Program
Piotr M. Hajac
10:00 - 10:45
Locality for quantum principal bundles
In topology, principal bundles are often assumed to be locally trivial. While the original Cartan definition of a principal bundle is inherently global, and has been successfully promoted to noncommutative geometry in the compact/unital setting, the standard concept of local triviality is hard to implement for quantum principal bundles. The goal of this talk is to review the state of the art of the concept of locality for quantum principal bundles. We begin with a guided tour concerning the topology of compact Hausdorff principal bundles. In particular, the difference between the piecewise and local triviality shall be explained. In the main part of the talk, the local triviality of compact quantum principal bundles will be compared with the piecewise triviality of principal comodule algebras. (This is a review talk based on joint works with many collaborators.)
Stéphane Launois
11:15 - 12:00
Automorphisms and derivations of quantum grassmannians
I will present recent results about automorphisms and derivations of quantum grassmannians. This is based on joint work with Tom Lenagan.
Elisa Ercolessi
12:00 - 12:45
From Quantum Doubles to Error Correction Codes
We will review Kitaev Toric code, enlightening the mathematical structures of its algebra of observables and connecting it to error correction codes used in fault tolerant quantum computation.
Pierre Bieliavsky
14:30 - 15:15
Una passeggiata nel giardino dei gruppi quantistici : smoothness versus measurability
After briefly reviewing certain aspects of the theory of locally compact quantum groups à la Kustermans - Vaes, I will present a simple deformation quantization-based geometrical construction which produces such objects in the context of Fröbenius type groups (in the Lie case, a Lie group is of Fröbenius type if it admits an open coadjoint orbit). I will then show that the theory of symmetric spaces yields smooth analogues of those. I will then comment on their respective behaviour when attending the question of deforming various categories of topological module-algebras. This work is partly joint with Victor Gayral [P.B.,V. Gayral; Mem. AMS 2015] and Victor Gayral, Sergei Neyshveyev and Lars Tuset [P.B.,V. Gayral, S. Neyshveyev, L.Tuset; J. Funct. Analysis 2022 ].
Zoran Škoda
15:15 - 16:00
Torsors for geometrically admissible actions of monoidal categories
In Tannakian formalism, groups and generalizations like groupoids and Hopf algebras can be reconstructed from the fiber functor which is a forgetful strict monoidal functor from its category of modules to the category of vector spaces. Can we do something similar for the actions of groups, their properties and generalizations ? If a Hopf algebra H (generalizing the algebra of functions on a group) coacts on an algebra A by a Hopf action (that is, A becomes an H-comodule algebra) then the category of H-modules acts on the category of A-modules, this action strictly lifts the trivial action of the category of vector spaces on A-modules and also H lifts to a comonoid in H-modules inducing a comonad on the category of A-modules. (This lifted comonad is related to a distributive law; entwining structures appear in some related examples.) The comodules over this comonad are the analogues of H-equivariant sheaves and the Galois condition can be stated in terms of affinity in the sense of Rosenberg. We propose taking these properties as defining for a general framework allowing for the definition of Galois condition/principal bundles/torsors in a number of geometric situations beyond the cases of Hopf algebras coacting on algebras. We also sketch how many other examples like coalgebra-Galois extensions and locally trivial nonaffine noncommutative torsors fit into this framework.
Marvin Dippell
16:00 - 16:45
Hochschild Cohomology for Reduction in Deformation Quantization
I will present a framework for incorporating coisotropic reduction into deformation quantization. To this end we will consider so-called constraint algebras and explore their deformation theory using a modification of Hochschild cohomology. In particular, I will present first results on the way to a Hochschild-Kostant-Rosenberg theorem which is compatible with reduction.