Sur la théorie des représentations et les algèbres d'opérateurs des produits en couronnes libres

Abstract : In this thesis, we study the combinatorial and operator algebraic properties of certain free compact quantum groups. We prove in chapter 2 that the duals of the quantum reflexion groups have, in most cases, the Haagerup property. In chapter 3, we describe the fusion rules of the free wreath product of a discrete group by the quantum permutation group. To do this, we describe the interrwinner spaces berween certain “basic” corepresentations of these free wreath products in terms of non-crossing partitions decorated by the elements of the group . This provides a whole new class of compact quantum groups whose fusions rules are explicitly computed. We give several applications of this result.We prove that, in most cases, the reduced C*-algebras associates with these free wreath products are simple with unique trace. We also prove that the associated II 1 factors are full. To conclude, we extend the result of chapter 2 to the free wreath products of finite groups by the quantum permutation group.
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Francois Lemeux. Sur la théorie des représentations et les algèbres d'opérateurs des produits en couronnes libres. Algèbres d'opérateurs [math.OA]. Université de Franche-Comté, 2014. Français. ⟨NNT : 2014BESA2008⟩. ⟨tel-01140897⟩

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