Free Orthogonal Quantum Groups Research Articles

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras

AbstractWe find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

A short note on strong convergence of q-Gaussians

In this paper, we prove strong convergence of [Formula: see text]-Gaussians with respect to a parameter [Formula: see text], which implies the spectrum of any self-adjoint non-commutative polynomial in [Formula: see text]-Gaussians is continuously deformed with respect to [Formula: see text]. With Bożejko’s Haagerup-type inequality, we follow Brannan’s approach to proving the asymptotic strong convergence of the free orthogonal quantum group.

Noncommutative Furstenberg boundary

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

Strong Haagerup inequalities on non-Kac free orthogonal quantum groups

We present natural analogues of strong Haagerup inequalities on non-Kac free orthogonal quantum groups OF+ in which Lp-analytic problems are harder due to their non-tracial nature. Furthermore, we prove optimality of the inequalities, and apply the obtained results to compute the optimal time for ultracontractivity of the heat semigroup and to distinguish the complex interpolation space Lp(OF+) and the real interpolation space Lp,p(OF+).

On the von Neumann algebra of class functions on a compact quantum group

We study analogues of the radial subalgebras in free group factors (called the algebras of class functions) in the setting of compact quantum groups. For the free orthogonal quantum groups we show that they are not MASAs, as soon as we are in a non-Kac situation. The most important notion to our present work is that of a (quasi-)split inclusion. We prove that the inclusion of the algebra of class functions is quasi-split for some unitary quantum groups – in this case the subalgebra is non-abelian and we also obtain a result concerning its relative commutant. In the positive direction, we construct certain bicrossed products from the quantum group SUq(2) for which the algebra of class functions is a MASA.

Cutoff profiles for quantum Lévy processes and quantum random transpositions

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time \(N\ln (N)\). Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.

Noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of Drinfeld doubles

We clarify the relation between noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of quantum groups. Specifically, given a compact quantum group G, we show that in many cases where the Poisson boundary of the dual discrete quantum group Gˆ has been computed, the underlying topological boundary either coincides with the Furstenberg–Hamana boundary of the Drinfeld double D(G) of G or is a quotient of it. This includes the q-deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C⁎-algebras. In particular, the boundary of D(Gq) for the q-deformation of a compact connected semisimple Lie group G is Gq/T (for q≠1), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of GC.We show also that the construction of the Furstenberg–Hamana boundary of D(G) respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of G. This leads to a notion of the Furstenberg–Hamana boundary of a rigid C⁎-tensor category.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

Braided Free Orthogonal Quantum Groups

Abstract We construct some braided quantum groups over the circle group. These are analogous to the free orthogonal quantum groups. They generalise the braided quantum SU(2) groups for a complex deformation parameter. We describe their irreducible representations and fusion rules and study when they are monoidally equivalent. A key tool here is to describe the bosonisation of our braided compact quantum groups through ordinary free orthogonal quantum groups.

Gradient forms and strong solidity of free quantum groups

Consider the free orthogonal quantum groups O_N^+(F) and free unitary quantum groups U_N^+(F) with N ge 3. In the case F = text {id}_N it was proved both by Isono and Fima-Vergnioux that the associated finite von Neumann algebra L_infty (O_N^+) is strongly solid. Moreover, Isono obtains strong solidity also for L_infty (U_N^+) . In this paper we prove for general F in GL_N(mathbb {C}) that the von Neumann algebras L_infty (O_N^+(F)) and L_infty (U_N^+(F)) are strongly solid. A crucial part in our proof is the study of coarse properties of gradient bimodules associated with Dirichlet forms on these algebras and constructions of derivations due to Cipriani–Sauvageot.

Property RD and Hypercontractivity for Orthogonal Free Quantum Groups

Abstract We prove that the twisted property RD introduced in [ 2] fails to hold for all non-Kac type, non-amenable orthogonal free quantum groups. In the Kac case we revisit property RD, proving an analogue of the $L_p-L_2$ non-commutative Khintchine inequality for free groups from [ 29]. As an application, we give new and improved hypercontractivity and ultracontractivity estimates for the generalized heat semigroups on free orthogonal quantum groups, both in the Kac and non-Kac cases.

Classification of globally colorized categories of partitions

Set partitions closed under certain operations form a tensor category. They give rise to certain subgroups of the free orthogonal quantum group [Formula: see text], the so-called easy quantum groups, introduced by Banica and Speicher in 2009. This correspondence was generalized to two-colored set partitions, which, in addition, assign a black or white color to each point of a set. Globally colorized categories of partitions are those categories that are invariant with respect to arbitrary permutations of colors. This paper presents a classification of globally colorized categories. In addition, we show that the corresponding unitary quantum groups can be constructed from the orthogonal ones using tensor complexification.

Cut-off phenomenon for random walks on free orthogonal quantum groups

We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cut-off at $$N\ln (N)/2(1-\cos (\theta ))$$ . This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.

Entropic uncertainty relations under localizations on discrete quantum groups

The uncertainty principle has been established within the framework of locally compact quantum groups in recent years. This paper demonstrates that entropic uncertainty relations can be strengthened under localizations on discrete quantum groups, which is the case if the dual compact quantum group G is the free orthogonal quantum group ON+ with N ≥ 3 or if G admits an infinite Λ(p) set with p > 2. On the other hand, this paper explains the reason why such phenomena do not appear when G is one of the connected semisimple compact Lie groups, O2+, and the quantum SU(2) groups. Also, we discuss the divergence of entropic uncertainty relations together with some explicit explanations.

Quantum Bernstein’s theorem and the hyperoctahedral quantum group

We study an extension of Bernstein theorem to the setting of quantum groups. For a d-tuple of free identically distributed random variables, we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group. For a subset not contained in the hyperoctahedral quantum group, we prove that preservation of freeness characterizes Wigner’s semicircle law. We show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group. We provide examples of quantum subsets which show that our result is an extension of results known in the literature.

Amenability and subexponential spectral growth rate of Dirichlet forms on von Neumann algebras

In this work we apply Noncommutative Potential Theory to characterize (relative) amenability and the (relative) Haagerup Property (H) of von Neumann algebras in terms of the spectral growth of Dirichlet forms. Examples deal with (inclusions of) countable discrete groups and free orthogonal compact quantum groups.

Highly Entangled, Non-random Subspaces of Tensor Products from Quantum Groups

In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels. An application to the construction of $d$-positive maps on matrix algebras is also presented.

Howe–Moore type theorems for quantum groups and rigid -tensor categories

We formulate and study Howe–Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast }$-tensor categories. We say that a rigid $C^{\ast }$-tensor category ${\mathcal{C}}$ has the Howe–Moore property if every completely positive multiplier on ${\mathcal{C}}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe–Moore property. As an immediate consequence, we deduce the Howe–Moore property for Temperley–Lieb–Jones standard invariants with principal graph $A_{\infty }$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan–Wenzl categories. Additionally, in the specific case of the quantum groups $\text{SU}_{q}(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\text{SU}_{q}(N)$, which coincide with the completely positive multipliers on the representation category of $\text{SU}_{q}(N)$.

Hypercontractivity of heat semigroups on free quantum groups

In this paper we study two semigroups of completely positive unital self-adjoint maps on the von Neumann algebras of the free orthogonal quantum group $O_N^+$ and the free permutation quantum group $S_N^+$. We show that these semigroups satisfy ultracontractivity and hypercontractivity estimates. We also give results regarding spectral gap and logarithmic Sobolev inequalities.

Actions of compact quantum groups

These lecture notes, prepared for the summer school Topological quantum groups, Bedlewo 2015, deal with aspects of the theory of actions of compact quantum groups on C*-algebras ('locally compact quantum spaces'). After going over the basic notions of isotypical components and reduced and universal completions, we look at crossed and smash product C*-algebras, up to the statement of the Takesaki-Takai-Baaj-Skandalis duality (in the algebraic setting). We then look at two special types of actions, namely homogeneous actions and free actions. We study the actions which combine both types, the quantum torsors, and show that more generally any homogeneous action can be completed to a free action with a discrete, classical set of `quantum orbits'. We end with a combinatorial description of the homogeneous actions for the free orthogonal quantum groups.