algebra and generalised quotients of the constructed Hopf algebra rather than H. In my PhD I showed how to build a Galois connection directly without the need of an additional Hopf algebra. My construction was not only innovative and original but it made possible to prove a generalised version of Schauenburg theorem on closedness, as well as to give new interesting results. Finally, applying my results to the Hopf algebra H itself, as it is a basic example of an H-comodule algebra itself (the way a group G is an example of a G-set, in fact the free object with one generator). My techniques allowed for a new simple proof of Takeuchi correspondence between generalised Hopf subalgebras and generalised quotients of H when H is finite dimensional. Bijectivity of Takeuchi correspondence was also proved in 2007 by Serge Skryabin with another approach. We should also mention that he proved a final corollary which fills a missing brick in a old and well known way of proving this theorem, while I have propose an entierly new proof. In this proposal we want to generalised the statements of my PhD to the theory of bialgebroids (Hopf algebroids). The theory of Hopf algebroids is an established field through research done by many Hopf algebraists including Gabriella Böhm, Tomasz Brzeziński, Peter Schauenburg, Gigel Militaru, K. Szachanányi, Lars Kadison (University of Porto) and many others. In particular there is a very important paper: ‘Cleft extensions of Hopf algebroids’ by Gabriella Böhm and Tomasz Brzeziński (published in Applied Categorical Structures, 2006), which indicates that the aim of this project is feasible and should bring new significant results. As I showed in my PhD thesis, the Galois theory of cleft extension has a very interesting structure and one can classify closed elements on both sides of the Galois connection. We want to generalise this result to cleft extensions of bialgebroids/Hopf algebroids.