Description du projet
La conjecture de Baum-Connes pour les groupes quantiques discrets présentant des phénomènes de torsion
La conjecture de Baum-Connes suggère que la K-théorie de la C*-algèbre réduite d’un groupe est identique à la K-homologie équivariante d’une certaine sorte d’espace de classification du groupe. Meyer et Nest ont reformulé cette conjecture en utilisant le langage des catégories triangulées. Cette reformulation fonctionne à la fois pour les groupes classiques localement compacts et pour les groupes quantiques discrets sans torsion. Le projet CONCOQUANT, financé par l’UE, entend répondre à plusieurs questions clés liées aux phénomènes de torsion dans les groupes quantiques discrets. Le projet étudiera également comment obtenir les propriétés de stabilité de Baum-Connes pour les constructions pertinentes de groupes quantiques, les produits quantiques semi-directs et les produits en couronne libre.
Objectif
This project focuses on the Baum-Connes conjecture formulation for discrete quantum groups. The work of R. Meyer and R. Nest in the second half of 2000's has lead to a categorial formulation of the Baum-Connes conjecture in the context of triangulated categories. This reformulation works for both classical locally compact groups and torsion-free discrete quantum groups. Thus one of the main questions that the project aims to understand is the torsion phenomena for discrete quantum groups in relation with the categorical framework of Meyer-Nest. This will allow to manipulate conveniently the corresponding homological algebra for two main purposes. First, introducing a new insight for a proper formulation of the Baum-Connes conjecture for arbitrary discrete quantum groups. Second, carrying out explicit K-theory computations of C*-algebras defining relevant examples of quantum semi-direct products and free wreath products. The compact bicrossed product construction will be studied in detail in this framework in order to classify its torsion actions and to obtain the corresponding stability result of BC. Moreover, this construction will provide a vast class of new examples satisfying the quantum BC conjecture coming from recent constructions by several authors involving approximation properties such as property (T) or Haagerup property. The project aims also to carry out further developments in the quantum setting. One the one hand, defining and developping a quantum equivariant Künneth formula theory using the notion of Künneth functor. On the other hand, studying the recently discovered connections between compact quantum groups and non-local games, in the framework of quantum information theory, in order to address relevant open questions concerning the Connes' embedding conjecture with potential applications and consequences within the area of algorithm theory in computer science.
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinateur
1165 Kobenhavn
Danemark