Discrete Quantum Groups Research Articles

SAT actions of discrete quantum groups and minimal injective extensions of their von Neumann algebras

We introduce a natural generalization of the notion of strongly approximately transitive (SAT) states for actions of locally compact quantum groups. In the case of discrete quantum groups of Kac type, we show that the existence of unique stationary SAT states entails rigidity results concerning injective extensions of quantum group von Neumann algebras.

Topological boundaries of representations and coideals

For a locally compact quantum group G, a (left) coideal is a (left) G-invariant von Neumann subalgebra of L∞(G). We introduce and analyze various generalizations of amenability and coamenability to coideals of discrete and compact quantum groups. We focus on a particular class of coideals found in the category of compact quantum groups, which are associated with a compact quasi-subgroup. This class includes all coideals of the quotient type. We also introduce the notion of a Furstenberg-Hamana boundary for representations of discrete quantum groups and use it to study amenability and coamenability properties of coideals. We then prove that a coideal of a compact quantum group that is associated with a compact quasi-subgroup is coamenable if and only if its codual coideal is G-injective. If G is a unimodular or an exact discrete quantum group, we can replace G-injectivity in the latter statement with the weaker condition of relative amenability. This result leads to a complete characterization of the unique trace property. Specifically, a unimodular discrete quantum group G has the unique trace property if and only if the action of G on its noncommutative Furstenberg boundary is faithful. We also demonstrated that if a unimodular discrete quantum group G is C⁎-simple then it has the unique trace property. These findings are the quantum analogs of the groundbreaking results of Breuillard, Kalantar, Kennedy, and Ozawa and they provide answers to questions posted by Kalantar, Kasprzak, Skalski, and Vergnioux.

Actions of Compact and Discrete Quantum Groups on Operator Systems

Abstract We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.

The approximation property for locally compact quantum groups

We study the Haagerup–Kraus approximation property for locally compact quantum groups, generalising and unifying previous work by Kraus–Ruan and Crann. Along the way we discuss how multipliers of quantum groups interact with the C⁎-algebraic theory of locally compact quantum groups. Several inheritance properties of the approximation property are established in this setting, including passage to quantum subgroups, free products of discrete quantum groups, and duals of double crossed products. We also discuss a relation to the weak⁎ operator approximation property. For discrete quantum groups, we introduce a central variant of the approximation property, and relate this to a version of the approximation property for rigid C⁎-tensor categories, building on work of Arano–De Laat–Wahl.

Tracial states and $\mathbb G$-invariant states of discrete quantum groups

Tracial states and $\mathbb G$-invariant states of discrete quantum groups

Infinite quantum permutations

We define and study quantum permutations of infinite sets. This leads to discrete quantum groups which can be viewed as infinite variants of the quantum permutation groups introduced by Wang. More precisely, the resulting quantum groups encode universal quantum symmetries of the underlying sets among all discrete quantum groups.We also discuss quantum automorphisms of infinite graphs, including some examples and open problems regarding both the existence and non-existence of quantum symmetries in this setting.

Weak expectations of discrete quantum group algebras and crossed products

Weak expectations of discrete quantum group algebras and crossed products

Hecke algebras and the Schlichting completion for discrete quantum groups

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions.

Amenable fusion algebraic actions of discrete quantum groups on compact quantum spaces

In this paper, we introduce actions of fusion algebras on unital $$C^*$$ -algebras, and define amenability for fusion algebraic actions. Motivated by S. Neshveyev et al.’s work, considering the representation ring of a compact quantum group as a fusion algebra, we define the canonical fusion algebraic (for short, CFA) form of a discrete quantum group action on a compact quantum space. Furthermore, through the CFA form, we define FA-amenability of discrete quantum group actions, and present some basic connections between FA-amenable actions and amenable discrete quantum groups. As an application, thinking of a state on a unital $$C^*$$ -algebra as a “probability measure” on a compact quantum space, we show that amenability for a discrete quantum group is equivalent to both of FA-amenability for an action of this discrete quantum group on a compact quantum space and the existence of this kind of “probability measure” that is FA-invariant under this action.

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.

Modular properties of type I locally compact quantum groups

The following paper is devoted to the study of type I locally compact quantum groups. We show how various operators related to the modular theory of the Haar integrals on G and ˆG act on the level of direct integrals. Using these results we derive a web of implications between properties such as unimodularity or traciality of the Haar integrals. We also study in detail two examples: discrete quantum group ˆSUq(2) and the quantum az+b group.

Quantum Galois groups of subfactors

For a finite-index [Formula: see text] subfactor [Formula: see text], we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on [Formula: see text] in a trace-preserving fashion and fixing [Formula: see text] pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.

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.

On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C ∗ \mathrm {C}^{*} -algebra of a countable discrete quantum group \mathbbl {\Gamma } implies that \mathbbl {\Gamma } is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.

Fields of locally compact quantum groups: Continuity and pushouts

We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of [Formula: see text]-algebras are again free via a Fell-topology characterization for [Formula: see text]-field continuity, recovering a result of Blanchard’s in a somewhat more general setting.

Weak Multiplier Hopf Algebras II: Source and Target Algebras

Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

Discrete quantum subgroup asymptotically fixing a sequence of finite subsets

Motivated by generalizing Szemerédi’s theorem, we define the elements in a discrete quantum group asymptotically fixing a sequence of finite subsets and prove that the set of these elements is a quantum subgroup. Using this we obtain a version of mean ergodic theorem for discrete quantum groups.

Property (T), property (F) and residual finiteness for discrete quantum groups

We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg’s theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness.We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (factorization, residual finiteness, property (T)) under extensions of discrete quantum groups.

The Baum–Connes property for a quantum (semi-)direct product

The well known “associativity property” of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete quantum groups. This decomposition allows to define an appropriate triangulated functor relating the Baum–Connes property for the quantum semi-direct product to the Baum–Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum–Connes property generalizes the result [5] of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. TheK-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena. The analogous strategy can be applied for the dual of a quantum direct product. In this case, we obtain, in addition, a connection with the Künneth formula , which is the quantum counterpart to the result [7] of J. Chabert, S. Echterhoff and H. Oyono-Oyono. Again the K -amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena.

From Hopf algebras to topological quantum groups. A short history, various aspects and some problems

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.