Abstract. Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C * -algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group O + n . We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of U + n . In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups.We also study the notion of easy groups. IntroductionIn order to provide a new notion of symmetries adapted to the situation in operator algebras, Woronowicz introduced compact matrix quantum groups in 1987 in [Wor87]. The idea is roughly to take a compact Lie group G ⊆ M n (C) and to pass to the algebra C(G) of continuous functions over it. It turns out that this algebra fulfills all axioms of a C * -algebra with the special feature that the multiplication is commutative. If we now also dualize main properties of the group law µ : G×G → G to properties of a map ∆ : C(G) → C(G×G) ∼ = C(G)⊗C(G) (the comultiplication), and consider noncommutative C * -algebras equiped with such a map ∆, we are right in the heart of the definition of a compact matrix quantum group (see Section 3.1 for details). It generalizes the notion of a compact matrix group. The reader not familiar with quantum groups might also first jump to Section 5 and read it as a motivation.Date: December 2, 2015. 2010 Mathematics Subject Classification. 20G42 (Primary); 05A18, 46L54 (Secondary).

Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica and Speicher's orthogonal easy quantum groups. In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary easy quantum groups, which is done in a separate article. The present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the article.

Abstract. In this paper, we find the fusion rules for the free wreath product quantum groups G * S + N for all compact matrix quantum groups of Kac type G and N ≥ 4. This is based on a combinatorial description of the intertwiner spaces between certain generating representations of G * S + N . The combinatorial properties of the intertwiner spaces in G * S + N then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between G * S + N and a compact quantum group whose dual is a discrete quantum subgroup of the free product G * SUq(2), for some 0 < q ≤ 1. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.

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By a construction of Vaughan Jones, the bipartite graph Γ(A) associated with the natural inclusion of C inside a finite-dimensional C * -algebra A gives rise to a planar algebra P Γ(A) . We prove that every subfactor planar subalgebra of P Γ(A) is the fixed point planar algebra of a uniquely determined action of a compact quantum group G on A. We use this result to introduce a conceptual framework for the free wreath product operation on compact quantum groups in the language of planar algebras/standard invariants of subfactors. Our approach will unify both previous definitions of the free wreath product due to Bichon and Fima-Pittau and extend them to a considerably larger class of compact quantum groups. In addition, we observe that the central Haagerup property for discrete quantum groups is stable under the free wreath product operation (on their duals).Recall that π 0 (resp. π 1 ) is the pair partition of 2k with blocks {(2i, 2i + 1)} (resp. {2i + 1, 2i + 2)}). Definition 6.7. A free pair (T, T ) of planar tangles is called reduced if T = T π and T = T kr (π) for some non-crossing partition π such that π ≥ π 0 .An example of reduced free pair is given in Figure 17 with π = {{1, 6}, {2, 3, 4,

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We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.Résumé. -Nous déterminons le comportement asymptotique de la fonction de Green pour des marches aléatoires à dérive nulle confinées dans des cônes convexes multidimensionnels. Comme conséquence, nous prouvons qu'il existe une unique fonction harmonique discrète et positive pour ces processus aléatoires, c'est-à-dire que leur frontière de Martin se réduit à un singleton.