#### Room 326

#### The categorical framework for the Baum-Connes conjecture

#### Ruben Martos

##### University of Paris 7

*The axiomatization of compact quantum groups (in the setting of C*-algebras) made*

by S. L. Woronowicz (in 1987) is being very fruitful since it’s allowed (and it still allows) to

translate a large number of classical results in group theory like the classical Peter-Weyl

theory for representations of compact groups or classical approximation properties

(Haagerup property or amenability). Works of S. Baaj, G. Skandalis and R. Vergnioux show

that we can even define an equivariant KK-theory with respect to a quantum group and

generalize some results in this context.

This is why we can think about a “quantum” formulation of the Baum-Connes

conjecture. For this purpose, the well theoretic framework is the triangulated categories

(firstly studied by Jean-Louis Verdier and A. Grothendieck).

In this talk, we are going to introduce in a simple way the tools used for being able to

formulate the Baum-Connes conjecture in the categories language. More precisely, we’ll

define a triangulated category and some objets of interest, afterwards we’ll translate these

very general constructions to the category KK and we’ll explain how to formulate the Baum-

Connes conjecture in this language. Finally, we’ll do an overview of the current situation of

the conjecture for quantum groups.

Done