Abstract. We consider the hypercube in R n, and show that its quantum symmetry group is a q-deformation of On at q = −1. Then we consider the graph formed by n segments, and show that its quantum symmetry group is free in some natural sense.

Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law.

Abstract. To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not one-to-one, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the construction: compatibility to product operations, characterization of matrices which give usual groups, explicit computations for small matrices.

Abstract. This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments. 1.

Abstract. We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If K is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra K n: this is a refinement of Wang’s universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-ergodic coaction on the diagonal algebra K n, on which we determine the possible group gradings when K is algebraically closed and has characteristic zero. 1.

Abstract. We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group Q4. Our main tool is a new presentation for the algebra C(Q4), corresponding to an isomorphism of type Q4 ≃ SO−1(3). The quantum subgroups of Q4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants. 1.

Abstract. We compute the spectral measure of the standard generators uij of the Wang algebra Ao(n). We show in particular that this measure has support [−2 / √ n + 2, 2 / √ n + 2], and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra Ao(F), a representation of SU q 2 due to Woronowicz, and several calculations with orthogonal polynomials.

Abstract. We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π: As(n) → B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n = 6.

Abstract. We introduce and study a remarkable family of real probability measures πst, that we call free Bessel laws. These are related to the free Poisson law π via the formulae πs1 = π ⊠s and π1t = π ⊞t. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.