Le produit en couronne libre d'un groupe quantique compact par un groupe quantique d'automorphismes

摘要

In this thesis, we define and study the free wreath product of a compact quantum group by a quantum automorphism group and, in this way, we generalize the previous notion of free wreath product by the quantum symmetric group introduced by Bichon.Our investigation is divided into two part. In the first, we define the free wreath product of a discrete group by a quantum automorphism group. We show how to describe its intertwiners by making use of decorated noncrossing partitions and from this, thanks to a result of Lemeux, we deduce the irreducible representations and the fusion rules. Then, we prove some properties of the operator algebras associated to this compact quantum group, such as the simplicity of the reduced C*-algebra and the Haagerup property of the von Neumann algebra.The second part is a generalization of the first one. We start by defining the notion of free wreath product of a compact quantum group by a quantum automorphism group. We generalize the description of the spaces of the intertwiners obtained in the discrete case and, by adapting a monoidal equivalence result of Lemeux and Tarrago, we find the irreducible representations and the fusion rules. Then, we prove some stability properties of the free wreath product operation. In particular, we find under which conditions two free wreath products are monoidally equivalent or have isomorphic fusion semirings. We also establish some analytic and algebraic properties of the dual quantum group and of the operator algebras associated to a free wreath product. As a last result, we prove that the free wreath product of two quantum automorphism groups can be seen as the quotient of a suitable quantum automorphism group.