Free Wreath Product Research Articles

Note on bi-exactness for creation operators on Fock spaces

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

Free wreath products with amalgamation

We define and study a notion of free wreath product with amalgamation for compact quantum groups. These objects were already introduced in the case of duals of discrete groups under the name “free wreath products of pairs” in a previous work of ours. We give several equivalent descriptions and use them to establish properties like residual finiteness, the Haagerup property or a smash product decomposition.

RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

AbstractOne of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups.We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient-${\mathcal {S}}_2$condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

On the Representation Theory of some Noncrossing Partition Quantum Groups

We compute the representation theory of two families of noncrossing partition quantum groups connected to amalgamated free products and free wreath products. This illustrates the efficiency of the methods developed in our previous joint work with M. Weber.

Quantum reflections, random walks and cut-off

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of reflections. We also study random walks on quantum reflection groups and more generally on free wreath products of finite groups by quantum permutation groups.

Free wreath product quantum groups and standard invariants of subfactors

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 PΓ(A). We prove that every subfactor planar subalgebra of PΓ(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).

Torsion and K-theory for Some Free Wreath Products

Abstract We classify torsion actions of free wreath products of arbitrary compact quantum groups by $S_{N}^{+}$ and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum–Connes property then $\mathbb{G}\wr _{\ast }S_{N}^{+}$ also satisfies the strong Baum–Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by $SO_{q}(3)$.

Wreath products of finite groups by quantum groups

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon.We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.

L2-Betti numbers of rigid C∗-tensor categories and discrete quantum groups

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.

The free wreath product of a compact quantum group by a quantum automorphism group

Let G be a compact quantum group and Gaut(B,ψ) be the quantum automorphism group of a finite dimensional C*-algebra (B,ψ). In this paper, we study the free wreath product G≀⁎Gaut(B,ψ). First of all, we describe its space of intertwiners and find its fusion semiring. 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 G≀⁎Gaut(B,ψ). 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.

The free wreath product of a discrete group by a quantum automorphism group

Let $\mathbb{G}$ be the quantum automorphism group of a finite dimensional C*-algebra $(B,\psi)$ and $\Gamma$ a discrete group. We want to compute the fusion rules of $\widehat{\Gamma}\wr_* \mathbb{G}$. First of all, we will revise the representation theory of $\mathbb{G}$ and, in particular, we will describe the spaces of intertwiners by using noncrossing partitions. It will allow us to find the fusion rules of the free wreath product in the general case of a state $\psi$. We will also prove the simplicity of the reduced C*-algebra, when $\psi$ is a trace, as well as the Haagerup property of $L^\infty(\widehat{\Gamma}\wr_* \mathbb{G})$, when $\Gamma$ is moreover finite.

Free wreath product quantum groups: The monoidal category, approximation properties and free probability

In this paper, we find the fusion rules of the free wreath product quantum groups G≀⁎SN+ 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≀⁎SN+. The combinatorial properties of the intertwiner spaces in G≀⁎SN+ allow us to obtain several probabilistic applications. We prove also the monoidal equivalence between G≀⁎SN+ and a compact quantum group whose dual is a discrete quantum subgroup of the free product Gˆ⁎SUq(2)ˆ, 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 the ACPAP property and exactness.

The Full Classification of Orthogonal Easy Quantum Groups

We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.

Quantum isometry group of dual of finitely generated discrete groups - II

As a continuation of the programme of [13], we carry out explicit computations of ℚ(Γ,S), the quantum isometry group of the canonical spectral triple on C r * (Γ) coming from the word length function corresponding to a finite generating set S, for several interesting examples of Γ not covered by the previous work [13]. These include the braid group of 3 generators, ℤ 4 *n etc. Moreover, we give an alternative description of the quantum groups H s + (n,0) and K n + (studied in [3], [4]) in terms of free wreath product. In the last section we give several new examples of groups for which ℚ(Γ) turns out to be a doubling of C * (Γ).

Existence and examples of quantum isometry groups for a class of compact metric spaces

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X,d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X,d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X,d) which admits a one-to-one continuous map f:X→Rn for some n such that d0(f(x),f(y))=ϕ(d(x,y)) (where d0 is the Euclidean metric) for some homeomorphism ϕ of R+.As concrete examples, we obtain Wang's quantum permutation group Sn+ and also the free wreath product of Z2 by Sn+ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].

A note on reduced and von Neumann algebraic free wreath products

We study operator algebraic properties of the reduced and von Neumann algebraic versions of the free wreath products $\mathbb{G}\wr_{*}S_{N}^{+}$, where $\mathbb{G}$ is a compact matrix quantum group. Based on recent results on their corepresentation theory by Lemeux and Tarrago in [Lemeux and Tarrago (2014)], we prove that $\mathbb{G}\wr_{*}S_{N}^{+}$ is of Kac type whenever $\mathbb{G}$ is, and that the reduced version of $\mathbb{G}\wr_{*}S_{N}^{+}$ is simple with unique trace state whenever $N\geq8$. Moreover, we prove that the reduced von Neumann algebra of $\mathbb{G}\wr_{*}S_{N}^{+}$ does not have property $\Gamma$.

The fusion rules of some free wreath product quantum groups and applications

In this article we find the fusion rules of the free wreath product quantum groups Γˆ≀⁎SN+ for any discrete group Γ. To do this we describe the spaces of intertwiners between basic corepresentations which allows us to identify the irreducible corepresentations. We then apply the knowledge of the fusion rules to prove, in most cases, several operator algebraic properties of the associated reduced C⁎-algebras such as simplicity and uniqueness of the trace. We also prove that the associated von Neumann algebra is a full type II1-factor and that the dual of Γˆ≀⁎SN+ has the Haagerup approximation property for all finite groups Γ.

Representations of quantum permutation algebras

We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π : A s ( 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 .

FREE PRODUCT FORMULAE FOR QUANTUM PERMUTATION GROUPS

Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_{\mathrm{w}}G(Y)$, where $*_{\mathrm{w}}$ is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula $\mu(G*_{\mathrm{w}}H)=\mu(G)\boxtimes\mu(H)$, where $\mu$ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.