Advances in Hopf Algebroids

On the gauge groups and bialgebroids

For starters, we will recall the fundamental concept of a Hopf-Galois extension, and instantiate it through quantum principal SU(2)-bundles with noncommutative seven-spheres as total spaces and noncommutative four-spheres as base spaces, together with monopole bundle over the quantum sphere.

Then we will recall the construction of the Ehresmann-Schauenburg bialgebroid of a Hopf-Galois extension, which is a noncommutative analogue of the Ehresmann groupoid of a classical principal bundle. Next, we will show that, the gauge group of any Hopf Galois extension is isomorphic to the group of bisections of its Ehresmann-Schauenburg bialgebroid. Then, we will prove that the group of bisections and the group of automorphisms of the bialgebroid form a crossed module. In particular, we will consider and Galois objects (non-trivial noncommutative principal bundles over a point), whose base-space subalgebra is the ground field and the corresponding Ehresmann-Schauenburg bialgebroid becomes a Hopf algebra. Examples will include monopole bundle and Galois objects of Taft algebras.

A noncommutative calculus on the cyclic dual of Ext

We show that if the cochain complex computing Ext groups in the category of modules over a Hopf algebroid admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products.

Presentation

From homology theories to Hopf algebroids - and back!

An important source of examples of Hopf algebroids is a construction due to Adams which starts with a suitable homology theory and constructs a Hopf algebroid, together with a lift of the homology theory to the category of comodules of that Hopf algebroid.

This construction only works if the homology theory satisfies a certain flatness condition. Pstragowski has found a way to describe the category of comodules as a category of additive sheaves on a site, and this construction works in much greater generality. Moreover, if one can show by abstract means that the resulting category of sheaves is equivalent to the category of comodules of some Hopf algebroid, then one obtains a new homology theory which does satisfy the flatness condition.

This motivates the following recognition question: when is a category equivalent to the category of comodules of a Hopf algebroid? I will explain some general recognition results and how they can be applied to the category constructed by Pstragowski.

FRT type construction of Hopf algebroids

The Faddeev-Reshetikhin-Takhtajan (FRT) construction associates a coquasitriangular bi- or Hopf algebra to a solution of the Yang-Baxter equation. There are various generalizations of this construction. What is important in this talk among them is Hayashi's generalization. He established a way of constructing a weak Hopf algebra from a Yang-Baxter face model, which is, mathematically, a braided object in the category of bimodules over the finite product of the base field. Being motivated by his work, and also hoping applications to dynamical quantum groups, I investigated category-theoretic aspects of the FRT construction. In this talk, I will introduce a way to construct a bialgebroid over an algebra A from a strong monoidal functor from a small monoidal category, D, to the category of rigid A-bimodules. The resulting bialgebroid, B, may be called the FRT bialgebroid over A since the category of B-comodules is lax braided if D is. Furthermore, it turns out that B is a Hopf algebroid if the monoidal category D is rigid. The construction and the proof of these results are based on the fact that a bialgebroid over A is identified with a left adjoint bimonad on the category of A-bimodules. The FRT bialgebroid is, in fact, obtained as a special case of a Tannaka theoretic construction of coend bimonads.

Twisting Hopf algebroids and beyond

Xu has introduced Drinfeld twists for associative bialgebroids.

General proof that appropriate formulas twist the antipode as well has been missing for 20 years.

I shall sketch my result from 2019 which resolves the question if the antipode is invertible. Then I will discuss twisting examples for Hopf algebroids of differential operators and a relation of these examples to multiplicative unitaries.

Proper treatment requires extensions of the formalism to internal Hopf algebroids in monoidal categories with completed tensor products (after works of Böhm and Stojić). Further examples suggest going beyond Hopf algebroids to a nonassociative version or other examples toward nonaffine quantum groupoids, which are both a work in progress.