### Research group:

The Foundations of Peter-Weyl-Galois Theory

#### 5 - 9 March, 2012, Warsaw

Let F be a field, G a finite group, and H the Hopf algebra of all set-theoretic maps from G to F. If E is a finite field extension of F and G is its Galois group, the extension is Galois if and only if the canonical map from the tensor product over F of E with itself to the tensor product over F of E with H (resulting from viewing E as an H-comodule) is an isomorphism. Similarly, a finite covering space is regular if and only if the analogous canonical map is an isomorphism. We extend this point of view to actions of compact quantum groups on unital C*-algebras. We prove that such an action is free if and only if the canonical map (obtained using the underlying Hopf algebra of the compact quantum group) is an isomorphism. In particular, we are able to express the freeness of a compact Hausdorff group action on a compact Hausdorff space in algebraic terms. This formulation of the freeness of such actions allows us to generalize the associated vector bundle construction to the case of actions whose field of isotropy groups is continuous.

### Organizer

- Piotr M. Hajac

### Programme

**MONDAY, 05th OF MARCH, room 322, 3rd floor of IMPAN**

**10:15-12:00** Kenny De Commer *(Universite de Cergy-Pontoise, France)*

A FINITENESS CONDITION FOR COMPACT QUANTUM GROUP ACTIONS

By a theorem due to Hoegh-Krohn, Landstad and Stormer, the isotypical components of an ergodic action of a compact group on a unital C*-algebra are finite dimensional. This was later generalized to compact quantum groups by F. Boca. On the other hand, since recently we know that the isotypical components of a free action of a compact quantum group on a unital C*-algebra are finitely-generated modules over the fixed-point subalgebra. In this talk, we give several characterizations, in terms of Galois maps, of actions for which such a finite generation property holds in the general setting of compact quantum groups acting on unital C*algebras. In the classical case, they are precisely the actions whose field of isotropy groups is continuous. This is joint work with M. Yamashita, and a joint project with P. F. Baum and P. M. Hajac.

**14:15-16:00** Kenny De Commer *(Universite de Cergy-Pontoise, France)*

GROUP-DUAL SUBGROUPS OF COMPACT QUANTUM GROUPS

There are two special classes of compact quantum groups, namely the classical compact groups and the Fourier duals of discreet groups. In this talk, we discuss what can be said about the discrete group duals sitting inside a given compact quantum group. We will consider several examples such as Wang's quantum automorphism groups and Goswami's quantum isometry groups of connected compact Riemannian manifolds. This is joint work with T. Banica and J. Bhowmick.

**THURSDAY, 08th OF MARCH, Warsaw, 74 Hoża street, 5th floor**

**10:15-12:00** Paul F. Baum *(Pennsylvania State/ IMPAN)*

THE PETER-WEYL-GALOIS-THEOREM FOR COMPACT PRINCIPAL BUNDLES

**13:15-15:00** Paul F. Baum *(Pennsylvania State/ IMPAN)*

CYCLES FOR K-HOMOLOGY AND EXACT CROSSED PRODUCTS

### Participants

- Paul F. Baum (Pennsylvania State/ IMPAN)
- Kenny De Commer (University de Cergy-Pontoise, France)
- Piotr M. Hajac (IMPAN/University of Warsaw)
- Tomasz Maszczyk (IMPAN/ University of Warsaw)
- Jan Rudnik (IMPAN)
- Andrzej Sitarz (IMPAN/ Jagiellonian University)