In 1987, Woronowicz gave a definition of compact matrix quantum groups generalizing compact Lie groups G ⊆ M n (C) in the setting of noncommutative geometry. About twenty years later, Banica and Speicher isolated a class of compact matrix quantum groups with an intrinsic combinatorial structure. These so called easy quantum groups are determined by categories of partitions. They have been proven useful in order to understand various aspects of quantum groups, in particular linked with Voiculescu's free probability theory. Furthermore, they exhibit a way to find examples of compact quantum groups besides q-deformations and quantum isometry groups. These characteristics naturally motivated attempts to fully classify them. This is completed in the present article.
We consider homogeneous compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial symmetries with central support. We show that such quantum groups have a simple presentation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy homogeneous orthogonal quantum groups. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.
We show that all orthogonal free quantum groups are isomorphic to variants of the free orthogonal Wang algebra, the hyperoctahedral quantum group or the quantum permutation group. We also obtain a description of their free complexification. In particular we complete the calculation of fusion rules of all orthogonal free quantum groups and their free complexifications.
International audienceWe investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras M1 * B M2 over an amenable von Neumann subalgebra B. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra (M1, ϕ1) * (M2, ϕ2) with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in [Io12a]. Next, we prove that any countable nonsingular ergodic equivalence relation R defined on a standard measure space and which splits as the free product R = R1 * R2 of recurrent subequivalence relations gives rise to a nonamenable factor L(R) with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors L ∞ (X) ⋊ Γ arising from nonsingular free ergodic actions Γ (X, µ) on standard measure spaces of amalgamated groups Γ = Γ1 * Σ Γ2 over a finite subgroup Σ
Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group Sn and the orthogonal group On as well as Wang's quantum permutation group S + n and his free orthogonal quantum group O + n . In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.
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