Monoidal equivalence of compact quantum groupsu0018768/students/derijdt-phd-thesis.… · In this...

KATHOLIEKE UNIVERSITEIT LEUVEN

Faculteit Wetenschappen

Departement Wiskunde

Monoidal equivalence of compactquantum groups

An De Rijdt

Promotor :

Prof. dr. Stefaan Vaes

Proefschrift ingediend tot het

behalen van de graad van

Doctor in de Wetenschappen

juli 2007

Dankwoord

Ongelooflijk maar waar, mijn thesis is af! Voor we met de wiskunde vanstart gaan, wil ik de mensen die hun steentje hebben bijgedragen tocheven bedanken.

Eerst en vooral wil ik mijn promotor, Stefaan Vaes, hartelijk bedanken.Dankzij jouw hulp, vertrouwen en oeverloos enthousiasme is deze thesistot een goed einde gekomen.

De andere leden van de jury, Johan Quaegebeur, Fons Van Daele, KarelDekimpe, Roland Vergnioux en Sergey Neshveyev, wil ik bedanken voorhet nalezen van de thesis en de beoordeling ervan.

De blok B-ers hebben ervoor gezorgd dat de afgelopen jaren niet gauw ver-geten zullen worden. Het was altijd fijn vertoeven op de vijfde verdieping.Een bijzondere vermelding voor Lies, mede-strijder aan het vrouwenfront,voor het regelen van de laatste praktische beslommeringen. Aangezienjullie deze thesis nu aan het lezen zijn, heeft ze haar werk goed gedaan!

Een welgemeende dankjewel ook aan de (ex)-collega’s van Functionaal-analyse: Stefaan, Johan Q. en K., Tom, Kenny, Nikolas, Arnoud, Pieter,Sebastien, Anselm en Fons. Bureau 05.27 is de voorbije 4 jaar zowat eentweede thuis geweest, niet in het minst door de medebewoners, het eerstejaar Arnoud, daarna Nikolas.

Nikolas, colleeg, bedankt voor de fijne samenwerking, voor het gezelschapen voor je steun en hulp als ik weer es een keertje overstresst was. Merciook voor de vele legendarische uitjes en congressen. Een ding mag gezegd,saai is het met jou nooit! Ik ga ons bureautje missen!

Uiteraard mag ik mijn vrienden, zowel wiskundig als niet-wiskundig hierniet vergeten. Bedankt voor de fijne sporturen, spelletjes- en andereavonden, fietsvakanties en nog veel meer.

Tot slot mag ook mijn familie, in het bijzonder mijn ouders, broer en zusniet in het rijtje ontbreken. Bedankt voor jullie geloof in mij en voor hetmeesupporteren en duimen voor een goede afloop!

An De Rijdt, 4 juli 2007

Contents

Introduction 1

1 Compact and Discrete Quantum Groups 9

1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 The definition of a compact quantum group . . . . . . . . 10

1.3 Representation Theory . . . . . . . . . . . . . . . . . . . . 11

1.4 Discrete quantum groups and duality . . . . . . . . . . . . 16

1.5 Example: Ao(F ) and Au(F ) . . . . . . . . . . . . . . . . . 19

2 Actions of quantum groups 25

2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Bimodular structure of the spectral subalgebra . . . . . . 29

2.3 Ergodic actions . . . . . . . . . . . . . . . . . . . . . . . . 35

2.4 Invariant subalgebras and Homogeneous spaces . . . . . . 39

2.5 Quantum automorphism groups . . . . . . . . . . . . . . . 42

3 Ergodic actions with large multiplicity 45

3.1 Monoidal equivalence of quantum groups . . . . . . . . . . 47

3.2 Ergodic actions of full quantum multiplicity . . . . . . . . 50

3.2.1 An alternative look at monoidal equivalence . . . . 59

3.3 Unitary fiber functors preserving the dimension . . . . . . 62

3.4 Monoidal equivalence for Ao(F ) . . . . . . . . . . . . . . . 67

ii CONTENTS

3.5 Monoidal equivalence for Au(F ) . . . . . . . . . . . . . . . 73

3.6 Monoidal equivalence for Aaut(B,ϕ) . . . . . . . . . . . . 77

4 Actions through monoidally equivalence 85

4.1 Correspondence between actions . . . . . . . . . . . . . . 86

4.2 Homogeneous spaces through monoidal equivalence . . . . 97

4.3 Application: Invariant subalgebras of Ao(F ) . . . . . . . . 98

5 Poisson boundaries 107

5.1 The Poisson boundary of a discrete quantum group . . . . 108

5.2 Poisson boundaries and monoidal equivalence . . . . . . . 110

5.3 Concrete computations . . . . . . . . . . . . . . . . . . . . 116

5.4 Example: Quantum automorphism groups . . . . . . . . . 118

5.4.1 The canonical Kac subgroup of Aaut(M2(C),Tr(·D)) 119

5.4.2 The Poisson boundary of Aaut(B,ϕ) . . . . . . . . 120

Nederlandse samenvatting 123

N.1 Compacte en discrete kwantumgroepen . . . . . . . . . . 123

N.1.1 Notaties . . . . . . . . . . . . . . . . . . . . . . . . 123

N.1.2 Compacte kwantumgroepen . . . . . . . . . . . . . 124

N.1.3 Discrete kwantumgroepen . . . . . . . . . . . . . . 126

N.1.4 De universele orthogonale compacte kwantumgroepen127

N.2 Acties van kwantumgroepen en spectrale deelruimten . . . 127

N.3 Ergodische acties met grote multipliciteit . . . . . . . . . 130

N.3.1 Monoidale equivalentie van kwantumgroepen . . . 130

N.3.2 Ergodische acties van volle kwantummultipliciteit . 131

N.3.3 Monoidale equivalentie voor Ao(F ) . . . . . . . . . 132

N.3.4 Monoidale equivalentie voor Aaut(B,ω) . . . . . . 133

N.4 Acties van monoidaal equivalente kwantumgroepen . . . . 135

N.5 Poissonranden via monoidale equivalentie . . . . . . . . . 137

CONTENTS iii

Bibliography 141

Introduction

Abstract

In this thesis, we tackle two different types of problems on com-pact and discrete quantum groups. On the one hand, we providethe first examples of ergodic actions of compact quantum groupswith large multiplicities. On the other hand, we calculate the Pois-son boundaries for some non-amenable discrete quantum groupswith commutative fusion rules. Although these problems are of adifferent kind, we solve them with the same tool: monoidal equiva-lence. We perform an in-depth study of monoidal equivalence forcompact quantum groups to answer the above stated questions.

In the 1980’s, Woronowicz introduced the notion of a compact quan-tum group and generalized the classical Peter-Weyl representation theory.Many fascinating examples of compact quantum groups are available bynow: Drinfel’d and Jimbo [20, 32] introduced the q-deformations of com-pact semi-simple Lie groups and Rosso [40] showed that they fit into thetheory of Woronowicz. The universal orthogonal and unitary quantumgroups were introduced by Van Daele and Wang [50] and studied in detailby Banica [2, 3]. For any F ∈ GL(n,C) satisfying FF = ±1, one definesthe compact quantum group Ao(F ) as the universal quantum group gen-erated by the coefficients of a unitary n by n matrix U with relationsU = FUF−1. The comultiplication on Ao(F ) is (uniquely) defined insuch a way that U becomes a representation. Without being exhaustive,we remark that other examples of compact quantum groups are relatedwith graphs and metric spaces as been constructed by Bichon [13] and byBanica [4, 5].

2 Introduction

Discrete quantum groups arose first as duals of compact quantum groups.However they admit an intrinsic axiomatization as discovered by VanDaele in [48] and by Effros and Ruan in [22].

The results in the first two chapters of this thesis are not new, althoughchapter 2 gives a personal presentation of the theory of actions of compactquantum groups. The results of our research are bundled in chapters 3,4 and 5, that we briefly present now.

Ergodic actions with large multiplicity

[Chapter 3]

By a well known theorem of Høegh-Krohn, Landstad and Størmer [29],compact groups only admit ergodic actions on tracial C∗-algebras. In-deed, the (unique) invariant state is necessarily a trace. Moreover, givena compact group G acting ergodically on a C∗-algebra B, one studies theso-called spectral subspaces: the action of G on B yields a unitary repre-sentation of G which can be decomposed into a direct sum of irreduciblerepresentations. The number of times an irreducible representation ofthe group G appears in this composition is called its multiplicity. It wasproven in [29] that the multiplicity of an irreducible representation isnecessarily bounded by the dimension of the irreducible representation.

A deeper analysis of the spectral structure of ergodic actions of compactgroups has been made by A. Wassermann [56, 57, 58]. In the culmina-tion of his work, Wassermann shows that the compact group SU(2) onlyadmits ergodic actions on von Neumann algebras of finite type I. It is anopen problem if this is the case for an arbitrary compact group G. Infact, this problem is even not solved for the group SU(n) with n ≥ 3.

The abstract theory of ergodic actions of compact quantum groups on C∗-algebras has been initiated by Boca [15] and Landstad [34]. The majordifference with the compact group case, is the following: the (unique)invariant state is no longer a trace. Indeed, Wang [54] gave examples ofergodic actions of universal unitary quantum groups on type III factors.

Moreover, it could not be proven that the multiplicity of an irreduciblerepresentation is still bounded by its dimension. However, Boca obtainedthat the multiplicity of an irreducible representation is bounded by itsquantum dimension. Landstad refined this result by introducing a new

3

invariant, the quantum multiplicity, which he showed to be bounded bythe quantum dimension.

In [58], A. Wassermann gives a complete classification of the ergodic ac-tions of SU(2), essentially labeling them by the finite subgroups of SU(2).It would, of course, be great to give a complete classification of ergodicactions of the deformed SUq(2). In [44], Tomatsu provides a first stepin this direction: he computes all ergodic actions of SUq(2) on ‘virtual’quotient spaces SUq(2)/Γ. (More precisely, he describes all the invariantsubalgebras of the quantum group SUq(2).) By construction, the ergodicactions of SUq(2) on this virtual quotient spaces are such that the multi-plicity of an irreducible representation is bounded by its dimension.

Although the classical multiplicity bound theorem could not be proven,in all the examples known until recently, the multiplicity of an irreduciblerepresentation was nevertheless always bounded by the ordinary dimen-sion.

In chapter 3 of this thesis, we provide the first concrete examples ofergodic actions where the multiplicity of an irreducible representation isstrictly larger than the ordinary dimension of the representation. Thisimplies in particular that there are much more ergodic actions of SUq(2)than the ones studied by Tomatsu. The contents of chapter 3 mostlycoincide with the paper [14], which is joint work with Stefaan Vaes andJulien Bichon.

We focus on a special type of ergodic actions, namely the ones with fullquantum multiplicity. Studying these, we find examples of ergodic actionswhich do not satisfy the multiplicity bound theorem. The tool we use isthe notion of monoidal equivalence of quantum groups, that we explainnow.

One can look at a compact quantum group with several degrees of preci-sion. At first, we study only the fusion rules in the representation theory:we label the irreducible representations and describe how a tensor pro-duct of irreducibles breaks up into irreducibles. Taking into account onlythese fusion rules, we loose a lot of information: for example, the q-deformed compact Lie groups have the same fusion rules as their classicalcounterparts. In a next approximation, one studies the representationtheory of a compact quantum group as a monoidal category, but withoutits concrete realization (the so-called forgetful functor to the category ofHilbert spaces). This is crucial: by the Tannaka-Krein reconstruction

4 Introduction

theorem [63], the concrete monoidal category of (finite-dimensional) rep-resentations essentially determines the compact quantum group. Notethat knowing the representation theory of a compact quantum group as amonoidal category, comes down to knowing the fusion rules together withthe 6j-symbols, see Remark 3.1.3.

Closely related to the notion of monoidally equivalent quantum groups,is the notion of a unitary fiber functor on a compact quantum group.Essentially, a unitary fiber functor gives another concrete realization, dif-ferent from the tautological realization, of the representation theory of acompact quantum group.

We are able to describe ergodic actions of full quantum multiplicity asunitary fiber functors on the representation category. This provides uswith a powerful categorical tool for constructing ergodic actions. More-over, these ergodic actions of full quantum multiplicity were among thefirst examples of ergodic actions where the multiplicity of the irreduciblerepresentations is strictly greater than the dimension of the representa-tion space. To be more precise, we find that the quantum groups Ao(F )and SUq(2) are monoidally equivalent for the appropriate q. The uni-tary fiber functors corresponding to this monoidal equivalences provideus with ergodic actions of SUq(2) where the multiplicity of the fundamen-tal representation is strictly greater than 2.

In the theory of Hopf algebras, ergodic actions of full quantum multi-plicity correspond to Hopf-Galois extensions. In this algebraic setting,several results related to ours have been obtained. The relation betweenHopf-Galois extensions and fiber functors is due to Ulbrich [47] and the re-lation between monoidal equivalence of Hopf algebras and Hopf-bi-Galoisextensions has been established by Schauenburg [42]. Fiber functors pre-serving the dimension and 2-cocycles have been studied by Etingof andGelaki [24]. The main difference between these Hopf algebraic results andour work, lies in dealing with the ∗-structure and positivity. In a sense,we are dealing with the real forms of (certain) Hopf algebras. This al-lows us to construct Hilbert space representations and C∗-algebras. Thecompatibility of fiber functors with ∗-structures is a severe restriction.Indeed, there exist many fiber functors on the representation category ofSU(2) (see [16]), but the forgetful functor is the only one compatible withthe ∗-structure.

5

Actions of monoidally equivalent quantum groups

[Chapter 4]

Chapter 4 is an extension of the preprint [21], which consists of jointwork with Nikolas Vander Vennet. In this part of the thesis, monoidalequivalence is viewed in another way. Rather than studying it as unitaryfiber functors, we examine which are the similarities between monoidallyequivalent compact quantum groups. In particular, we study actions ofmonoidally equivalent quantum groups.

In, [39], Pinzari and Roberts obtained a categorical description of allergodic actions of a compact quantum group. Inspired by [14] (i.e. thefirst part of this thesis), they describe an ergodic action (not necessarily offull quantum multiplicity) of a compact quantum group as a special kindof functor on the representation category. When the ergodic action is offull quantum multiplicity, the corresponding functor is just a unitary fiberfunctor. In particular, Pinzari and Roberts’ categorical characterizationof ergodic actions yields a bijective correspondence between the ergodicactions of monoidally equivalent compact quantum groups.

We generalize this result and obtain a concrete and constructive bijectivecorrespondence between (not necessarily ergodic) actions of monoidallyequivalent quantum groups on unital C∗-algebras. Moreover, the cor-respondence is of that kind that it preserves the spectral subspaces ofthe actions. Restricting this to ergodic actions, this just means that themultiplicities of the irreducible representations are preserved through thiscorrespondence. It should be emphasized that our approach is not cate-gorical. The correspondence is obtained in a concrete, constructive way.

Poisson boundaries and monoidal equivalence

[Chapter 5]

A major application of the bijective correspondence between actions ofchapter 4 is found in the study of Poisson boundaries for discrete quantumgroups. Group invariant random walks on countable groups have beenstudied intensively and a natural problem arises as the identification ofthe associated Poisson boundaries and Martin boundaries. The quantumcase was first studied by Biane ([9]) for duals of compact groups and later

6 Introduction

by Izumi ([30]) for arbitrary discrete quantum groups. In his paper, heidentified the Poisson boundary of the dual of SUq(2) with the Podlessphere. Later, Neshveyev and Tuset ([38]) associated a Martin bound-ary with a random walk on a discrete quantum group and proved thatthe Martin boundary of the dual of SUq(2) is still given by the Podlessphere, hence generalizing Biane’s work [9] on the dual of SU(2). Izumi,Neshveyev and Tuset then computed the Poisson boundary of the dualof SUq(n) in their paper [31]. Another type of boundary was studied byVaes and Vergnioux in [53].

A next step in the identification of Poisson boundaries was made byTomatsu. Recently, he managed to identify the Poisson boundaries ofall amenable discrete quantum groups G when G has commutative fusionrules. This Poisson boundary appears to be the homogeneous space of thecompact quantum group G with respect to the maximal closed quantumsubgroup of Kac type ([45]). In particular, this method yields an ex-plicit description of the Poisson boundary for the duals of all q-deformedclassical compact Lie groups.

All the computations of Poisson boundaries described above, deal withamenable quantum groups. The first identification of the Poisson bound-ary of a non-amenable discrete quantum group was given by Vander Ven-net and Vaes in [52]. They proved that the Poisson boundary of Ao(F )can be identified with a higher dimensional Podles sphere. Their methodmakes extensively use of the monoidal equivalence of Ao(F ) with SUq(2)for the appropriate q. Hence, they reduce the identification problem toan already solved one, namely that of SUq(2).

The results in the final chapter of this thesis again were found in colla-boration with Nikolas Vander Vennet. We provide a systematic methodto relate Poisson boundaries for the duals of monoidally equivalent quan-tum groups. The relation goes as follows. The Poisson boundary of Gadmits a natural action of G. If G1 and G2 are monoidally equivalent,the boundaries of G1 and G2 are related through the bijective correspon-dence between the actions of G1 and G2 that was obtained in chapter4. This means that if we know the Poisson boundary of the dual of acompact quantum group G, we at the same time know it for the duals ofall compact quantum groups which are monoidally equivalent with G.

Combining our result with Tomatsu’s work, we can identify the Pois-son boundary of the dual of some non-coamenable compact quantum

7

groups which are monoidally equivalent with coamenable compact quan-tum groups. A first class of examples of this kind are the universal or-thogonal quantum groups Ao(F ). If the dimension of F is greater than 3,then Ao(F ) is not coamenable. As remarked above, the quantum groupsAo(F ) and SUq(2) are monoidally equivalent for the right q. Moreover,the Poisson boundary of SUq(2) was identified by Izumi and also, in adifferent way, by Tomatsu (SUq(2) is coamenable). Through the con-struction, we can thus identify the Poisson boundary of Ao(F ). As statedbefore, this result was already obtained in [52] by another method. Asecond class of examples of the above type come from quantum auto-morphism groups Aaut(B,ϕ). These are coamenable if and only if thedimension of the C∗-algebra B equals 4. Using the fact that every quan-tum automorphism group is monoidally equivalent with a coamenableone, we explicitly identify the Poisson boundary of the duals of all suchquantum automorphism groups.

Chapter 1

Compact and Discrete

Quantum Groups

In this chapter, we give an overview of the theory of compact quantumgroups which was developed by Woronowicz in [60]. We refer to [37] fora survey of basic results.

1.1 Notations

Consider a subset S of a C∗-algebra. We denote by 〈S〉 the linear span ofS and by [S] the closed linear span of S. We use the notation ωη,ξ(a) =〈η, aξ〉 and we use inner products that are linear in the second variable.Moreover we denote by ξ∗ : H → C : η 7→ 〈ξ, η〉 and denote by H thedual Hilbert space of H, i.e. H := ξ∗ | ξ ∈ H.

The symbol ⊗ denotes tensor products of Hilbert spaces and minimal ten-sor products of C∗-algebras. We use the symbol ⊗alg for algebraic tensorproducts of ∗-algebras and ⊗ for the tensor product of von Neumann alge-bras. We also make use of the leg numbering notation in multiple tensorproducts: if a ∈ A⊗ A, then a12, a13, a23 denote the obvious elements inA⊗A⊗A, e.g. a12 = a⊗ 1.

The adjointable operators between C∗-modules or bounded operators be-tween Hilbert-spaces H and K are denoted by L(H,K).

Let B be a unital *-algebra. We call a linear map ω : B → C such that

10 Chapter 1. Compact and Discrete Quantum Groups

ω(1) = 1 a faithful state if ω(a∗a) ≥ 0 for all a ∈ B and ω(a∗a) = 0 if andonly if a = 0.

1.2 The definition of a compact quantum group

For a compact group G, the relevant information of this group can berecovered from its corresponding C∗-algebra of continuous functions C(G)and the comultiplication. First, the multiplication on the group can belifted to a *-homomorphism on the function algebra, given by

∆ : C(G) → C(G)⊗ C(G) : ∆(f)(s, t) = f(st)

for s, t ∈ G. Notice that we identified C(G) ⊗ C(G) with C(G ⊗ G).The associativity of the product translates into co-associativity of the *-homomorphism, this means (∆⊗ id)∆ = (id⊗∆)∆. Such a co-associative*-homomorphism is called a comultiplication. Extra conditions are neededto ensure that a commutative unital C∗-algebra with comultiplication isthe function algebra of a compact group. In [37] it was noticed that theleft and right cancellation property of a compact semi-group G can beexpressed by the density conditions

[∆(C(G))(1⊗ C(G))] = [∆(C(G))(C(G)⊗ 1)] = C(G)⊗ C(G) .

Moreover, a compact semi-group with cancellation is a compact group.Leaving out the commutativity of the C∗-algebra C(G), we arrive at thefollowing definition.

Definition 1.2.1. A compact quantum group G is a pair (C(G),∆),where

• C(G) is a unital C∗-algebra;

• ∆ : C(G) → C(G) ⊗ C(G) is a unital ∗-homomorphism satisfyingthe co-associativity relation

(∆⊗ id)∆ = (id⊗∆)∆ ;

• G satisfies the left and right cancellation property expressed by

[∆(C(G))(1⊗ C(G))] = [∆(C(G))(C(G)⊗ 1)] = C(G)⊗ C(G) .

1.3 Representation Theory 11

The map ∆ is called the comultiplication on G.

The notation C(G) suggests the analogy with the basic example given bycontinuous functions on a compact group. Just adding commutativity tothe definition, brings us back to the classical case of a compact group. Inthe quantum case however, there is no underlying space G and C(G) is anon-abelian C∗-algebra.

This definition appears to be a good one in the sense that almost ev-ery result for compact groups can be generalized for compact quantumgroups.

A basic example, next to the commutative one, is that of the dual of adiscrete group.

Example 1.2.2. Let Γ be a discrete group. The group C∗-algebra C∗(Γ)is a compact quantum group for the comultiplication ∆(λs) = λs ⊗ λs.

An important property of a compact group is that there always exists aHaar measure. For a compact quantum group, this remains true.

Theorem 1.2.3 (Woronowicz, [62]). Let G be a compact quantum group.There exists a unique state h on C(G) which satisfies

(id⊗ h)∆(a) = h(a)1 = (h⊗ id)∆(a)

for all a ∈ C(G). The state h is called the Haar state of G.

1.3 Representation Theory

The Peter-Weyl Theory of representations for compact groups, is verysimilar for compact quantum groups. Although the proofs are sometimesmore difficult, a great part of the theory can be generalized. We gatherthe main definitions and results in this section.

Definition 1.3.1. A unitary representation U of a compact quantumgroup G on a Hilbert space H is a unitary element U ∈ L(H ⊗ C(G))satisfying

(id⊗∆)(U) = U12U13 . (1.1)

The reason we mostly work with unitary representations, is that everyinvertible representation is equivalent to a unitary one.


Definition 1.3.2. Whenever U1 and U2 are unitary representations ofG on the respective Hilbert spaces H1 and H2, we define

Mor(U1, U2) := T ∈ L(H2,H1) | U1(T ⊗ 1) = (T ⊗ 1)U2 .

The elements of Mor(U1, U2) are called intertwiners. We use the notationEnd(U) := Mor(U,U).

Intertwiners determine on which subspaces representations behave thesame, or which are the invariant subspaces for a certain representation.Hence, they express the possibility to break up a representation into sub-representations. If a representation has no non-trivial subrepresentations,it is called irreducible.

Definition 1.3.3. A unitary representation U is said to be irreducible ifEnd(U) = C1.

If Mor(U1, U2) contains a unitary operator, the representations U1 andU2 are said to be unitarily equivalent. This means exactly that the rep-resentations are the same up to a change of base of the Hilbert space.

We have the following essential result.

Theorem 1.3.4. Every irreducible representation of a compact quan-tum group is finite-dimensional. Every unitary representation is unitarilyequivalent to a direct sum of irreducibles.

So it is clear that the irreducible representations are the building blocksfor the representation theory. Moreover, we almost exclusively deal withfinite-dimensional representations. By choosing an orthonormal basis ofthe Hilbert space H, a finite-dimensional unitary representation of G canbe considered as a unitary matrix (Uij) with entries in C(G) and (1.1)becomes

∆(Uij) =∑k

Uik ⊗ Ukj .

The product in the C∗-algebra C(G) yields a tensor product on the levelof unitary representations.

Definition 1.3.5. Let U1 and U2 be unitary representations of G on therespective Hilbert spaces H1 and H2. We define the tensor product

U1T© U2 := U1

13U223 ∈ L(H1 ⊗H2 ⊗ C(G)) .


It is easy to check that U1T© U2 is again a representation.

Notation 1.3.6. Let G be a compact quantum group. We denote byIrred(G) the set of equivalence classes of irreducible unitary representa-tions. We choose representatives Ux on the Hilbert space Hx for everyx ∈ Irred(G). Whenever x, y ∈ Irred(G), we use x ⊗ y to denote theunitary representation Ux T© Uy. The class of the trivial unitary repre-sentation 1 ∈ C(G) is denoted by ε.

Because of Theorem 1.3.4, the tensor product of two irreducible repre-sentations can be written as the direct sum of irreducible representations.Hence, we define the natural numbers mult(z, x⊗ y) such that

x⊗ y ∼=⊕

z∈Irred(G)

mult(z, x⊗ y) · U z .

The collection of natural numbers mult(z, x ⊗ y) are called the fusionrules of G.

For every x ∈ Irred(G), there is a unique x ∈ Irred(G) such that

Mor(x⊗ x, ε) 6= 0 6= Mor(x⊗ x, ε) .

The irreducible representation x is called the contragredient of x. Up toequivalence, a representative of x can be constructed as follows. Choosea basis for Hx and write Ux = (Uxij). Then ((Uxij)

∗)ij is an invertible,but possibly non-unitary representation of G. Its unitarization yieldsa representative of x. The matrix F ∈ GL(n,C) (determined up to aunitary) unitarizing ((Uxij)

∗)ij plays a special role in the modular theoryof G, as we discuss below.

For every x ∈ Irred(G), we take a non-zero element tx ∈ Mor(x ⊗ x, ε).Write the antilinear map

jx : Hx → Hx : ξ 7→ (ξ∗ ⊗ 1)tx (1.2)

and define Qx := j∗xjx. We normalize tx in such a way that Tr(Qx) =Tr(Q−1

x ). This uniquely determines Qx and fixes tx up to a numberof modulus 1. If we take the unique tx ∈ Mor(x ⊗ x, ε) such that(t∗x ⊗ 1)(1⊗ tx) = 1, then jx = j−1

x and Qx = (jxj∗x)−1.

Definition 1.3.7. For x ∈ Irred(G), the value Tr(Qx) is called the quan-tum dimension of x and denoted by dimq(x).


Observe that t∗xtx = dimq(x)1 and that dimq(x) = dimq(x) ≥ dim(x),with equality holding if and only if Qx = 1. Indeed, the first equalityfollows from the property that Tr(AB) = Tr(BA) if A and B are anti-linear. The inequality then is a consequence of that Qx ⊗Q−1

x has tracebigger or equal than dim(x)2.

The irreducible representations of G and the Haar state h are connectedby the orthogonality relations (see [60]).

(id⊗ h)(Ux(ξη∗ ⊗ 1)(Uy)∗) =δx,y1

dimq(x)〈η,Qxξ〉 ,

(id⊗ h)((Ux)∗(ξη∗ ⊗ 1)Uy) =δx,y1

dimq(x)〈η,Q−1

x ξ〉(1.3)

for ξ ∈ Hx and η ∈ Hy.

The first equation can be seen as follows: (ι ⊗ h)(Ux(ξη∗ ⊗ 1)(Uy)∗) ∈Mor(x, y), so it is zero unless x = y. Because tx ∈ Mor(x⊗ x, ε),

Ux13Ux23(tx ⊗ 1) = tx ⊗ 1 ,

so Ux23(t⊗ 1) = (Ux13)∗(t⊗ 1). If we multiply this equation by ξ∗ ⊗ 1⊗ 1

or η∗ ⊗ 1⊗ 1, where ξ, η ∈ Hx and rewrite everything, we get

jx(ξ)∗jx(η)⊗ 1 = (t∗x ⊗ 1)Ux13(ξη∗ ⊗ 1⊗ 1)(Ux13)

∗(tx ⊗ 1) .

Applying the Haar functional gives us

〈η,Qxξ〉1 = dimq(x)(ι⊗ h)(Ux(ξη∗ ⊗ 1)(Ux)∗) ,

which proves the result.

Notation 1.3.8. Let G be a compact quantum group. We denote byC(G) the linear span of the coefficients of finite dimensional representa-tions of G. Hence,

C(G) = 〈(ωξ,η ⊗ id)(Ux) | x ∈ Irred(G), ξ, η ∈ Hx〉

Also, for x ∈ Irred(G), denote by

C(G)x = 〈(ωξ,η ⊗ id)(Ux) | ξ, η ∈ Hx〉


Note that ∆ : C(G)x → C(G)x ⊗alg C(G)x and that C(G)∗x = C(G)x.

Moreover, C(G) is a unital dense ∗-subalgebra of C(G). Restricting ∆ toC(G), C(G) becomes a Hopf ∗-algebra. There exists hence a co-unit onC(G), i.e. a unital *-homomorphism ε : C(G) 7→ C such that

(ε⊗ id)∆(a) = a

for all a ∈ C(G). The co-unit ε is determined by (id⊗ ε)(U) = 1 for everyfinite dimensional representation U .

In fact C(G) carries all the information of the compact quantum group G.It is hence possible to handle with a great deal of problems concerningcompact quantum groups in a purely algebraic way.

There are on the C∗-algebraic level, different appearances of the samequantum group. There is also a von Neumann algebraic version.

Definition 1.3.9. The reduced C∗-algebra Cr(G) is defined as the normclosure of C(G) in the GNS-representation with respect to h. The uni-versal C∗-algebra Cu(G) is defined as the enveloping C∗-algebra of C(G).The von Neumann algebra L∞(G) is defined as the von Neumann algebragenerated by Cr(G).

Note that if G is the dual of a discrete group Γ, we have Cr(G) = C∗r (Γ),

Cu(G) = C∗(Γ) and L∞(G) = L(Γ). So these notions coincide withthe full, respectively reduced group C∗-algebra, respectively group vonNeumann algebra.

Given an arbitrary compact quantum group G, we have surjective homo-morphisms Cu(G) → C(G) → Cr(G), but most of the time we are onlyinterested in Cr(G) and Cu(G). So, given the underlying Hopf∗-algebra,there exist different C∗-versions. From this point of view, we only considertwo quantum groups as being different if the underlying Hopf∗-algebrasare non-isomorphic.

Definition 1.3.10. A compact quantum group G is said to be coamenable(and the discrete quantum group G is said to be amenable) if the homo-morphism Cu(G) → Cr(G) is an isomorphism.

Equivalently, a compact quantum group is coamenable if we have on thesame C∗-algebra a bounded co-unit and a faithful Haar state.


Proposition 1.3.11. The Haar state h is a KMS-state on both Cr(G)and Cu(G) and the modular group is determined by

(id⊗ σht )(Ux) = (Qitx ⊗ 1)Ux(Qitx ⊗ 1) (1.4)

for every x ∈ Irred(G).

Denote that the Haar measure is a trace if and only if for all x ∈ Irred(G),Qx = 1.

Definition 1.3.12. We call a compact quantum group G of Kac type ifthe Haar state h on G is a trace.

Let us remark that the original definition of a Kac algebra is much moreinvolved, but in the compact case it just reduces to this definition.

1.4 Discrete quantum groups and duality

For an abelian locally compact group G, the set of characters can be madeinto an abelian locally compact group G in a natural way. Repeating this

procedure brings us back to our original group, so G ∼= G. This is the

content of the Pontryagin-Van Kampen duality Theorem. If G is compactabelian, its dual will be discrete and vice versa. We now explain how thisis generalized to the setting of compact and discrete quantum groups.

Following Van Daele ([48]), a discrete quantum group is a multiplier Hopf*-algebra whose underlying *-algebra is a direct sum of matrix algebras.The dual of a compact quantum group is such a discrete quantum groupand is defined as follows.

Definition 1.4.1. Let G be a compact quantum group. We define thedual (discrete) quantum group G as follows.

c0(G) =⊕

x∈Irred(G)

L(Hx) , `∞(G) =∏

x∈Irred(G)

L(Hx) .

We regard here a c0-, respectively `∞-direct sum.

The notation introduced above is aimed to suggest the basic examplewhere G is the dual of a discrete group Γ, given by C(G) = C∗(Γ) and

1.4 Discrete quantum groups and duality 17

∆(λx) = λx ⊗ λx for all x ∈ Γ. The map x 7→ λx yields an identificationof Γ and Irred(G) and then, `∞(G) = `∞(Γ), while c0(G) = c0(Γ).

We denote the minimal central projections of `∞(G) and c0(G) by px,x ∈ Irred(G).

We have a natural unitary V ∈ M(c0(G)⊗ C(G)) given by

V =⊕

x∈Irred(G)

Ux . (1.5)

This unitary V implements the duality between G and G. We have anatural comultiplication

∆ : `∞(G) → `∞(G)⊗`∞(G) such that (∆⊗ id)(V) = V13V23 .

From the definition of V, it follows that (id⊗∆)(V) = V12V13. It is henceclear that the duality between G and G connects the multiplication onone algebra with the comultiplication on the other and vice versa.

One can deduce from this the following equivalent way to define theco-product structure on `∞(G).

∆(a)S = Sa for all a ∈ `∞(G), S ∈ Mor(y ⊗ z, x) . (1.6)

One sees that the comultiplication on the discrete quantum group G,and hence the multiplication on G, is determined by the structure of theintertwiners. This is the basis of the Tannaka-Krein Duality theory forcompact quantum groups. In [61], Woronowicz showed that a so-calledconcrete monoidal W*-category is always the representation category of aunique compact quantum group. He also reconstructs the quantum groupfrom a given category. We will adapt this technique later in the thirdchapter to come to more general results concerning monoidal categoriesand actions.

The unitary V is closely related to the right regular representation of thequantum group.

Both the algebras C(G) and c0(G) have two natural representations onthe same Hilbert space.

Using (1.3), we canonically identify the GNS Hilbert space L2(C(G), h)with

L2(G) :=⊕

x∈Irred(G)

(Hx ⊗Hx)


by taking

ρ : C(G) → L(L2(G)) : ρ((ωη,ξ ⊗ id)(Ux)

)ξ0 = ξ ⊗ (1⊗ η∗)tx , (1.7)

λ : C(G) → L(L2(G)) : λ((ωη,ξ ⊗ id)(Ux)

)ξ0 = (1⊗ η∗)tx ⊗ ξ

for all x ∈ Irred(G) and all ξ, η ∈ Hx. Here ξ0 denotes the canonical unitvector in Hε ⊗Hε = C.

Definition 1.4.2. The right regular representation V ∈ L(L2(G)⊗C(G))of G is defined as

V(ρ(a)ξ0 ⊗ 1) =((ρ⊗ id)∆(a)

)(ξ0 ⊗ 1)

Notation 1.4.3. We define

λ : `∞(G) → L(L2(G)) : λ(a)ξx = (ax ⊗ 1)ξx

ρ : `∞(G) → L(L2(G)) : ρ(a)ξx = (1⊗ ax)ξx

for all a ∈ `∞(G), ξx ∈ Hx ⊗Hx.

We define the unitary u ∈ L(L2(G)) by u(ξ ⊗ η) = η ⊗ ξ for ξ ∈ Hx,η ∈ Hx. Note that ρ = (Adu)λ and that u2 = 1.

Proposition 1.4.4. The right regular representation of G is given by

V = (λ⊗ id)(V) .

If we now write

V := (λ⊗ ρ)(V) = (id⊗ ρ)(V) and V = (ρ⊗ ρ)(V21) , (1.8)

the operators V and V are multiplicative unitaries on L2(G) in the senseof [1]. This means they satisfies the pentagonal equation

V12V13V23 = V23V12 and V12V13V23 = V23V12 . (1.9)

It holds that V = (u⊗ 1)V21(u⊗ 1).

The discrete quantum group `∞(G) comes equipped with a naturalmodular structure.

1.5 Example: Ao(F ) and Au(F ) 19

Notation 1.4.5. We have canonically defined states ϕx and ψx on L(Hx)related to (1.3) as follows.

ψx(A) =1

dimq(x)t∗x(A⊗ 1)tx =

Tr(QxA)Tr(Qx)

= (id⊗ h)(Ux(A⊗ 1)(Ux)∗) ,

ϕx(A) =1

dimq(x)t∗x(1⊗A)tx =

Tr(Q−1x A)

Tr(Q−1x )

= (id⊗ h)((Ux)∗(A⊗ 1)Ux) ,

for all A ∈ B(Hx).

Remark 1.4.6. The states ϕx and ψx are significant, since they providea formula for the invariant weights on `∞(G). The left invariant weight isgiven by

∑x∈Irred(G) dimq(x)2ψx, and the right invariant weight is given

by∑

x∈Irred(G) dimq(x)2ϕx.

There is, besides definition 1.3.10, a dual definition of amenability. Theequivalence of both definitions is non-trivial and was proven by Ruanin [41] for Kac algebras. Tomatsu generalized his results to arbitrarydiscrete quantum groups in [46].

Definition 1.4.7. A discrete quantum group G is amenable if there existsa left invariant mean on `∞(G), i.e a state m ∈ `∞(G)∗ s.t.m((ω ⊗ id)∆(x)) = m(x)ω(1) for all ω ∈ `∞(G)∗ and x ∈ `∞(G).

1.5 Example: the universal orthogonal and uni-

tary compact quantum groups

There are several natural examples of compact quantum groups. As wementioned, there are the obvious ones coming from compact and discretegroups. Drinfel’d and Jimbo [20, 32] introduced the q-deformations ofcompact semi-simple Lie groups and Rosso [40] showed that they fit intothe theory of Woronowicz. The universal orthogonal and unitary quan-tum groups were introduced by Van Daele and Wang [50] and studied indetail by Banica [2, 3]. These compact quantum groups can in generalnot be obtained as deformations of classical objects.

Definition 1.5.1. Let F ∈ GL(n,C) satisfying FF = ±1. We define thecompact quantum group G = Ao(F ) as follows.


• Cu(G) is the universal C∗-algebra with generators (Uij) and rela-tions making U = (Uij) a unitary element of Mn(C) ⊗ Cu(G) andU = FUF−1, where (U)ij = (Uij)∗.

• ∆(Uij) =∑

k Uik ⊗ Ukj .

In these examples, the unitary matrix U is a representation, called thefundamental representation. The definition of G = Ao(F ) makes sensewithout the requirement FF = ±1, but the fundamental representationis irreducible if and only if FF ∈ R1. We then normalize such thatFF = ±1.

Remark 1.5.2. Different matrices F can lead to isomorphic compactquantum groups. It is easy to classify the quantum groups Ao(F ). ForF1, F2 ∈ GL(n,C) with FiF i = ±1, we write

F1 ∼ F2 ⇔ there exists a unitary v ∈ U(n) such that F1 = vF2vt.

(1.10)where vt is the transpose of v. Then it holds that Ao(F1) ∼= Ao(F2)if and only if F1 ∼ F2. We now describe a fundamental domain for theequivalence relation ∼ on the matrices F ∈ GL(n,C) satisfying FF = ±1.This provides a precise enumeration of the quantum groups Ao(F ).

Let F ∈ GL(n,C) with FF = ±1. Denote H = Cn and J : H → H thecomplex conjugation. We rather look at the anti-linear operator F = JF ,satisfying F2 = ±1. In the case where F2 = 1, our data come down togiving a real vector space together with a Hilbert space structure on thecomplexification. In the case where F2 = −1, H becomes a right moduleon the quaternions such that, restricting the quaternions to C, we get aHilbert space. In particular H is even-dimensional.

It is then straightforward to provide a fundamental domain for the equi-valence relation (1.10) (see e.g. [64]). Take F ∈ GL(n,C) with FF = ±1.Let F = J |F | be the polar decomposition of F . Then, J is an anti-unitary, J 2 = ±1 and J |F |J ∗ = |F |−1. Define H< as the subspace of Hspanned by the eigenvectors of |F | with eigenvalue λ < 1. Define H> =JH<, which is as well the subspace of H spanned by the eigenvectors of|F | with eigenvalue λ > 1. Finally, let H1 be the subset of eigenvectorsof |F | with eigenvalue 1. Take an orthonormal basis ξ1, . . . , ξk of H< ofeigenvectors of |F | with eigenvalues 0 < λ1 ≤ · · · ≤ λk < 1.


If FF = 1, we have J 2 = 1 and we take an orthonormal basis µ1, . . . , µn−2k

for H1 of real vectors: J µi = µi. If (ei) denotes the standard basis for Cn

and w : (ei) → (ξi,J ξi, µi) denotes the transition unitary, we find that

wtFw =

0 D(λ1, . . . , λk) 0D(λ1, . . . , λk)−1 0 0

0 0 1n−2k

. (1.11)

Here, D(λ1, . . . , λk) denotes the diagonal matrix with the λi along thediagonal.

If FF = −1, we have J 2 = −1, H1 has even dimension and we takean orthonormal basis µ1, . . . , µr, J µ1, . . . ,J µr for H1. If w : (ei) →(ξi, µi,J ξi,J µi) denotes the transition unitary, we find that

wtFw =(

0 D(λ1, . . . , λn/2)−D(λ1, . . . , λn/2)−1 0

), (1.12)

where 0 < λ1 ≤ · · · ≤ λn/2 ≤ 1.

Since the spectrum of F ∗F is invariant under the equivalence relation(1.10), a fundamental domain is given by the matrices in (1.11) with2k ≤ n and 0 < λ1 ≤ · · · ≤ λk < 1 and the matrices in (1.12) with0 < λ1 ≤ · · · ≤ λn/2 ≤ 1.

If F ∈ GL(2,C), we get up to equivalence, the matrices

Fq =(

0 |q|1/2−sgn(q)|q|−1/2 0

)(1.13)

for q ∈ [−1, 1], q 6= 0, with corresponding quantum groups Ao(Fq) ∼=SUq(2), see [59].

This last remark means that if the matrix F has dimension 2, we preciselyobtain the quantized versions of the classical Lie group SU(2), as consi-dered by Woronowicz in [59]. So SUq(2) ∼= Ao(Fq) and dimq(U) =

∣∣q+ 1q

∣∣.Banica [2] has shown that the irreducible representations of Ao(F ) canbe labeled by N, in such a way that the fusion rules are identical to thefusion rules of the compact Lie group SU(2). In particular, Mor(U T©U, ε)is one-dimensional and generated by

tF :=∑i

ei ⊗ Fei ,

where (ei) is the standard basis of Cn.


Theorem 1.5.3 (Banica). Let F ∈ GL(n,C) and FF = ±1. Let G =Ao(F ). Then Irred(G) can be identified with N in such a way that

x⊗ y ∼= |x− y| ⊕ (|x− y|+ 2)⊕ · · · ⊕ (x+ y) ,

for all x, y ∈ N.

Moreover, all quantum groups G having the same fusion rules as SU(2)are isomorphic with one of the Ao(F ).

Definition 1.5.4. For all n ∈ N and F ∈ GL(n,C), we define Au(F ) asthe universal compact quantum group generated by the coefficients of therepresentation U ∈Mn(C)⊗Au(F ) with relations

U and (F ⊗ 1)U(F−1 ⊗ 1) are unitary .

Then, Au(F ) is a compact quantum group.

Banica [3] has shown that the irreducible representations of Au(F ) canbe labeled by the free monoid N ? N generated by α and β. We putan involution on N ? N by defining α := β. Defining Uα := U andUβ := (F⊗1)U(F−1⊗1), the spaces Mor(Uα T©Uβ, ε) and Mor(Uβ T©Uα, ε)are one-dimensional and generated by

tF :=∑i

ei ⊗ Fei resp. sF :=∑i

ei ⊗ F−1ei .

He also computed the corresponding fusion rules.

Theorem 1.5.5 (Banica). Let F ∈ GL(n,C) and let G = Au(F ). ThenIrred(G) can be identified with N ? N in such a way that

x⊗ y ∼=⊕

a,b,g∈N?N|x=ag,y=gb

ab ,

for all x, y ∈ N ? N.

The quantum groups Ao(F ) and Au(F ) are of course universal by their de-finition, but also in a more conceptual way. By this we mean the following.Consider a compact matrix quantum group G. This is a compact quantumgroup generated by the coefficients of one fundamental finite dimensional


unitary representation V ∈Mn(C(G)) (see [62]). The adjoint representa-tion of V has, as we mentioned V as a representant. As V is unitarizable,there exists a matrix F ∈ GL(n,C) such that (F ⊗ 1)V (F−1 ⊗ 1) is uni-tary. This leads to a surjective homomorphism π : C(Au(F )) → C(G)given by (id⊗ π)(U) = V . So every compact matrix quantum group is aquantum subgroup of some Au(F ) in the sense of definition 2.4.1.

For the orthogonal universal quantum groups an analogous story holds.Suppose we have a compact matrix quantum group G with fundamentalrepresentation V ∈Mn(C(G)) equivalent with its contragredient V . Thenthere exists a matrix F ∈ GL(n,C) such that (F ⊗ 1)V (F−1 ⊗ 1) =V . Hence every compact matrix quantum group with self-contragredientfundamental representation is a quantum subgroup of some Ao(F ).

In the next chapter, we will discuss another class of universal quantumgroups, namely the quantum automorphism groups of finite dimensionalC∗-algebras with a state. Before we can do so, we first need to introducethe notion of an action of a quantum group.

Chapter 2

Actions of quantum groups

and spectral subspaces

This chapter is also an introductory one. We recall the basic notions andresults about actions of compact quantum groups. Most of the content isnot new, although we give a quite new and convenient presentation. Atthe end we consider some examples and applications.

In a way analogous to groups acting on operator algebras, compact anddiscrete quantum groups can act on C∗- and von Neumann algebras. Theabstract theory of ergodic actions of compact quantum groups on C∗-algebras has been initiated by Boca [15] and Landstad [34].

In the first section we recall some basic definitions concerning actions ofquantum groups and their spectral structure. The second section focusseson the structure of the spectral subalgebra and the spectral subspaces.The next part takes care of the special case of ergodic actions. We con-tinue with examining the important class of invariant subalgebras andhomogeneous spaces. In the last section the class of quantum automor-phism groups is introduced.

2.1 Basic definitions

An action of a compact group G on a C∗-algebra B is a group homomor-phism α : G → Aut(B) such that the map G → B : g 7→ α(g)(a) is con-tinuous for all a ∈ B. Here we denote by Aut(B) the *-automorphisms of

26 Chapter 2. Actions of quantum groups

B. To quantize this notion, we must lift it to the level of C(G). Well, theaction induces a map α : B → B ⊗ C(G) by defining α(a)(g) = α(g)(a).Note that we identified B ⊗ C(G) and the continuous functions from G

to B. It is straightforward to check that α is a *-homomorphism. More-over, denoting by ∆ the comultiplication on C(G), (id⊗∆)α = (α⊗ id)αbecause α is an action.

This brings us to the following definition.

Definition 2.1.1. Let B be a unital C∗-algebra. A (right) action of Gon B is a unital ∗-homomorphism δ : B → B ⊗ C(G) satisfying

(δ ⊗ id)δ = (id⊗∆)δ and [δ(B)(1⊗ C(G))] = B ⊗ C(G) .

We denote by Bδ = a ∈ B | δ(a) = a⊗ 1 the fixed point algebra of theaction. There is always a canonical conditional expectation E : B → Bδ

given by E(a) = (id⊗ h)δ(a), where h is the Haar state on C(G). By itsdefinition, E is invariant under δ. Indeed,

(E ⊗ id)δ(a) = (id⊗ h⊗ id)(δ ⊗ id)δ(a) = (id⊗ h⊗ id)(id⊗∆)δ(a)

= (id⊗ h)δ(a)⊗ 1 = E(a)⊗ 1

by invariance of the Haar state h. Moreover, E is the unique conditionalexpectation onto the fixed point algebra which is invariant under theaction δ. Indeed, suppose that E1 : B → Bδ is a conditional expectationthat is invariant under δ. Then for a ∈ B,

E1(a) = (E1 ⊗ h)δ(a) = E1E(a) = E(a)

and hence E1 = E.

The action δ is said to be ergodic if the fixed point algebra Bδ equals C1.In that case, B admits a unique invariant state ω given by ω(b)1 = E(b).

The standard example of a compact quantum group action is the actionof a compact quantum group G on itself by comultiplication. Indeed,∆ : C(G) → C(G) ⊗ C(G) satisfies all properties. Moreover, this actionis always ergodic and has the Haar state as invariant state.

We can now examine the relation between an action of a compact quan-tum group G and its irreducible representations. For an ergodic action δ,this is clear. It can be viewed as a representation of G on the GNS space

2.1 Basic definitions 27

Hω, where ω is the invariant state of δ. Because every representationcan be split up in irreducible representations of G, we can look at thedecomposition of δ with respect to Irred(G).

But for a more general action, the following definition also makes sense.

Definition 2.1.2. Let δ : B → B ⊗ C(G) be an action of the compactquantum group G on the unital C∗-algebra B. For every x ∈ Irred(G),We define the spectral subspace associated with x by

Kx = X ∈ Hx ⊗B | (id⊗ δ)(X) = X12Ux13 .

Defining

Mor(δ, x) = S : Hx → B | S linear and δ(Sξ) = (S ⊗ id)(Ux(ξ ⊗ 1)) ,

we have Kx∼= Mor(δ, x), associating to every X ∈ Kx the operator

SX : Hx → B : ξ 7→ X(ξ ⊗ 1). Hence it is clear that Kx is given by theintertwiners between x and the action δ.

Definition 2.1.3. We define the subspaces Bx as

Bx := 〈X(ξ ⊗ 1) | X ∈ Kx, ξ ∈ Hx〉 .

Remark that δ : Bx → Bx ⊗alg C(G)x. In the next section, it will appearthat the map

Hx ⊗Kx → Bx : ξ ⊗X 7→ X(ξ ⊗ 1)

is actually a bijection. After this identification, we have δ(b) = Ux13(b⊗1)for b ∈ Bx, so the action really behaves as the irreducible representationx on the space Bx.

We define B as the subspace of B generated by the natural subspaces, i.e.

B := 〈X(ξ ⊗ 1) | x ∈ Irred(G), X ∈ Kx, ξ ∈ Hx〉 .

Proposition 2.1.4 (Boca, [15]). B is a dense unital *-subalgebra of Band the restriction δ : B → B ⊗alg C(G) defines an action of the Hopf∗-algebra (C(G),∆) on B.


Proof. From the observation

BxBy ⊂ 〈Bz | z ∈ Irred(G) and Mor(x⊗ y, z) 6= 0〉 ,

it follows that B is an algebra. Moreover, as C(G)∗x = C(G)x, it holds thatB∗x = Bx, so B is indeed a *-algebra.

The density follows from the fact that elements of the form

(ι⊗ h)((ι⊗ δ)(X)(Ux13)∗) ,

with X ∈ Hx⊗B, are in Kx. Now by definition δ(B)(1⊗C(G)) is densein B ⊗ C(G) and the coefficients of the Ux are dense in C(G). Hence, ifwe let X run over all Hx⊗B for all x ∈ Irred(G), the second leg is densein B.

By definition, δ : B → B ⊗alg C(G) and hence the restriction of δ to Bgives a Hopf*-action of C(G) on B.

Terminology 2.1.5. We call B the spectral subalgebra of δ.

We saw that a compact quantum group (C(G),∆) has many C∗-versions,while the underlying Hopf∗-algebra is the same. The same remark appliesto actions.

Terminology 2.1.6. An action δ : B → B ⊗ C(G) of G on B is said tobe universal if B is the universal enveloping C∗-algebra of B. It is saidto be reduced if the conditional expectation (id ⊗ h)δ of B on the fixedpoint algebra Bδ is faithful.

In the special case where B = C(G) and δ = ∆, the ∗-algebra B coin-cides with the underlying Hopf ∗-algebra C(G) consisting of coefficientsof finite-dimensional representations. So, we obtain the usual notions ofuniversality and reducedness.

Every action δ : B → B ⊗ C(G) has a natural reduced and a universalcompanion. We only use this for ergodic actions. We define the reducedversion Br as the GNS-representation of B with respect to the invariantstate ω and the universal version Bu as the universal enveloping C∗-algebra of B.

Again, we have surjective homomorphisms Bu → B → Br. As before, weonly consider two actions to be different if the underlying Hopf ∗-algebraactions are different. We make extensively use of this fact.

Actions on von Neumann algebras are defined as follows.

2.2 Bimodular structure of the spectral subalgebra 29

Definition 2.1.7. A right action of a compact (resp. discrete) quantumgroup G (resp. G) on a von Neumann algebra N is an injective normalunital ∗-homomorphism

δ : N → N⊗L∞(G) resp. δ : N → N⊗`∞(G)

satisfying (δ ⊗ id)δ = (id⊗∆)δ, resp. (δ ⊗ id)δ = (id⊗ ∆)δ.

Remark that we do not require the density condition like for C∗-algebraicactions. The reason is that this is automatically fulfilled for von Neumannalgebras. This is a quite deep result and we refer to [51], theorem 2.6 for aproof. This implies that proposition 2.1.4 remains true for von Neumannalgebraic actions. We will although obtain this result in a more directway in the next section.

2.2 Bimodular structure of the spectral subalge-

bra

In this section, we take a closer look at the spectral subalgebra and spec-tral subspaces.

Consider a compact quantum group G which acts on a von Neumannalgebra N via an action δ : N → N⊗L∞(G). Recall that we denote byN the spectral subalgebra of δ.

Take x ∈ Irred(G). For a ∈ N δ and b ∈ N , left and right multiplicationturn N into a N δ-N δ-bimodule.

Definition 2.2.1. We define the following N δ-valued scalar products:

〈a, b〉l = E(ab∗) and 〈a, b〉r = E(a∗b) for all a, b ∈ N . (2.1)

These scalar products are compatible with the bimodule structure, in thesense that

a〈b, c〉l = 〈ab, c〉l and 〈b, c〉ra = 〈b, ca〉r ,〈b, c〉la = 〈b, a∗c〉l and a〈b, c〉r= 〈a∗b, c〉r

for a ∈ N δ and b, c ∈ N .


Using the orthogonality relations 1.3, it is straightforward to check thatthe spaces Nx are mutually orthogonal for these inproducts.

Closely related to the spectral subspaces, we introduce the followingspaces:

Definition 2.2.2. For all x ∈ Irred(G), we define

Kx := X ∈ L(Hx)⊗N | (id⊗ δ)(X) = X12Ux13 .

It is clear that Kx ∼= Hx ⊗Kx as a linear space. We can turn Kx into aN δ-N δ-bimodule by defining

a •X = (1⊗ a)X and X • a = X(1⊗ a)

for all a ∈ N δ and X ∈ Kx.

Definition 2.2.3. Again, we define two N δ-valued scalar products

〈X,Y 〉l = (ψx ⊗ id)(XY ∗) and 〈X,Y 〉r = (ϕx ⊗ id)(X∗Y ) (2.2)

where ϕx and ψx are the states defined in 1.4.5.

Remark that an equivalent definition for the second scalar product is1⊗ 〈X,Y 〉r = (id⊗ E)(X∗Y ).

We now prove that Kx is isomorphic to Nx as a bimodule and moreoverin a way that respects the scalar products.

Proposition 2.2.4. The linear map

Ix : Kx → N : Ix(X) = (Tr⊗id)(X)

is N δ-bimodular and respects the left and right N δ-inner products. Forboth inner products, the adjoint is given by

Jx : N → Kx : Jx(a) = dimq(x)(Q−1x ⊗ 1)(id⊗ id⊗ h)(δ(a)23(Ux13)

∗) .

In particular Jx Ix = id.


Proof. It is clear that Ix is bimodular. Take X,Y ∈ Kx and eidim(x)i=1

an orthonormal basis of Hx. Then

〈Ix(A), Ix(Y )〉l = E((Tr⊗id)(X)(Tr⊗id)(Y )∗)

=dim(x)∑i,j=1

(id⊗ h)δ((e∗i ⊗ 1)X(eie∗j ⊗ 1)Y ∗(ej ⊗ 1))

=dim(x)∑i,j=1

(e∗i ⊗ 1)X((id⊗ h)(Ux(eie∗j ⊗ 1)(Ux)∗)⊗ 1)Y ∗(ej ⊗ 1)

=dim(x)∑i,j=1

ψx(eie∗j )(e∗i ⊗ 1)(XY ∗)(ej ⊗ 1)

= (ψx ⊗ id)(XY ∗) = 〈X,Y 〉l

Moreover

〈Ix(X), Ix(Y )〉r = E((Tr⊗id)(X)∗(Tr⊗id)(Y ))

=dim(x)∑i,j=1

(id⊗ h)δ((e∗i ⊗ 1)X∗(eie∗j ⊗ 1)Y (ej ⊗ 1))

=dim(x)∑i,j=1

(e∗i ⊗ 1)(id⊗ id⊗ h)((Ux13)∗X∗

12(eie∗j ⊗ 1⊗ 1)Y12U

x13)(ej ⊗ 1)

=dim(x)∑i,j=1

(e∗i ⊗ 1)(1⊗ (ϕx ⊗ id)(X∗(eie∗j ⊗ 1)Y ))(ej ⊗ 1)

=dim(x)∑i=1

(ϕx ⊗ id)(X∗(eie∗i ⊗ 1)Y )

= (ϕx ⊗ id)(X∗Y ) = 〈X,Y 〉r ,

so Ix leaves both inner products invariant.

The following calculation

〈Ix(X), a〉l = E((Tr⊗id)(X)a∗)

= (Tr⊗id⊗ h)(X12Ux13δ(a)

∗23)

=1

dimq(x)(Tr⊗id)(XJx(a)∗(Qx ⊗ 1))

= (ψx ⊗ id)(XJx(a)∗) = 〈X,Jx(a)〉l


proves that Jx is the adjoint of Ix for the right inner product.

Also,

〈Ix(X), a〉r = E((Tr⊗id)(X)∗a)

= (Tr⊗id⊗ h)((Ux13)∗X∗

12δ(a)23)

= (Tr⊗id⊗ h)(X∗12δ(a)23(Q

−1x ⊗ 1)(Ux13)

∗(Q−1x ⊗ 1)) (2.3)

= (ϕx ⊗ id)(X∗Jx(a)) = 〈X,Jx(a)〉r ,

where in (2.3) we used the KMS-property of the Haar state. Hence Jx isthe adjoint of Ix for both inner products.

As for all x ∈ Irred(G), Jx Ix = id, we can define the idempotentsPx : N → Nx : a 7→ Ix Jx(a).

As we remarked before, we want to prove in a direct way that N is densein N . The following proposition leads us to the final result.

Proposition 2.2.5. For all a ∈ N

E(a∗a) =∑

x∈Irred(G)

〈Jx(a),Jx(a)〉r .

andE(aa∗) =

∑x∈Irred(G)

〈Jx(a),Jx(a)〉l

with all sums converging strongly.

Proof. Take an orthonormal basis eidim(x)i=1 of Hx and set

fi = (dimq(x)Qx)12 ei. Define bxij = (f∗i ⊗ 1)Ux(ej ⊗ 1). It is straight-

forward to check that bxijx,i,j is an orthonormal basis for L2(G) for theinproduct 〈a, b〉 = h(a∗b).

It follows that

E(a∗a) = (id⊗ h)(δ(a)∗δ(a))

=∑x,i,j

(id⊗ h)(δ(a)∗(1⊗ bxij))(id⊗ h)((1⊗ (bxij)∗)δ(a)) .


We calculate

(id⊗ h)((1⊗ bxij)∗δ(a))

= (id⊗ h)(δ(a)(1⊗ σh−i(bxij)

∗))

= (id⊗ h)(δ(a)(1⊗ (e∗jQ−1x ⊗ 1)(Ux)∗(Q−1

x fi ⊗ 1))

= (e∗jQ−1x ⊗ 1)(id⊗ id⊗ h)(δ(a)23(Ux13)

∗)(Q−1x fi ⊗ 1)

= dimq(x)−12 (e∗j ⊗ 1)Jx(a)(Q

− 12

x ei ⊗ 1) .

Hence

E(a∗a) =∑x,i

dimq(x)−1(e∗iQ− 1

2x ⊗ 1)Jx(a)∗Jx(a)(Q

− 12

x ei ⊗ 1)

=∑

x∈Irred(G)

(ϕx ⊗ id)(Jx(a)∗Jx(a)) =∑

x∈Irred(G)

〈Jx(a),Jx(a)〉r .

The equality for the other inner product is proven analogously by using

the basis fi = dimq(x)12Q

− 12

x ei for L2(G) with inner product 〈a, b〉 =h(ab∗). This ends the proof.

We have now enough information to prove the main result.

Corollary 2.2.6. For all a ∈ N ,

(∑x∈F

Px(a))F⊂Irred(G) ,Ffinite → a

in the left 2-norm and the right 2-norm with respect to every state.

Proof. For a state ω on N , set ‖a‖l,ω = ω(〈a, a〉l)12 .

Take now a finite set F ⊂ Irred(G).

‖a−∑x∈F

Px(a)‖2l,ω

= ω(〈a−∑x∈F

Px(a), a−∑y∈F

Px(a)〉l)

= ω(〈a, a〉l −∑y∈F

〈a,Py(a)〉l −∑x∈F

〈Px(a), a〉l +∑x,y∈F

〈Px(a),Py(a)〉l) .


But 〈a,Px(a)〉l = 〈Jx(a),Jx(a)〉l and 〈Px(a),Py(a)〉l = δx,y〈Jx(a),Jy(a)〉lbecause the spaces Nx are mutually orthogonal. Hence we obtain

‖a−∑x∈F

Px(a)‖2l,ω = ω(E(aa∗)−

∑x∈F

〈Jx(a),Jx(a)〉l)

and this converges to 0 by the previous proposition. This completes theproof.

Corollary 2.2.7. The spectral subalgebra N is dense in N .

Spectral subspaces

The spectral subspaces are closely related to the spaces Nx and Kx above.We can in an analogous way as above, turn the spectral subspaces intobimodules over the fixed point algebra.

Moreover, one can check easily that XY ∗ ∈ N δ for X,Y ∈ Kx, so

〈·, ·〉r : Kx ×Kx → N δ : (X,Y ) 7→ XY ∗ . (2.4)

defines an N δ-valued inner product.

We can also put a right inner product on Kx by means of the conditionalexpectation E onto the fixed point algebra. For X,Y ∈ Kx, an a state ηon Bδ, (id⊗ηE)(X∗Y ) is an intertwiner for Ux and hence scalar. Indeed,

Ux∗((id⊗ ηE)(X∗Y )⊗ 1)Ux = (id⊗ ηE ⊗ id)((Ux13)∗X∗

12Y12Ux13)

= (id⊗ ηE)(id⊗ δ)(X∗Y )

= (id⊗ ηE)(X∗Y )⊗ 1 , (2.5)

where in the last step we used the invariance of E with respect to δ.

This means that we can define

〈·, ·〉r : Kx ×Kx → N δ by 1⊗ 〈X,Y 〉r := (id⊗ E)(X∗Y ) , (2.6)

which makes Kx a right Hilbert C∗-module over N δ.

In the case where δ is ergodic with invariant state ω, Kx is in fact aHilbert space because N δ = C. There are two scalar products on Kx,namely 〈X,Y 〉l1 = Y X∗ and 〈X,Y 〉r1 = (id ⊗ ω)(X∗Y ). Remark thatwe switched orders in the first scalar product to have conjugate linearityin the first variable.

2.3 Ergodic actions 35

Orthogonality relations

If we look at the mutual position of two spectral subspaces, it appearsthat they are in fact orthogonal.

Take x, y ∈ Irred(G), T ∈ L(Hx,Hy) and X ∈ Kx, Y ∈ Ky. It holds that

E(Y (T ⊗ 1)X∗) = (id⊗ h)δ(Y (T ⊗ 1)X∗)

= (id⊗ h)(Y12Uy13(T ⊗ 1⊗ 1)(Ux13)

∗X∗12)

= Y ((id⊗ h)(Uy(T ⊗ 1)Ux∗)⊗ 1)X∗ = δx,y Tr(QxT )Y X∗

using in the last step the orthogonality relations 1.3.

Moreover, analogously as in 2.5, for every state η, (id ⊗ ωE)(X∗Y ) ∈Mor(x, y), so is zero unless x = y. The spectral subspaces are hence alsoorthogonal for the second inner product.

2.3 Ergodic actions

We now turn towards ergodic actions. For ordinary groups, they werethoroughly studied by Høegh-Krohn, Landstad and Størmer in [29]. Animportant result of theirs says that the invariant state of an ergodic ac-tion is always a trace. This yields several consequences, for example thata compact group can never act ergodically on a type III factor. Also,study of the spectral subspaces yields a decomposition of the action intoirreducible representations. It was proven that the multiplicity of an ir-reducible representation is never greater than its dimension. A. Wasser-mann explored the spectral structure of ergodic actions even further in[56, 57, 58], leading to a classification of the ergodic actions of SU(2).

Although most notions can be generalized and a spectral analysis can bedone, things are much more complicated in the case of quantum groups.Even worse, none of the results above remain true!

Firstly, there still is a unique invariant state, but it does no longer have tobe a trace. Take for example the quantum group Au(F ) acting on itselfby comultiplication. The invariant state is given by the Haar state, andthis is not a trace if F is not a multiple of a unitary.

Moreover, the multiplicity bound theorem is no longer true either. Itis possible to construct actions where the multiplicity of an irreducible


action exceeds its dimension. The first such examples will be presentedin the next chapter. There is however a generalization of the multiplicitynotion, quantum multiplicity, which is never bigger than the quantumdimension of an irreducible representation.

All these changes between group actions and quantum group actions makethe classification of actions of the quantum deformations of SU(2) muchmore difficult than the non-quantum case.

Consider a compact quantum group G, a unital C∗-algebra B and anaction δ : B → B⊗C(G). Let Kx, x ∈ Irred(G) be the spectral subspacesof δ as defined before. We already noted that in the ergodic case, Kx isa Hilbert space. Since the spectral subspaces are nothing else than theintertwiners between the action and an irreducible representation, thefollowing definition is a very natural one:

Terminology 2.3.1. Let δ be an ergodic action of G on B. The dimen-sion of Kx is called the multiplicity of x in δ and denoted by mult(x, δ).If there is no ambiguity, we often write mult(x).

Of course, it follows that the spaces Bx are in this case also finite dimen-sional and have dimension dim(x) ·mult(x, δ).

Define, for every x ∈ Irred(G), the element Xx ∈ L(Hx,Kx) ⊗ B suchthat (Xx)∗(Y ⊗ 1) = Y ∗ for all Y ∈ Kx. On the left hand side of thisequation, Y is viewed as element of Kx, while on the right hand side Y ∗ isviewed as an element of Hx ⊗B. Observe that Xx(Xx)∗ = 1. Therefore,(Xx)∗Xx ∈ L(Hx)⊗B is a projection.

Let x ∈ Irred(G). Take t ∈ Mor(x ⊗ x, ε), normalized in such a waythat t∗t = dimq(x). Analogous as in equation (1.2), we can define theantilinear map

Rx : Kx → Kx : v 7→ (t∗ ⊗ 1)(1⊗ v∗) . (2.7)

Since t is fixed up to a number of modulus one, Lx := R∗xRx is a well

defined positive element of L(Kx).

Like the quantum dimension, we now define

Definition 2.3.2. multq(x) :=√

Tr(Lx) Tr(Lx) and we call multq(x) thequantum multiplicity of x in δ.

2.3 Ergodic actions 37

By definition, multq(x) = multq(x). Analogous as or the quantum dimen-sion, it is straightforward to check Rx = R−1

x and hence Lx = (RxR∗x)−1,

so multq(x) =√

Tr(Lx) Tr(L−1x ). Note that invertibility of Rx this im-

plies also that mult(x, δ) = mult(x, δ), but that was already clear by theequation Bx = B∗x.The generalization of the multiplicity bound theorem ([29]) is given bythe following result.

Theorem 2.3.3 (Boca, [15] and Landstad, [35]). Let δ : B → B ⊗C(G)be an ergodic action of a compact quantum group G on a unital C∗-algebraB. For every irreducible representation x ∈ Irred(G),

mult(x) ≤ multq(x) ≤ dimq(x) .

If multq(x) = dimq(x) for all x ∈ Irred(G), we call δ of full quantummultiplicity. With Xx defined as above, the ergodic action is of fullquantum multiplicity if and only if Xx is unitary for all x ∈ Irred(G).

Proof. Let x ∈ Irred(G). We have that

multq(x) =√

Tr(Lx) Tr(L−1x ) ≥ dim(Kx) = mult(x) ,

for all x ∈ Irred(G). By definition

〈X,LxY 〉l1 = 〈X,R∗xRxY 〉l1

= 〈(t∗ ⊗ 1)(1⊗ Y ∗), (t∗ ⊗ 1)(1⊗X∗)〉l1 (2.8)

= (t∗ ⊗ 1)(1⊗X∗Y )(t⊗ 1) .

Observe that t∗(1 ⊗ a)t = Tr(Q−1x a) for all a ∈ L(Hx). Taking an or-

thonormal basis Eii=1...mult(x) of Kx, it then holds that

Tr(Lx)1 =mult(x)∑i=1

(t∗ ⊗ 1)(1⊗ E∗i Ei)(t⊗ 1)

=mult(x)∑i=1

(Tr(Q−1x ⊗ id)(E∗

i Ei)

=mult(x)∑i=1

(Tr(Q−1x ·)⊗ id)((Xx)∗EiEi

∗Xx)

= (Tr(Q−1x ) · ⊗id)((Xx)∗Xx) .


Since (Xx)∗Xx is a projection, it follows that Tr(Lx) ≤ Tr(Q−1x ) =

dimq(x) for all x ∈ Irred(G). Applying this inequality to x and x, weconclude that multq(x) ≤ dimq(x) for all x ∈ Irred(G).

Moreover, multq(x) = dimq(x) for all x ∈ Irred(G) if and only if

(Tr(Q−1x ·)⊗ id)((Xx)∗Xx) = Tr(Q−1

x )1

for all x ∈ Irred(G). This last statement holds if and only if (Xx)∗Xx = 1for all x ∈ Irred(G), i.e. when Xx is unitary for all x ∈ Irred(G).

In the next chapter, ergodic actions of full quantum multiplicity will bethoroughly studied. We will characterize them in terms of unitary fiberfunctors.

Like the Haar state, the invariant state ω is a KMS state, at least foruniversal and for reduced ergodic actions.

Proposition 2.3.4. If the ergodic action δ of a compact quantum groupG on B is either universal or reduced, the invariant state ω is a KMSstate. The elements of the spectral subalgebra B ⊂ B (Definition 2.1.3)are analytic with respect to the modular group, given by

σωt(Y (ξ⊗1)

)= (L−itx Y )(Qitx ξ⊗1) for all x ∈ Irred(G), Y ∈ Kx, ξ ∈ Hx .

Proof. Let y, z ∈ Irred(G) and Y ∈ Ky, Z ∈ Kz. We already knew that(id ⊗ ω)(Y ∗Z) equals 0 if y 6= z and is scalar if y = z. Applying ω to(2.8), we conclude that

(id⊗ ω)(Y ∗Z) =δy,z1

dimq(y)〈Y, LyZ〉 .

If moreover ξ ∈ Hy, η ∈ Hz and if we put a = Y (ξ⊗ 1) and b = Z(η⊗ 1),we get

ω(a∗b) =δy,z

dimq(y)〈ξ, η〉〈Y, LyZ〉 . (2.9)

Using (1.3), one checks that

ω(ba∗) = (id⊗ ω)(Z(ηξ∗ ⊗ 1)Y ∗)

= (id⊗ h)(id⊗ ω ⊗ id)(id⊗ δ)(Z(ηξ∗ ⊗ 1)Y ∗)

= (id⊗ ω)(Z(id⊗ id⊗ h)(U z13(ηξ∗ ⊗ 1⊗ 1⊗ 1)(Uy)∗13)Y

∗)

=δy,z1

dimq(y)〈ξ,Qyη〉〈Y, Z〉 .

2.4 Invariant subalgebras and Homogeneous spaces 39

As a linear space, B ∼=⊕

x∈Irred(G)(Kx ⊗ Hx). So, we can define linearmaps σωt : B → B by the formula

σωt(Y (ξ ⊗ 1)

):= L−ity (Y )(Qity ξ ⊗ 1) .

It is clear that (σωt ) is a one-parameter group of linear isomorphisms ofB. Observe that all elements of B are analytic with respect to (σωt ) andthat, by the computations of ω(a∗b) and ω(ba∗) above,

ω(σωi (a)b∗) = ω(b∗a) = ω(aσωi (b)∗)

for all a, b ∈ B. Since (2.9) implies that ω is faithful on B, it follows thatσωi : B → B is multiplicative and that σωi (a)∗ = σω−i(a

∗) for all a ∈ B. Forall a, b, c ∈ B,

z 7→ ω(σω−z((σωz (a)σωz (b))− ab)c)

satisfies f(z+i) = f(z), is entire and is bounded along a strip 0 ≤ Im(z) ≤1 because of the definition. By Liouville ’s theorem, it is constant 0.Hence σωz is multiplicative, and in an analogous way it is proven that itis *-preserving. This allows to conclude that the σωt are ∗-automorphismsof B. It is also clear that ω is invariant under σωt .

If δ is a universal action, the one-parameter group (σωt ) extends to B byuniversality. If δ is a reduced coaction, we can extend σωt to B becauseω is invariant under σωt and ω is faithful on B. In both cases, it followsthat (σωt ) satisfies the KMS condition with respect to ω and so, ω is aKMS state.

Finally observe that ω is a trace if and only if Lx = 1 and Qx = 1 for allx ∈ G with Kx 6= 0.

2.4 Invariant subalgebras and Homogeneous spaces

In this section, we consider a special type of ergodic actions, given by theinvariant subalgebras and homogeneous spaces. If H is non-trivial closedsubgroup of a compact group G, right multiplication gives an ergodicaction of G on the homogeneous space H\G. On the function level,

∆ : C(H\G) → C(H\G)⊗ C(G)

gives an action.


Quantum subgroups and homogeneous spaces

In the last chapter, we will need the notion of a quantum subgroup.

Definition 2.4.1. Let (G,∆G) and (H,∆H) be compact quantum groups.We call H a closed quantum subgroup of G whenever there is given asurjective *-homomorphism rH : C(G) → C(H) satisfying ∆H rH =(rH ⊗ rH)∆G.

This definition doesn’t come out of thin air. If H is a closed subgroupof a compact group G, there is an obvious surjective *-homomorphismrH : C(G) → C(H). Indeed, just take the *-homomorphism which re-stricts continuous functions on G to the subgroup H. This satisfies theconditions. The converse is also true. Every surjective *-homomorphismfrom C(G) onto C(H) commuting with the comultiplication is given byan embedding of H as a closed subgroup of G.

Definition 2.4.2. Let (G,∆G) a compact quantum group with quan-tum subgroup (H,∆H). Define the Hopf*-algebra action γH : C(G) →C(H) ⊗alg C(G) : x 7→ (rH ⊗ id)∆G(x). Define the homogeneous spaceC(H\G) as the fixed point subalgebra of C(G) under γH.

Since the action γH is invariant under the Haar measure of G, we canextend it to Cr(G) and L∞(G) and hence define Cr(H\G) and L∞(H\G).By universality, γH is also extendable to Cu(G), which gives us Cu(H\G)

The restriction of the comultiplication to Cr(H\G), respectively Cu(H\G),respectively L∞(H\G) or gives an action of G on H\G.

Proposition 2.4.3. The restriction of rH to the quotient C(H\G) is theco-unit εG.

Proof. For a ∈ C(H\G),

∆H(rH(a)) = (rH ⊗ rH)∆G(a) = 1⊗ rH(a) ,

so rH(a) is scalar. We now apply the (id⊗εH) to both sides of the equationand use the fact that εHrH = εG ([45]). This give us that

rH(a) = (id⊗ εH)∆H(rH(a)) = εG(a)1 ,

which ends the proof.

2.4 Invariant subalgebras and Homogeneous spaces 41

In the last chapter, we will need a special kind of subgroup.

Definition 2.4.4. Consider a compact quantum group (G,∆G). We calla quantum subgroup (H,∆H) of Kac type maximal, if for any quantumsubgroup K of Kac type, L∞(H\G) ⊂ L∞(K\G).

Every compact quantum group has a unique maximal quantum subgroupof Kac type (see [43]). We call it the canonical Kac subgroup of thequantum group.

Invariant subalgebras

A more general notion is that of an invariant subalgebra.

Definition 2.4.5. Consider a compact quantum group G with comul-tiplication ∆. A right invariant subalgebra of G is a unital C∗-algebraB ⊂ C(G) such that ∆(B) ⊂ B ⊗ C(G).

If G is an ordinary group, by the Gelfand-Naimark theorem, a invariantsubalgebra is always of the form C(H\G) for a closed subgroup H ⊂ G.In in the quantum case however, such correspondence does not necessarilyhold. A quantum subgroup gives rise to a right invariant subalgebra, butthe converse is no longer true.

We can define an ergodic action δ of G on B by just restricting ∆ to B.

Proposition 2.4.6. Consider a compact quantum group G and a rightinvariant subalgebra B of C(G). Denote the action of G on B by δ. Forall x ∈ Irred(G), mult(x, δ) ≤ dim(x) and equality in all x is only reachedwhen B = C(G).

Proof. Let x ∈ Irred(G). From the definition of a spectral subspace, weget

Kx = X ∈ Hx ⊗B | (id⊗∆)(X) = X12Ux13

It is clear that

Kx ⊂ Kx := X ∈ Hx ⊗ C(G) | (id⊗∆)(X) = X12Ux13

with Kx the spectral subspace of the comultiplication ∆. Now Kx∼= Hx

where the bijection is given by Hx → Kx : ξ 7→ (ξ∗ ⊗ 1)Ux. Thenmult(x, δ) = dim(Kx) ≤ dim(Hx) = dim(x).


Equality for all x ∈ Irred(G) means that Kx = Kx, so Bx = C(G)x andhence B = C(G).

Inspired by the results of Wassermann in [58], Tomatsu calculated all theinvariant subalgebras of SUq(2) in [44]. We remark that some of themare not given as homogeneous spaces of closed quantum subgroups. Inthe next chapter however, we show that these invariant subalgebras giveus far from all the possible ergodic actions of SUq(2). As we constructactions that do not satisfy the multiplicity bound, they cannot arise asinvariant subalgebras.

2.5 Quantum automorphism groups

In this section we consider a class of universal quantum groups, namelythe quantum automorphism groups as studied by Wang in [55] and Banicain [5, 6, 7].

If we are given a unital, finite dimensional C∗-algebra B and a state ϕ,we can pose the following question. What is the most general compactquantum group, acting on this C∗-algebra B in such a way ϕ is invariantunder the action? Or in another way, what are the minimal conditions acompact quantum group must satisfy to act on B with invariant state ϕ?

Suppose we have a compact quantum group G and an action

α : B → B ⊗ C(G)

under which ϕ is invariant. As ϕ induces an inner product on B by〈a, b〉 = ϕ(a∗b), we can regard B as a Hilbert-space. We know thatU ∈ L(B)⊗C(G) such that U(x⊗ 1) = α(x) is a unitary representation.Denote by µ : B⊗B → B the multiplication and by η : C → B the linearunital map. The fact that α is multiplicative and unital is equivalentto µ ∈ Mor(U,U T© U) and η ∈ Mor(U, ε). Moreover, the involutionon B is completely determined by µ, η and the inner product. Indeed,for all a, b ∈ B, 〈a, b∗〉 = ϕ(a∗b∗) = 〈b ⊗ a, µ∗η(1)〉. Hence we denoteb∗ := 〈b ⊗ 1, µ∗η(1)〉B. Now unitarity of U and involutiveness of α areequivalent since µ∗η ∈ Mor(U2, ε). Indeed:

〈b⊗ 1⊗ 1, U23(µ∗η(1)⊗ 1)〉B = α(b∗) and

〈b⊗ 1⊗ 1, U∗13(µ

∗η(1)⊗ 1)〉B = α(b)∗ .

2.5 Quantum automorphism groups 43

So the fact that a representation U ∈ L(B)⊗C(G) gives rise to an actionof G on B can be expressed completely in terms of intertwiners.

We only consider C∗-algebras with a special kind of states.

Definition 2.5.1. Let (B,ϕ) be a finite dimensional C∗-algebra of dimen-sion ≥ 4 with a state. Take δ > 0. If for the inner product implementedby ϕ, one has that µµ∗ = δ21, we call ϕ a δ-form.

If B is a matrix-algebra, every state is of the form Tr(F ·) and a δ-form with δ2 = Tr(F−1). This can easily be checked by writing outµµ∗ in terms of the orthonormal basis (eijF− 1

2 )i,j=1···n of B. If B =⊕ki=1Mni(C), ϕ =

⊕ki=1 Tr(Fi·) is a δ-form if and only if Tr(F−1

i ) = δ2

for every i ∈ 1, . . . , k.

The fact that ϕ is a δ-state precisely corresponds to the representationU ∈ L(B) ⊗ C(G) being the direct sum of an irreducible representationand the trivial representation.

We can now give the definition of a quantum automorphism group:

Definition 2.5.2. [7] Let (B,ω) be an n-dimensional C∗-algebra witha δ-form. We define the compact quantum group G = Aaut(B,ω) asfollows.

• Cu(G) is defined as the universal C∗-algebra generated by the co-efficients of U ∈ L(B) ⊗ Cu(G) and relations making U unitary,η ∈ Mor(U, ε) and µ ∈ Mor(U,U T© U).

• (id⊗∆)(U) = U12U23.

For any m ≥ 2, Cu(Aaut(Mm(C),Tr)) is canonically isomorphic with theenveloping C∗-algebra of the *-subalgebra of Cu(Ao(Im)) generated bythe coefficients of the square of its fundamental representation ([7]).

Representation Theory

In [6], Banica has determined the irreducible representations and theirfusion rules for all quantum automorphism groups.

If B and ω are as above, the fusion rules of Aaut(B,ω) are those of SO(3).This means that the irreducible representations are labeled by N. We


choose Ui ∈ L(Hi) ⊗ C(Aaut(B,ω)) the representative of the irreduciblerepresentation with label i in such a way that U0 is the trivial representa-tion ε and that U = U0 ⊕ U1 ∈ L(B)⊗ C(Aaut(B,ω)) is the fundamentalrepresentation. The fusion rules are given by:

Ui ⊗ Uj = U|i−j| + U|i−j|+1 + · · ·+ Ui+j

We now can describe the *-algebra-structure of B in terms of the in-tertwiners. First of all, every element b of B can be seen as a couple(ω(b), b − ω(b)) = (λ, x) ∈ C ⊕ H1 = B. Also, the product m of twoelements has a scalar part, and a part in H1. By restricting m to H1⊗H1

and than cutting by the unique projection p1 ∈ Mor(U1, U) onto H1, weget a morphism θ ∈ Mor(U1, U

21 ). In the same way, we get a morphism

γ ∈ Mor(ε, U21 ). Actually, γ equals the restriction of ωm to H2

1 . Andas already remarked above, γ also implements the involution on B. Ofcourse the multiplication also consists of the following obvious morphisms:

T : H1 ⊗H0 → H1 : x⊗ 1 7→ x

S : H0 ⊗H1 → H1 : 1⊗ x 7→ x

R : H0 ⊗H0 → H0 : 1⊗ 1 7→ 1

If now a = (λ, x) and b = (µ, y) are elements of B, the multiplication isgiven by

m(a⊗ b) = (R(λ⊗ µ) + γ(x⊗ y), S(λ⊗ y) + T (x⊗ µ) + θ(x⊗ y)) .

Chapter 3

Ergodic actions with large

multiplicity and monoidal

equivalence of quantum

groups

This chapter is an extended version of the article [14], which is joint workwith Stefaan Vaes and Julien Bichon.

The results in this chapter can be summarized as follows.

• In theorem 2.3.3, we introduced ergodic actions of full quantum mul-tiplicity. These are precisely the ergodic actions for which the crossedproduct is isomorphic with the compact operators, see [34]. In the-orem 3.2.1 below, we provide a concrete bijective correspondence be-tween such ergodic actions of full quantum multiplicity and unitaryfiber functors.

• In Section 3.3, we study the special case of unitary fiber functors pre-serving the dimension. This leads to a bijective correspondence withunitary 2-cocycles on the dual, discrete, quantum group. The ideas forthis section come from the work of Wassermann [57].

• In Section 3.4, we establish the monoidal equivalence between theuniversal orthogonal quantum groups Ao(F ). Recall that, for any

46 Chapter 3. Ergodic actions with large multiplicity

F ∈ GL(n,C) satisfying FF = ±1, one defines the compact quan-tum group Ao(F ) as the universal quantum group generated by thecoefficients of a unitary n by n matrix U with relations U = FUF−1.The comultiplication on Ao(F ) is (uniquely) defined in such a waythat U becomes a representation. We show that Ao(F1) is monoidallyequivalent with Ao(F2) if and only if the signs of the FiFi agree andTr(F ∗

1F1) = Tr(F ∗2F2). In particular, if 0 < q ≤ 2−

√3, there is a con-

tinuous family of non-isomorphic Ao(F ) monoidally equivalent withSUq(2).

• In Section 3.5, we prove similar results for the universal unitary quan-tum groups Au(F ), defined as the universal quantum group generatedby the coefficients of a unitary n by n matrix U with the relation thatFUF−1 is unitary. Again, the comultiplication is defined such that Ubecomes a representation. We show that the quantum dimension of U ,i.e.

√Tr(F ∗F ) Tr((F ∗F )−1), is a complete invariant for the Au(F ) up

to monoidal equivalence.

• Using the previous results, we obtain a complete classification of theergodic actions of full quantum multiplicity of Ao(F ) and Au(F ), aswell as a computation of the 2-cohomology of their duals (Corollary3.4.6). In particular, we construct ergodic actions of SUq(2) such thatthe multiplicity of the fundamental representation is (for non-fixed q)arbitrarily large (Corollary 3.4.5).

• In the last section, we study monoidal equivalence for quantum auto-morphism groups Aaut(B,ϕ) with B a C∗-algebra of finite dimensionbigger than 4 and ϕ a δ-state on B. We prove that δ is a complete in-variant for Aaut(B,ϕ) up to monoidal equivalence and that this class ofquantum automorphism groups is closed under monoidal equivalence.

We choose not to use the abstract language of categories. We give ‘down-to-earth’ definitions of monoidally equivalent quantum groups and unitaryfiber functors, see 3.1.1 and 3.1.5. This makes the construction of asso-ciated C∗-algebras and actions straightforward. This concrete approachis well adapted to the language of representations of compact quantumgroups. In this way, using previous results of Banica [2, 3], we can showvery easily as well the monoidal equivalence of the universal orthogonaland universal unitary quantum groups and for the quantum automor-phism groups.

3.1 Monoidal equivalence of quantum groups 47

3.1 Monoidal equivalence of quantum groups

One can look at the representation category of a compact quantum groupwith several degrees of precision. The least accurate way is to look at thefusion rules. This does not determine the quantum group at all. Forinstance, all universal orthogonal quantum groups Ao(F ) have the samefusion rules as SU(2). But even at the level of ordinary groups, thereexist non-isomorphic compact groups with isomorphic representation rings.The dihedral group D4 and the quaternion group Q8 for example, havethe same fusion rules and are far from being the same group! However, atheorem of Handelman ([27]) ensures that this phenomenon cannot occurin the case of connected compact groups.

There exist of course reconstruction theorems that enable us to recovera compact group from its representation data. A compact group canbe reconstructed from its Krein-algebra (Tannaka-Krein reconstructiontheorem, [28], chapter 7). But also less concrete information, namelythe representation category, viewed as a rigid, monoidal, symmetric C∗-category suffices to recover the group completely (Doplicher-Roberts re-construction theorem, [19]).

Although the Tannaka-Krein reconstruction theorem could be generalizedto compact quantum groups in [63], the result of Doplicher and Roberts isno longer true. Indeed, contrary to the tensor C∗-category of an ordinarycompact group, the tensor C∗-category of representations of a compactquantum group can have several concrete realizations into the categoryof Hilbert spaces.

We define in this section what it means for two compact quantum groupsto be monoidally equivalent. In the rest of the chapter we will constructactions of full quantum multiplicity by means of this notion. In the nextchapter, it will be showed that monoidally equivalent quantum groupshave, in a sense, the same actions on C∗-algebras.

Definition 3.1.1. Two compact quantum groups G1 = (C(G1),∆1) andG2 = (C(G2),∆2) are said to be monoidally equivalent if there exists abijection ϕ : Irred(G1) → Irred(G2) satisfying ϕ(ε) = ε, together withlinear isomorphisms

ϕ : Mor(x1 ⊗ · · · ⊗ xr, y1 ⊗ · · · ⊗ yk)

→ Mor(ϕ(x1)⊗ · · · ⊗ ϕ(xr), ϕ(y1)⊗ · · · ⊗ ϕ(yk))


satisfying the following conditions:

ϕ(1) = 1 ϕ(S ⊗ T ) = ϕ(S)⊗ ϕ(T )

ϕ(S∗) = ϕ(S)∗ ϕ(ST ) = ϕ(S)ϕ(T )(3.1)

whenever the formulas make sense. In the first formula, we consider1 ∈ Mor(x, x) = Mor(x⊗ ε, x) = Mor(ε⊗x, x). Such a collection of mapsϕ is called a monoidal equivalence between G1 and G2.

Remark 3.1.2. To define a monoidal equivalence between G1 and G2, itsuffices to define a bijection ϕ : Irred(G1) → Irred(G2) satisfying ϕ(ε) =ε, together with linear isomorphisms

ϕ : Mor(y1 ⊗ · · · ⊗ yk, x) → Mor(ϕ(y1)⊗ · · · ⊗ ϕ(yk), ϕ(x))

for k = 1, 2, 3, satisfying

ϕ(1) = 1 (3.2)

ϕ(S)∗ϕ(T ) = ϕ(S∗T ) if S ∈ Mor(x⊗ y, a), T ∈ Mor(x⊗ y, b)(3.3)

ϕ((S ⊗ 1)T ) = (ϕ(S)⊗ 1)ϕ(T ) if T ∈ Mor(b⊗ z, a), S ∈ Mor(x⊗ y, b)(3.4)

ϕ((1⊗ S)T ) = (1⊗ ϕ(S))ϕ(T ) if T ∈ Mor(x⊗ b, a), S ∈ Mor(y ⊗ z, b)(3.5)

Indeed, such a ϕ admits a unique extension to a monoidal equivalence.Again, (3.2) should be valid for 1 ∈ Mor(x, x) = Mor(x ⊗ ε, x) =Mor(ε⊗ x, x).

Remark 3.1.3. Observe that the existence of the linear isomorphismsϕ : Mor(a⊗b, c) → Mor(ϕ(a)⊗ϕ(b), ϕ(c)) only says that G1 and G2 havethe same fusion rules. Adding (3.2)–(3.5) means that G1 and G2 moreoverhave the same 6j-symbols (see [17]). Indeed, taking orthonormal basesfor all Mor(a ⊗ b, c), we can write two natural orthonormal bases forMor(x ⊗ y ⊗ z, c), one given by elements (S ⊗ 1)T , the other given byelements (1 ⊗ S)T . The coefficients of the transition unitary betweenboth orthonormal bases are called the 6j-symbols of G1.

As we shall see in Sections 3.4 and 3.5, there are natural examples ofmonoidal equivalences where dim(ϕ(x)) 6= dim(x). On the other hand,

3.1 Monoidal equivalence of quantum groups 49

it is clear that dimq(ϕ(x)) = dimq(x) for all x ∈ G and all monoidalequivalences ϕ. Indeed, just apply (3.3) to t ∈ Mor(x ⊗ x, ε) such thatt∗t = dimq(x). We shall see in Section 3.5 that for a certain class ofcompact quantum groups (the universal unitary ones), this equality ofquantum dimension is the only constraint for monoidal equivalence.

Notation 3.1.4. If two compact quantum groups G1 and G2 are monoidallyequivalent, we write G1 ∼

monG2.

Monoidal equivalence of compact quantum groups can be expressed inseveral ways. Two equivalent descriptions, one of them being the equi-valence of C∗-tensor categories, are discussed in section 3.2.1.

Closely related to the notion of monoidal equivalence, is the followingnotion of a unitary fiber functor (see Proposition 3.2.4 for the relationbetween both notions).

Definition 3.1.5. Let G = (C(G),∆) be a compact quantum group. Aunitary fiber functor associates to every x ∈ Irred(G) a finite dimensionalHilbert space Hϕ(x) and consists further of linear maps

ϕ : Mor(y1 ⊗ · · · ⊗ yk, x1 ⊗ · · · ⊗ xr)

→ L(Hϕ(x1) ⊗ · · · ⊗Hϕ(xr),Hϕ(y1) ⊗ · · · ⊗Hϕ(yk))

satisfying equations (N.3) in Definition 3.1.1.

We make a remark analogous to 3.1.2. To define a unitary fiber functoron G, it suffices to associate to every x ∈ Irred(G) a finite-dimensionalHilbert space Hϕ(x), with Hϕ(ε) = C and to define linear maps

ϕ : Mor(y1 ⊗ · · · ⊗ yk, x) → L(Hϕ(x),Hϕ(y1) ⊗ · · · ⊗Hϕ(yk))

for k = 1, 2, 3, satisfying (3.2) – (3.5) as well as the non-degeneratenessassumption

ϕ(S)ξ | a ∈ Irred(G), S ∈ Mor(b⊗c, a), ξ ∈ Hϕ(a) spans Hϕ(b)⊗Hϕ(c)

for all b, c ∈ Irred(G).

The non-degeneracy is necessary to extend a map ϕ that satisfies (3.2) –(3.5) to a unitary fiber functor. Remark that a monoidal equivalence anda unitary fiber functor are always non-degenerate. Indeed, take b, c ∈


Irred(G) and consider the finite dimensional Hilbert space End(b ⊗ c).We can find minimal projections qk such that idb⊗c =

∑k qk. Now the

restriction of U b⊗c to qkHb⊗c gives an irreducible subrepresentation Ukof U b⊗c. Moreover, the embedding ιk : Hk → Hb⊗c is an element ofMor(U b⊗c, Uk). Take now xk ∈ Irred(G) such that Uk ∼ Uxk and takeOk ∈ Mor(Uk, Uxk) unitary. Set pk = ιkOk. Now

∑k ϕ(pk)ϕ(pk)∗ =

idϕ(b)⊗ϕ(c), so indeed

ϕ(S)ξ | a ∈ Irred(G), S ∈ Mor(b⊗ c, a), ξ ∈ Hϕ(a) = Hϕ(b) ⊗Hϕ(c) .

3.2 Ergodic actions of full quantum multiplicity

This section contains the main results of this chapter. We proof thecorrespondence between unitary fiber functors and ergodic actions of fullquantum multiplicity. This will enable us to give new examples of ergodicactions in further sections.

Theorem 3.2.1. Consider a compact quantum group G = (C(G),∆) andlet ϕ be a unitary fiber functor on G.

• There exists a unique unital ∗-algebra B equipped with a faithful stateω and unitary elements Xx ∈ L(Hx,Hϕ(x))⊗B for all x ∈ Irred(G),satisfying

1. Xy13X

z23(S ⊗ 1) = (ϕ(S)⊗ 1)Xx for all S ∈ Mor(y ⊗ z, x) ,

2. the matrix coefficients of the Xx form a linear basis of B,

3. (id⊗ ω)(Xx) = 0 if x 6= ε.

• There exists a unique action δ : B → B ⊗alg C(G) satisfying

(id⊗ δ)(Xx) = Xx12U

x13

for all x ∈ Irred(G).

• The state ω is invariant under δ. Denoting by Br the C∗-algebragenerated by B in the GNS-representation associated with ω anddenoting by Bu the universal enveloping C∗-algebra of B, the actionδ admits a unique extension to an action on Br, resp. Bu.

These actions are reduced, resp. universal and they are ergodic andof full quantum multiplicity.

3.2 Ergodic actions of full quantum multiplicity 51

• Every reduced, resp. universal, ergodic action of full quantum mul-tiplicity, arises in this way from a unitary fiber functor.

We have the same information as for the Tannaka-Krein Duality theorem[63], apart from the concreteness of the monoidal category. It is hencenatural to try to prove this result in the same spirit.

Proof. Let ϕ be a unitary fiber functor on G. Define the vector spaceB = ⊕x∈Irred(G)L(Hx,Hϕ(x))∗. We shall turn this vector space into a∗-algebra.

Define natural elements Xx ∈ L(Hx,Hϕ(x)) ⊗ B by (ωx ⊗ id)(Xx) =(δx,yωx)y∈Irred(G) for all ωx ∈ L(Hx,Hϕ(x))∗. By definition, the coefficientsof the Xx form a linear basis of B. Hence, it suffices to define a productand an involution on the level of the Xx.

It is clear that there exists a unique bilinear multiplication map B×B → Bsuch that

Xy13X

z23(S ⊗ 1) = (ϕ(S)⊗ 1)Xx for all S ∈ Mor(y ⊗ z, x) .

Indeed, take y, z ∈ Irred(G) and consider a basis Sxi ∈ Mor(y ⊗ z, x)for all x ∈ Irred(G). Then the bilinear map defined by

Xy13X

z23 :=

∑x,i

(ϕ(Sxi )⊗ 1)Xx(S∗i ⊗ 1)

will do the job.

But then

(Xa14X

b24)X

c34 ((S ⊗ 1)T ⊗ 1) = (ϕ(S)⊗ 1⊗ 1)Xy

13Xc23(T ⊗ 1)

= ((ϕ(S)⊗ 1)ϕ(T )⊗ 1)Xx = (ϕ((S ⊗ 1)T )⊗ 1)Xx

for all S ∈ Mor(a ⊗ b, y), T ∈ Mor(y ⊗ c, x). Since intertwiners of theform (S ⊗ 1)T linearly span Mor(a⊗ b⊗ c, x), we conclude that

(Xa14X

b24)X

c34 (S ⊗ 1) = (ϕ(S)⊗ 1)Xx

for all S ∈ Mor(a⊗ b⊗ c, x). Analogously,

Xa14(X

b24X

c34) (S ⊗ 1) = (ϕ(S)⊗ 1)Xx


for all S ∈ Mor(a⊗ b⊗ c, x). This proves the associativity of the producton B. It is clear that Xε provides the unit element of B.

Because for all b ∈ Irred(G) and T ∈ Mor(x⊗ y, b)

(ϕ(S)∗ ⊗ 1)Xx13X

y23(T ⊗ 1) = (ϕ(S∗T )⊗ 1)Xz = Xz(S∗T ⊗ 1) ,

it holds that(ϕ(S)∗ ⊗ 1)Xx

13Xy23 = Xz(S∗ ⊗ 1) (3.6)

for all S ∈ Mor(x⊗ y, z).

There exists an antilinear map b 7→ b∗ on B such that

(Xx)∗13(ϕ(t)⊗ 1) = Xx23(t⊗ 1) (3.7)

for all x ∈ Irred(G), t ∈ Mor(x⊗ x, ε). Indeed: taking t ∈ Mor(x⊗ x, ε)and t ∈ Mor(x⊗ x, ε), normalized in such a way that (t∗ ⊗ 1)(1⊗ t) = 1,we define

((ωη,ξ ⊗ id)(Xx)

)∗ := (ω(1⊗η∗)ϕ(t),(ξ∗⊗1)t ⊗ id)(Xx) (3.8)

for all ξ ∈ Hx and η ∈ Hϕ(x). Remark that equation (3.7) and (3.8) areequivalent.

For ξ ∈ Hx, η ∈ Hϕ(x), we compute(((ωη,ξ ⊗ id)(Xx)

)∗)∗ =((ω(1⊗η∗)ϕ(t),(ξ∗⊗1)t ⊗ id)(Xx)

)∗= (ω(1⊗((1⊗η∗)ϕ(t))∗ϕ(t)),(((ξ∗⊗1)t)∗⊗1)t ⊗ id)(Xx)

= (ωη,ξ ⊗ id)(Xx) ,

in the last step using our particular choice of t and t.

We also get(t∗ ⊗ 1)(Xx

23)∗ = (ϕ(t)∗ ⊗ 1)Xx

13 .

Because

(Xx(Xx)∗)13(ϕ(t)⊗ 1) = Xx13X

x23(t⊗ 1) = ϕ(t)⊗ 1

and because, by (3.6),

(t∗ ⊗ 1)((Xx)∗Xx)23 = t∗ ⊗ 1 ,

the elements Xx are unitaries.


Since for all x, y, z ∈ Irred(G) and S ∈ Mor(x⊗ y, z),

(Xx13X

y23(S ⊗ 1))∗ = ((ϕ(S)⊗ 1)Xz)∗ = (Xz)∗(ϕ(S)∗ ⊗ 1)

= (S∗ ⊗ 1)(Xy23)

∗(Xx13)

∗

by (3.6) and the fact that the Xx are unitary, our involution is anti-multiplicative. We conclude that B is a *-algebra.

Denote by ω the linear functional ω : B → C given by ω(1) = 1 and(id⊗ ω)(Xx) = 0 for all x 6= ε. We show that ω is a faithful state on B.

Let x, y ∈ Irred(G). Take t ∈ Mor(x⊗x, ε) such that t∗t = dimq(x). Taket ∈ Mor(x⊗ x, ε) such that (t∗ ⊗ 1)(1⊗ t) = 1. Then,

(ωρ,µ⊗ id)(Xy)(ωη,ξ ⊗ id)(Xx)∗ = (ωρ⊗(1⊗η∗)ϕ(t),µ⊗(ξ∗⊗1)t⊗ id)(Xy13X

x23) .

We know that (id⊗ω)(Xx13X

x23)t = ϕ(t). Because 1

dimq(x) tt∗ ∈ End(x⊗x)

is the projection onto the trivial representation and by definition of ω, itfollows that (id⊗ ω)(Xy

13Xx23) = δx,y

dimq(x)ϕ(t)t∗. We conclude that

ω((ωρ,µ ⊗ id)(Xy)(ωη,ξ ⊗ id)(Xx)∗

)= δx,y

1dimq(x)

〈ρ⊗ (1⊗ η∗)ϕ(t), ϕ(t)t∗(µ⊗ (ξ∗ ⊗ 1)t)〉

= δx,y1

dimq(x)〈(µ∗ ⊗ 1)t, (ξ∗ ⊗ 1)t〉〈ρ⊗ (1⊗ η∗)ϕ(t), ϕ(t)〉

= δx,y1

dimq(x)〈ξ,Qxµ〉〈η, ρ〉 .

Choose orthonormal bases (fxi ) for every space Hϕ(x). Any element a ∈ Badmits a unique decomposition

a =∑x,i

(ωfxi ,ξ

xi⊗ id)(Xx)

in terms of vectors ξxi ∈ Hx. Then,

ω(aa∗) =∑x,i

1dimq(x)

〈ξxi , Qxξxi 〉 .

It follows that ω(aa∗) ≥ 0 for all a and that ω(aa∗) = 0 if and only ifξxi = 0 for all x and i, i.e. if and only if a = 0.


We define an action δ : B → B ⊗alg C(G) by (id ⊗ δ)(Xx) = Xx12U

x13.

It is clear that ω is invariant under δ. It follows that we can extendδ to actions δr, resp. δu of (Cr(G),∆), respectively (Cu(G)∆) on Br,respectively Bu. Moreover, ω(x)1 = (id ⊗ h)δr(x) for all x ∈ Br andanalogously for x ∈ Bu. It follows that δr and δu are ergodic actions.

Remark that the spectral subspace Kx of δ is isomorphic with Hϕ(x) viathe map θ : Hϕ(x) → Kx : η 7→ (η∗ ⊗ 1)Xx. The unitary elements Xx

from section 2.3 are then linked with our Xx via θ. Hence by Theorem2.3.3, it follows that δr and δu are of full quantum multiplicity.

By definition, the action δr on Br is reduced and the action δu on Buis universal. Indeed, the canonical ∗-subalgebra of Bu generated by thespectral subspaces for δu, is exactly B.

It remains to show that any reduced, resp. universal ergodic action of fullquantum multiplicity arises as above from a unitary fiber functor. Letδ : B → B ⊗ C(G) be an ergodic action of full quantum multiplicity.Construct for x ∈ Irred(G), the unitary elements Xx ∈ L(Hx,Kx) ⊗ B

as in Section 2.3. Define Hϕ(x) := Kx. Let S ∈ Mor(y1 ⊗ · · · ⊗ yk,

x1 ⊗ · · · ⊗ xr). The element

Xy11,k+1 · · ·X

ykk,k+1(S ⊗ 1)(Xx1

1,r+1 · · ·Xxrr,r+1)

∗

is invariant under id⊗ δ. So, we can define ϕ(S) by the formula

Xy11,k+1 · · ·X

ykk,k+1(S ⊗ 1) = (ϕ(S)⊗ 1)Xx1

1,r+1 · · ·Xxrr,r+1

for all S ∈ Mor(y1⊗ · · · ⊗ yk, x1⊗ · · · ⊗xr). It is clear that ϕ is a unitaryfiber functor on G.

Denote by B the ∗-subalgebra of B generated by the spectral subspaces ofδ as defined in Section 2.3. By definition, B is generated by the coefficientsof the Xx. In order to show that B is isomorphic to the ∗-algebra definedby the unitary fiber functor ϕ, it suffices to show that the coefficients ofthe Xx form a linear basis of B. It is clear that the coefficients of theXx generate B. They are linearly independent by the observation that(ωY ,ξ ⊗ id)(Xx) = Y (ξ ⊗ 1) for Y ∈ Kx, ξ ∈ Hx and by (2.9).

We have the following notion of isomorphism for fiber functors.


Definition 3.2.2. Two unitary fiber functors ϕ and ψ on a compactquantum group G are said to be isomorphic if there exist unitaries ux ∈L(Hϕ(x),Hψ(x)) satisfying

ψ(S) = (uy1 ⊗ · · · ⊗ uyk)ϕ(S)(u∗x1

⊗ · · · ⊗ u∗xr)

for all S ∈ Mor(y1 ⊗ · · · ⊗ yk, x1 ⊗ · · · ⊗ xr).

If two unitary fiber functors are isomorphic, they give rise to essentiallythe same actions.

Proposition 3.2.3. Let ϕ and ψ be unitary fiber functors on G anddenote by δϕ, resp. δψ the associated actions on Bϕ, resp. Bψ. Then thefollowing statements are equivalent.

• The fiber functors ϕ and ψ are isomorphic.

• There exists a ∗-isomorphism π : Bϕ → Bψ satisfying (π ⊗ id)δϕ =δψπ.

Proof. Suppose first that ϕ and ψ are isomorphic. The map

π : L(Hx,Hϕ(x))∗ → L(Hx,Hψ(x))

∗ : f 7→ f u∗x

gives an isomorphism between Bϕ and Bψ. Denote byXx ∈ L(Hx,Hϕ(x))⊗Bϕ and Y x ∈ L(Hx,Hψ(x))⊗Bψ the canonical unitaries corresponding tothe unitary fiber functors ϕ and ψ. Now Y x = (ux ⊗ π)Xx, so

(id⊗ (π ⊗ id)δϕ)(Xx) = (id⊗ π ⊗ id)(Xx12U

x13) = (id⊗ π)(Xx)12Ux13

= (u∗x ⊗ 1⊗ 1)(id⊗ δψ)(Y x) = (id⊗ δψπ)(Xx)

Conversely, suppose that π : Bϕ → Bψ is a *-isomorphism satisfying(π ⊗ id)δϕ = δψπ. Let Xx and Y x be as above. Now Y x(id⊗ π)(Xx)∗ isinvariant under id⊗δψ, so it is of the form ux⊗1 with ux ∈ L(Hϕ(x),Hψ(x))unitary. For S ∈ Mor(y1 ⊗ · · · ⊗ yk, x1 ⊗ · · · ⊗ xr), it holds now that

ψ(S)⊗ 1 = Y y11,k+1 · · ·Y

ykk,k+1(S ⊗ 1)(Y x1

1,r+1 · · ·Yxrr,r+1)

∗

= (uy1 ⊗ · · · ⊗ uyk⊗ 1)(ϕ(S)⊗ 1)(u∗x1

⊗ · · · ⊗ u∗xk⊗ 1)

by definition of the uxi and uyi . We conclude that ϕ and ψ are isomorphic.


The following proposition follows immediately from the Tannaka-Kreinreconstruction theorem, but we give a detailed statement for clarity. Itsproof is completely analogous to the proof of Theorem 3.2.1.

Proposition 3.2.4. Let ϕ be a unitary fiber functor on a compact quan-tum group G = (C(G),∆).

• There exists a unique universal compact quantum group G2 withunderlying Hopf ∗-algebra (C(G2),∆2) and unitary representationsUϕ(x) ∈ L(Hϕ(x))⊗ C(G2) satisfying

1. Uϕ(y)13 U

ϕ(z)23 (ϕ(S)⊗ 1) = (ϕ(S)⊗ 1)Uϕ(x) for all

S ∈ Mor(y ⊗ z, x),

2. the matrix coefficients of the Uϕ(x) form a linear basis of C(G2).

• Uϕ(x) | x ∈ G is a complete set of irreducible representations ofG2 and ϕ is a monoidal equivalence of compact quantum groups.

The following result is then a corollary of Theorem 3.2.1.

Proposition 3.2.5. Consider two compact quantum groups G1 and G2.Let ϕ : G1 → G2 be a monoidal equivalence. In particular, ϕ is a unitaryfiber functor on G1.

Denote by Br, resp. Bu the C∗-algebras associated to ϕ as in Theorem3.2.1, with dense ∗-subalgebra B. Denote by δ1 the corresponding actionof (C(G1),∆1) on B. Denote by Xx ∈ L(Hx,Hϕ(x)) ⊗ B the unitariesgenerating B.

• There is a unique action δ2 : B → C(G2) ⊗alg B satisfying(id ⊗ δ2)(Xx) = U

ϕ(x)13 Xx

23 for all x ∈ Irred(G1). The action δ2commutes with δ1 and extends to Br, resp. Bu, yielding a reduced,resp. universal, ergodic action of full quantum multiplicity.

• Every pair of commuting reduced, resp. universal, ergodic actions offull quantum multiplicity arises in this way from a monoidal equi-valence.

Proof. Given the monoidal equivalence ϕ, it is obvious to construct theaction δ2.


It remains to show the second statement. Let δ1 : B → B ⊗ C(G1)and δ2 : B → C(G2) ⊗ B be commuting ergodic actions of full quantummultiplicity. Denote by B the unital ∗-subalgebra of B generated by thespectral subspaces of δ1. Using Theorem 3.2.1, we get a unitary fiberfunctor ϕ on G1 and we may assume that B and δ1 are constructed fromϕ as in Theorem 3.2.1. In particular, B is generated by the coefficients ofXx ∈ L(Hx,Hϕ(x))⊗ B.

Because δ1 and δ2 commute, the element (id⊗ δ2)(Xx)(Xx)∗13 is invariantunder (id⊗ id⊗ δ1). Since δ1 is ergodic and Xx unitary, we get a unitaryelement Uϕ(x) ∈ L(Hϕ(x)) ⊗ C(G2) such that (id ⊗ δ2)(Xx) = U

ϕ(x)12 Xx

13.Because δ2 is a action, we easily compute that Uϕ(x) is a unitary repre-sentation of G2.

It remains to show that Uϕ(x) | x ∈ Irred(G1) is a complete set ofirreducible unitary representations of G2 and that ϕ is a monoidal equi-valence.

Assume that S ∈ Mor(ϕ(x), ϕ(y)). The element (Xy)∗(S ⊗ 1)Xx ∈L(Hx,Hy) ⊗ B is invariant under id ⊗ δ2, so it has the form T ⊗ 1,with T ∈ L(Hx,Hy). It follows that T ∈ Mor(x, y) = δx,yC and hence,S ∈ δx,yC. So, the Uϕ(x) are mutually inequivalent irreducible represen-tations of G2.

In order to show that the set Uϕ(x) | x ∈ Irred(G1) exhausts all irre-ducible representations of G2, it suffices to show, for all a ∈ C(G2), that(id⊗ h2)((1⊗ a)Uϕ(x)) = 0 for all x ∈ Irred(G1), implies a = 0. Then forall x ∈ Irred(G), we get (id⊗h2⊗id)((1⊗a⊗1)Uϕ(x)

12 Xx13) = 0. But, given

the formula for δ2, and the fact that the coefficients of the Xx generateB, this means we get (h2 ⊗ id)((a⊗ 1)δ2(x)) = 0 for all x ∈ B. Since δ2is of full quantum multiplicity, δ2(B)(1⊗B) is total in C(G2)⊗B. Thisimplies that a = 0.

It remains to show that ϕ is a monoidal equivalence. For this, it suffices toshow that Mor(ϕ(y)⊗ϕ(z), ϕ(x)) = ϕ(Mor(y⊗z, x)). If S ∈ Mor(y⊗z, x),we use the multiplicativity of δ2 to obtain

(ϕ(S)⊗ 1⊗ 1)Uϕ(x)12 Xx

13 = (id⊗ id⊗ δ2)((ϕ(S)⊗ 1)Xx)

= (id⊗ id⊗ δ2)(Xy13X

z23(S ⊗ 1))

= Uϕ(y)13 Xy

14 Uϕ(z)23 Xz

24 (S ⊗ 1⊗ 1)

= Uϕ(y)13 U

ϕ(z)23 (ϕ(S)⊗ 1⊗ 1)Xx

13 .


It follows that S ∈ Mor(ϕ(y) ⊗ ϕ(z), ϕ(x)). The converse inclusion isshown analogously.

If two compact quantum groups G1 and G2 are given together with amonoidal equivalence ϕ : G1 → G2, we can prove the following proposi-tion.

Proposition 3.2.6. The orthogonality relations (1.3) generalize and takethe following form.

(id⊗ ω)(Xx(ξ1η∗1 ⊗ 1)(Xy)∗) =δx,y1

dimq(x)〈η1, Qxξ1〉 ,

(id⊗ ω)((Xx)∗((ξ2η∗2 ⊗ 1)Xy) =δx,y1

dimq(x)〈η2, Q

−1ϕ(x)ξ2〉 ,

(3.9)

for ξ1 ∈ Hx, η1 ∈ Hy, ξ2 ∈ Hϕ(x) and η2 ∈ Hϕ(y).

Proof. Denote by δ1 and δ2 the actions constructed from the monoidalequivalence. The first relation follows from the calculation below, wherein the last step we used (1.3).

(id⊗ ω)(Xx(ξ1η∗1 ⊗ 1)(Xy)∗)⊗ 1

= (id⊗ id⊗ h)(id⊗ δ1)(Xx(ξ1η∗1 ⊗ 1)(Xy)∗)

= (id⊗ id⊗ h)(Xx12U

x13(ξ1η

∗1 ⊗ 1⊗ 1)(Uy)∗13(X

y)∗12)

= Xx((id⊗ h)(Ux(ξ1η∗1 ⊗ 1)(Uy)∗)⊗ 1)(Xy)∗

=δx,y1

dimq(x)〈η1, Qxξ1〉1 .

The other orthogonality relation is proven in an analogous way, but nowusing δ2.

Given a monoidal equivalence ϕ : G1 → G2, we can consider the conversemonoidal equivalence ϕ−1 : G2 → G1. This gives then rise to a *-algebraB generated by the coefficients of unitaries Y x ∈ L(Hϕ(x),Hx)⊗ B.

Proposition 3.2.7. It holds that B ∼= Bop, where Bop is just B equippedwith the opposite multiplication.


Proof. The elements (Xx)∗ ∈ L(Hϕ(x),Hx)⊗Bop satisfy the requirementsin theorem 3.2.1 for the converse monoidal equivalence ϕ−1. Hence wedefine the vector space isomorphism π : B → B such that (id⊗ π)(Y x) =(Xx)∗. It is easily checked that π is an anti-isomorphism, proving thatB ∼= Bop.

3.2.1 An alternative look at monoidal equivalence

In this subsection, we describe two equivalent ways to look at monoidalequivalence. The first is inspired by the following observation, made tous by Fons Van Daele. He remarked that one can look at the concept ofmonoidal equivalence in a “dual” way of the one above. The other is thetranslation of our definition 3.1.1 to the language of tensor C∗-categories.

Proposition 3.2.8. Consider two compact quantum groupsG1 = (C(G1),∆1) and G2 = (C(G2),∆2) and a bijection

ϕ : Irred(G1) → Irred(G2) .

Set `∞(G1, G2) =∏x∈Irred(G) L(Hx,Hϕ(x)). There is a bijective corre-

spondence between:

1. Monoidal equivalences with bijection ϕ.

2. Coassociative linear maps

∆ : `∞(G1, G2) → `∞(G1, G2)⊗`∞(G1, G2)

satisfying

∆(ax) = ∆2(a)∆(x) and ∆(xb) = ∆(x)∆1(b) (3.10)

∆(b)∗∆(a) = ∆1(b∗a) and ∆(b)∆(a)∗ = ∆2(ba∗) (3.11)

for all x, y ∈ `∞(G1, G2), a ∈ `∞(G2), b ∈ `∞(G1).

Proof. Suppose ϕ : G1 → G2 is a monoidal equivalence. Define

∆ : `∞(G1, G2) → `∞(G1, G2)⊗`∞(G1, G2)

by∆(b)T = ϕ(T )bz for T ∈ Mor(x⊗ y, z) ,


where bz ∈ L(Hz,Hϕ(z)) is the z-component of b. Because ϕ is a monoidalequivalence, it is clear that ∆ is coassociative. Equation (3.10) followsimmediately from the definition of ∆ and of ∆1 and ∆2 like given in (1.6).Equation (3.11) follows from the fact that ϕ is a monoidal equivalence.

Conversely, suppose that ∆ : `∞(G1, G2) → `∞(G1, G2)⊗`∞(G1, G2) isa coassociative map satisfying (3.10). For all x, y, z ∈ Irred(G1) andT ∈ Mor(x⊗ y, z), we define a linear map

S : L(Hz,Hϕ(z)) → L(Hz,Hϕ(x) ⊗Hϕ(y)) : b 7→ ∆(b)T .

Now S satisfies the property that for all a ∈ L(Hz), b ∈ L(Hz,Hϕ(z)),S(ba) = S(b)a. Indeed, from equation (3.10), it follows that

S(ba) = ∆(ba)T = ∆(b)∆1(a)T = ∆(b)Ta = S(b)a .

Such a linear map has the property that there exists a unique ϕ(T ) ∈L(Hϕ(z),Hϕ(x) ⊗Hϕ(y)) such that S(b) = ϕ(T )b.

From (3.10), it follows that for all b ∈ `∞(G2) and T ∈ Mor(x⊗ y, z),

∆2(b)ϕ(T ) = ϕ(T )b

and hence ϕ(T ) ∈ Mor(ϕ(x) ⊗ ϕ(y), ϕ(z)). We extend ϕ to linear mor-phisms ϕ : Mor(y1⊗· · ·⊗yk, x) → Mor(ϕ(y1)⊗· · ·⊗ϕ(yk), ϕ(x)) by defin-ing ϕ((S ⊗ 1)T ) := (ϕ(S)⊗ 1)ϕ(T ) and ϕ((1⊗ S)T ) := (1⊗ ϕ(S))ϕ(T ).This is well-defined by the co-associativity of ∆.

Take S ∈ Mor(x⊗ y, z) and T ∈ Mor(x⊗ y, u). For all a ∈ L(Hz,Hϕ(z))and b ∈ L(Hu,Hϕ(u)), it holds

a∗ϕ(S)∗ϕ(T )b = S∗∆(a)∗∆(b)T = S∗∆1(a∗b)T = S∗Ta∗b .

Hence ϕ(S)∗ϕ(T ) = S∗T .

By using the counterpart of equation (3.10), we define in an analogousway as ϕ the map ψ : G2 → G1. It is clear that ϕψ = id = ψϕ. Weobtain a monoidal equivalence between G1 and G2.

Monoidal equivalence of G1 and G2 can be described in yet another way,as an equivalence of tensor C∗-categories Rep(G1) and Rep(G2). We


prefer not to introduce the whole categorical machinery, but just indicatehow things work. We refer to [25] and [19] for basic notions of (tensor)C∗-categories.

An equivalence ψ of the tensor C∗-categories Rep(G1) and Rep(G2) asso-ciates to every finite dimensional unitary representation U of G1 on HU ,a finite dimensional unitary representation ψ(U) of G2 on Hψ(U) andassociates to intertwiners for G1, intertwiners for G2. Several compatibi-lity relations should hold. In particular, for all finite dimensional unitaryrepresentations U, V of G1, there are isomorphisms

W(U, V ) : Hψ(U T©V ) → Hψ(U) ⊗Hψ(V ) ,

satisfying

(W(U1, U2)⊗ 1)W(U1 T© U2, U3) = (1⊗W(U2, U3))W(U1, U2 T© U2) ,

(ψ(T )⊗ ψ(S))W(U, V ) = W(U, V )ψ(T ⊗ S)

for T ∈ Mor(U ′, U) and S ∈ Mor(V ′, V ) for U ′, V ′ ∈ Rep(G1). One thenunambiguously defines

W(U1, . . . , Uk) : Hψ(U1 T©... T©Uk) → Hψ(U1) ⊗ . . .Hψ(Uk)

for all U1, . . . , Uk ∈ Rep(G1).

Given such an equivalence ψ of the tensor C∗-categories Rep(G1) andRep(G2), we take representatives Ux, x ∈ Irred(G1) and Uy, y ∈ Irred(G2)and a bijection

ϕ : Irred(G1) → Irred(G2)

such that ψ(Ux) = Uϕ(x).

For x1, . . . , xk, y1, . . . , yr ∈ Irred(G1), we define

ϕ : Mor(x1⊗. . .⊗xk, y1⊗. . .⊗yr) → Mor(ϕ(x1)⊗. . .⊗ϕ(xk), ϕ(y1)⊗. . .⊗ϕ(yr))

by ϕ(S) := W(Ux1 , . . . , Uxk)ψ(S)W(Uy1 , . . . , Uyr)∗. This gives us amonoidal equivalence of G1 and G2.

Conversely, starting form a monoidal equivalence ϕ : G1 → G2, we definean equivalence between the categories Rep(G1) and Rep(G2) as follows.First, observe that Rep(Gi) is isomorphic to the category of representa-tions of `∞(Gi) with tensor product π1 T© π2 = (π1 ⊗ π2)∆i. Consider`∞(G1, G2) as a W∗-`∞(G2)-`∞(G1)-bimodule.


Whenever π ∈ Rep(G1), we define the Hilbert space

Hψ(π) := `∞(G1, G2)⊗πH

with the representation

ψ(π) : `∞(G2) → L(Hψ(π)) given by ψ(π)(a)(b⊗ ξ) = ab⊗ ξ .

The natural isomorphism

W(π1, π2) : Hψ(π1 T©ψ2)) → Hψ(π1) ⊗Hψ(π2)

is given by the composition of

Hψ(π1 T©π2) = `∞(G1, G2) ⊗(π1⊗π2)∆1

(H1 ⊗H2)

∆⊗id−→ (`∞(G1, G2)⊗`∞(G1, G2)) ⊗π1⊗π2

(H1 ⊗H2)

−→ Hψ(π1) ⊗Hψ(π2) .

A unitary fiber functor on G translated to the category-theoretic language(see [25]) is a *-functor from the category Rep(G) to the category ofHilbert spaces. Such a thing is called a representation of Rep(G). Eachunitary fiber functor on G and hence each representation of Rep(G) givesvia 3.2.4 rise to a compact quantum group that is monoidally equivalentto G.

3.3 Unitary fiber functors preserving the dimen-

sion

We study in this section unitary fiber functors ϕ on a compact quantumgroup G preserving the dimension, i.e. satisfying dimHϕ(x) = dimHx forall x ∈ Irred(G). Taking into account Theorem 3.2.1, this comes downto the study of ergodic actions of full quantum multiplicity satisfyingmult(x) = dim(x) for all x ∈ Irred(G).

We establish a relation between unitary fiber functors preserving the di-mension (up to isomorphism) and the 2-cohomology of the dual, discretequantum group (c0(G), ∆). The following definition is due to Landstad[35] and Wassermann [57], who consider it for the dual of a compactgroup.

3.3 Unitary fiber functors preserving the dimension 63

Definition 3.3.1. A unitary element Ω ∈ M(c0(G)⊗ c0(G)) is said to bea 2-cocycle if it satisfies

(∆⊗ id)(Ω)(Ω⊗ 1) = (id⊗ ∆)(Ω)(1⊗ Ω) . (3.12)

Two 2-cocycles Ω1 and Ω2 are said to differ by a coboundary if there existsa unitary u ∈ M(c0(G)) such that Ω2 = ∆(u)Ω(u∗ ⊗ u∗). We denote thisrelation by Ω1 ∼ Ω2 and observe that ∼ is an equivalence relation on theset of 2-cocycles.

Remark 3.3.2. In the quantum setting, there is no reason that theproduct of two 2-cocycles is again a 2-cocycle. So, although we coulddefine the 2-cohomology of G as the set of equivalence classes of 2-cocycles,this set has no natural group structure.

Notation 3.3.3. Remember that we denote by px, x ∈ Irred(G), theminimal central projections of c0(G) = ⊕xL(Hx).

Remark 3.3.4. Up to coboundary, we can and will assume that a unitary2-cocycle is normalized, i.e.

(pε ⊗ 1)Ω = pε ⊗ 1 and (1⊗ pε)Ω = 1⊗ pε .

Let Ω be a normalized unitary 2-cocycle on G, the dual of G = (C(G),∆).Denote

Ω(2) := (∆⊗ id)(Ω)(Ω⊗ 1) = (id⊗ ∆)(Ω)(1⊗ Ω) .

It follows from Remark 3.1.2 that there is a unique unitary fiber functorϕΩ on G satisfying

HϕΩ(x) = Hx , ϕΩ(S) = Ω∗S , ϕΩ(T ) = Ω∗(2)T ,

for all S ∈ Mor(y ⊗ z, x) and T ∈ Mor(x ⊗ y ⊗ z, a). Observe that weimplicitly used that L(Hy ⊗ Hz) is an ideal in M(c0(G) ⊗ c0(G)) andhence, Ω∗S is a well defined element of L(Hx,Hy ⊗ Hz) whenever S ∈Mor(y ⊗ z, x).

From Proposition 3.2.4, we get a compact quantum group (C(GΩ),∆Ω),whose dual (c0(GΩ), ∆Ω) is given by

c0(GΩ) = ⊕xL(Hx) = c0(G) and ∆Ω(a)ϕΩ(S) = ϕΩ(S)a


for all a ∈ L(Hx), S ∈ Mor(y ⊗ z, x). Also, ϕΩ becomes a monoidalequivalence between (C(G),∆) and (C(GΩ),∆Ω). Observe that

∆Ω(a) = Ω∗∆(a)Ω for all a ∈ c0(GΩ) = c0(G) .

Proposition 3.3.5. Let ϕ be a unitary fiber functor on a compact quan-tum group G such that dimHϕ(x) = dimHx for all x ∈ Irred(G). Thenthere exists a normalized unitary 2-cocycle Ω on G, uniquely determinedup to coboundary, such that ϕ is isomorphic with ϕΩ.

Proof. Denote by δ : Br → Br ⊗ Cr(G) the reduced ergodic action asso-ciated with ϕ by Theorem 3.2.1. Consider the generating unitaries Xx ∈L(Hx,Hϕ(x))⊗ B satisfying (id⊗ δ)(Xx) = Xx

12Ux13 for all x ∈ Irred(G).

Since dimHϕ(x) = dimHx, we can take unitary elements ux : Hϕ(x) →Hx. Take uε = 1. Define Y x = (ux ⊗ 1)Xx and consider Y := ⊕xY x ∈M(c0(G)⊗B). Because the element (∆⊗ id)(Y )Y ∗

23Y∗13 is invariant under

(id⊗ id⊗ δ), we find a unitary element Ω ∈ M(c0(G)⊗ c0(G)) such that

(∆⊗ id)(Y ) = (Ω⊗ 1)Y13Y23 .

Applying ∆⊗ id⊗ id and id⊗ ∆⊗ id to this equality, we obtain that Ωis a unitary 2-cocycle on (c0(G), ∆).

It remains to show that ϕ and ϕΩ are isomorphic. Let S ∈ Mor(y⊗ z, x).Then,

(S ⊗ 1)Y x = (∆⊗ id)(Y )(S ⊗ 1)

= (Ω⊗ 1)Y13Y23(S ⊗ 1)

= (Ω(uy ⊗ uz)⊗ 1)Xy13X

z23(S ⊗ 1)

= (Ω(uy ⊗ uz)⊗ 1)(ϕ(S)⊗ 1)Xx

= (Ω(uy ⊗ uz)ϕ(S)u∗x ⊗ 1)Y x .

Hence, ϕΩ(S) = Ω∗S = (uy ⊗ uz)ϕ(S)u∗x for all S ∈ Mor(y ⊗ z, x).

It is obvious that ϕΩ1 is isomorphic with ϕΩ2 if and only if the 2-cocyclesΩ1 and Ω2 differ by a coboundary.

Fix a normalized unitary 2-cocycle Ω on G and consider the unitary fiberfunctor ϕΩ. Theorem 3.2.1 yields C∗-algebras BΩ

r and BΩu with ergodic

actions δr and δu of full quantum multiplicity. It is, of course, possible

3.3 Unitary fiber functors preserving the dimension 65

to describe these C∗-algebras directly in terms of Ω: they correspond tothe Ω-twisted group C∗-algebras of G. Such Ω-twisted group C∗-algebrashave been studied by Landstad [35] and Wassermann [57] when Ω is aunitary 2-cocycle on the dual of a compact group.

Definition 3.3.6. An Ω-representation of G on a Hilbert space K is aunitary X ∈ M(c0(G)⊗K(K)) satisfying

(∆⊗ id)(X) = (Ω⊗ 1)X13X23 .

The following lemma can be checked immediately using the formulas inNotation 1.8.

Lemma 3.3.7. Denoting Ω := (1 ⊗ u)Ω∗21(1 ⊗ u), the unitary ΩV ∈

M(c0(G) ⊗ K(H)) is an Ω-representation. It is called the right regularΩ-representation.

Proof. We make the following calculation:

(∆⊗ id)(Ω)

= (1⊗ 1⊗ u)(∆⊗ id)(Ω∗21)(1⊗ 1⊗ u)

= (1⊗ 1⊗ u)((id⊗ ∆)(Ω)∗312)(1⊗ 1⊗ u)

= (1⊗ 1⊗ u)((1⊗ Ω)(Ω∗ ⊗ 1)(∆⊗ id)(Ω∗))312(1⊗ 1⊗ u)

= (Ω⊗ 1)Ω13(1⊗ 1⊗ u)V31(1⊗ 1⊗ u)Ω23(1⊗ 1⊗ u)V ∗31(1⊗ 1⊗ u)

= (Ω⊗ 1)(ΩV )13Ω23V∗13 .

Together with the fact that (∆⊗id)(V ) = V13V23, this ends the proof.

The next lemma is crucial to define the twisted quantum group C∗-algebras.

Lemma 3.3.8. Let X be an Ω-representation of G on K. Then,[(µ⊗ id)(X) | µ ∈ c0(G)∗] is a unital C∗-algebra.

Proof. Write B := [(µ ⊗ id)(X) | µ ∈ c0(G)∗]. From the defining rela-tion for an Ω-representation, it follows that B is an algebra acting non-


degenerately on K. Since (∆⊗ id)(X)X∗23 = (Ω⊗ 1)X13, we have

B = [(µ⊗ η ⊗ id)((∆⊗ id)(X)X∗

23

)| µ, η ∈ c0(G)∗]

= [(µ⊗ η ⊗ id)(V12X13V

∗12X

∗23

)| µ, η ∈ c0(G)∗]

= [(µ⊗ η ⊗ id)(X13V

∗12X

∗23

)| µ, η ∈ c0(G)∗]

= [(µ⊗ η ⊗ id)(X13

((1⊗K(H))V ∗(K(H)⊗ 1)

)12X∗

23

)| µ, η ∈ c0(G)∗]

= [(µ⊗ η ⊗ id)(X13(K(H)⊗K(H)⊗ 1)X∗

23

)| µ, η ∈ c0(G)∗]

= [(µ⊗ id)(X)(η ⊗ id)(X)∗ | µ, η ∈ c0(G)∗] = [BB∗] .

Here we used the regularity of the multiplicative unitary V . It followsthat B is a C∗-algebra. Since Ω is normalized, we have (pε⊗1)X = pε⊗1and hence, B is unital.

Let X be an Ω-representation of G and set B = [(µ⊗id)(X) | µ ∈ c0(G)∗].Observe that X ∈M(c0(G)⊗B)).

Definition 3.3.9. We define the unital C∗-algebras

C∗r (G,Ω) := [(µ⊗ id)(ΩV ) | µ ∈ c0(G)∗] and

C∗u(G,Ω) := [(µ⊗ id)(X) | µ ∈ c0(G)∗] ,

where X denotes a universal Ω-representation.

Remark that an Ω-representation X on K is said to be universal if forany Ω-representation Y on K1, there exists a *-homomorphism

π : [(µ⊗ id)(X) | µ ∈ c0(G)∗] → [(µ⊗ id)(Y ) | µ ∈ c0(G)∗]

satisfying (id ⊗ π)(X) = Y . Although it can be shown directly that auniversal Ω-representation exists, it also follows from the following propo-sition.

Proposition 3.3.10. Denote by B the unital ∗-algebra associated byTheorem 3.2.1 with the unitary fiber functor ϕΩ and by δ the associatedergodic action. Consider the unitaries Xx ∈ L(Hx) ⊗ B generating B.Denote by BΩ

r and BΩu the associated reduced and universal C∗-algebra.

For any Ω-representation X of G on a Hilbert space K, we obtain a re-presentation π of B on K given by (id ⊗ π)(Xx) = (px ⊗ 1)X for allx ∈ Irred(G). Taking X = ΩV , we get an isomorphism BΩ

r∼= C∗

r (G,Ω).Taking X to be a universal Ω-representation, we get an isomorphismBΩu∼= C∗

u(G,Ω).

3.4 Monoidal equivalence for Ao(F ) 67

Proof. The formula (id⊗ π)(Xx) = (px⊗ 1)X defines a one-to-one corre-spondence between representations of B and Ω-representations of G. Thisalready shows the isomorphism BΩ

u∼= C∗

u(G,Ω).

Consider π : B → L(H) given by (id ⊗ π)(Xx) = (px ⊗ 1)(ΩV ). Denoteby ω the unique invariant state on B. To prove the isomorphism BΩ

r∼=

C∗r (G,Ω), it suffices to define a faithful state ω1 on C∗

r (G,Ω) such thatω1π = ω. Define α : C∗

r (G,Ω) → M(K(H)⊗Cr(G)) : α(a) = V (a⊗ 1)V ∗.Now

(id⊗ α)(ΩV ) = V23(ΩV )12V ∗23 = Ω12V23V12V

∗23

= (ΩV )12V13 = (ΩV )12V13 .

As a consequence

(id⊗ απ)(Xx) = (px ⊗ α)(ΩV ) = (px ⊗ 1⊗ 1)((ΩV )12V13)

= (id⊗ π)(Xx)12Ux13 = (id⊗ π ⊗ id)(id⊗ δ)Xx

Hence, α : C∗r (G,Ω) → C∗

r (G,Ω) ⊗ Cr(G) is an action satisfying απ =(π ⊗ id)δ. It follows that (id⊗ h)α(a) ∈ C1 for all a ∈ C∗

r (G,Ω). So, wecan define ω1 by the formula ω1(a)1 = (id⊗h)α(a). Clearly, ω1π = ω.

3.4 Monoidal equivalence for Ao(F )

In this section, we will apply the results of this chapter to the class ofuniversal orthogonal compact quantum groups. The definition of Ao(F )was given in section 1.5.

Recall that we denoted

tF :=∑i

ei ⊗ Fei ,

where (ei) is the standard basis of Cn.

We can express the conditions for monoidal equivalence completely interms of the matrix F .


Theorem 3.4.1. Let F ∈ GL(n,C) with FF = c1 and c ∈ R. ConsiderG = Ao(F ).

• Take F1 ∈ GL(n1,C) satisfying F1F 1 = c11 and cTr(F ∗F ) = c1

Tr(F ∗1 F1) .There exists a unitary fiber functor ϕF1 on G, uniquely determinedup to isomorphism, such that

ϕF1

( 1√Tr(F ∗F )

tF)

=1√

Tr(F ∗1F1)

tF1 . (3.13)

• Every unitary fiber functor ϕ on G is isomorphic with one of theform ϕF1. Moreover, ϕF1 is isomorphic with ϕF2 if and only ifn1 = n2 and there exists a unitary v ∈ U(n1) and a λ 6= 0 such thatF2 = λvF1v

t.

Proof. Take a parameter β ∈ R \ 0. Consider the universal gradedC∗-algebra (A(n,m))n,m∈N satisfying An,m = 0 if n − m is odd andgenerated by elements v(r, s) ∈ A(r+s, r+s+2) with relations (denoting1n the unit of the C∗-algebra A(n) := A(n, n))

v(r, s)∗v(r, s) = 1r+sv(r, s+ 1)∗v(r + 1, s) = β 1r+s+1

v(r, k + l + 2)v(r + k, l) = v(r + k + 2, l)v(r, k + l)

v(r, k + l + 2)∗v(r + k + 2, l) = v(r + k, l)v(r, k + l)∗

Take F ∈ GL(n,C) with FF = c1 and c ∈ R. Put β = cTr(F ∗F ) . Let G =

Ao(F ) and denote by Un the n-fold tensor product of the fundamentalrepresentation, with the convention that U0 = ε. Take the isometrict ∈ Mor(U2, ε) given by t = 1√

Tr(F ∗F )tF . Then (t∗ ⊗ 1)(1⊗ t) = β 1.

We get a natural ∗-homomorphism

π : (A(n,m))n,m∈N → (Mor(Um, Un))n,m∈N

given by π(v(r, s)) = 1r ⊗ t ⊗ 1s. Because of Proposition 1 in [2], π issurjective. It follows from the comments after Theoreme 4 in [3], that πis faithful on A(n, n) for all n. But then π is faithful on A(n,m) because

π(T ) = 0 ⇔ π(T ∗T ) = 0 ⇔ T ∗T = 0 ⇔ T = 0 .


We conclude that π is a ∗-isomorphism.

Take F1 ∈ GL(n1,C) satisfying F1F 1 = c11 and cTr(F ∗F ) = c1

Tr(F ∗1 F1) .Write K = Cn1 and denote by Kn the n-fold tensor product of K, withthe convention that K0 = C. From the preceding discussion, we obtain afaithful ∗-homomorphism

π : (Mor(Um, Un))n,m∈N → (L(Kn,Km))n,m∈N

satisfying π(t) = 1√Tr(F ∗1 F1)

tF1 , π(1) = 1 and π(1⊗T ⊗ 1) = 1⊗ π(T )⊗ 1

for all T .

We choose a concrete identification Irred(G) = N as follows. We definePn ∈ Mor(Un, Un) as the unique projection satisfying PnT = 0 for allr < n and all T ∈ Mor(Un, U r). We define Un as the restriction of Un tothe image of Pn. We then identify

Mor(m1 ⊗ · · · ⊗mk, n1 ⊗ · · · ⊗ nr)

= (Pm1 ⊗ · · · ⊗ Pmk) Mor(Um1+···+mk , Un1+···+nr)(Pn1 ⊗ · · · ⊗ Pnr) .

Define Hϕ(n) := π(Pn)Kn and define ϕ(S) by restricting π toMor(m1 ⊗ · · · ⊗ mk, n1 ⊗ · · · ⊗ nr). It is now obvious that ϕ is a uni-tary fiber functor on G.

Suppose conversely that ϕ is a unitary fiber functor on Ao(F ). We con-tinue to use the concrete identification Irred(G) = N introduced above.Up to isomorphism, we may assume that Hϕ(1) = Cn1 and we denoteK = Cn1 . We define the ∗-homomorphism

π : (Mor(Un, Um))n,m → (L(Kn,Km))n,m

by restricting ϕ. Define the matrix F1 such that ϕ(tF ) = tF1 . ThenF1F 1 = c11 with c1 = c and Tr(F ∗

1F1) = Tr(F ∗F ). Since tF gene-rates the graded C∗-algebra (Mor(Um, Un))n,m, π coincides with the ∗-homomorphism constructed in the first part of the proof starting withF1. Denoting by Tn ∈ Mor(1⊗ · · · ⊗ 1, n) the embedding, we get unitaryoperators ϕ(Tn) : Hϕ(n) → π(Pn)Kn that implement the isomorphismbetween ϕ and ϕF1 .

As an immediate consequence, we have the following.


Corollary 3.4.2. Let F ∈ GL(n,C) with FF = c1 and c ∈ R. ConsiderG = Ao(F ). A compact quantum group G1 is monoidally equivalent withG if and only if there exists F1 ∈ GL(n1,C) satisfying F1F 1 = c11 and

cTr(F ∗F ) = c1

Tr(F ∗1 F1) such that G1∼= Ao(F1).

Remark that even without knowing about Theorem 3.4.1, a compactquantum group that is monoidally equivalent to some Ao(F ) must beof the form Ao(F1). Indeed, monoidal equivalence requires having thesame fusion rules.

Proof. The unitary fiber functor ϕF1 constructed in Theorem 3.4.1 yieldsa monoidal equivalence Ao(F ) ∼

monAo(F1). Since these fiber functors ϕF1

are, up to isomorphism, the only unitary fiber functors on Ao(F ), we aredone.

So, we exactly know when the compact quantum groups Ao(F1) andAo(F2) are monoidally equivalent. If this is the case, Proposition 3.2.5provides us with a universal C∗-algebra Bu and a pair of ergodic actionsof full quantum multiplicity. It is possible to give an intrinsic descriptionof this C∗-algebra Bu.

Theorem 3.4.3. Let Fi ∈ GL(ni,C) by such that FiF i = ci1 for ci ∈ R(i = 1, 2). Assume that c1 = c2 and Tr(F ∗

1F1) = Tr(F ∗2F2).

• Denote by Cu(Ao(F1, F2)) the universal unital C∗-algebra generatedby the coefficients of

Y ∈Mn2,n1(C)⊗ Cu(Ao(F1, F2)) with relations

Y unitary and Y = (F2 ⊗ 1)Y (F−11 ⊗ 1) .

Then, Cu(Ao(F1, F2)) 6= 0 and there exists a unique pair of com-muting universal ergodic actions of full quantum multiplicity, δ1 ofAo(F1) and δ2 of Ao(F2), such that

(id⊗ δ1)(Y ) = Y12(U1)13 and (id⊗ δ2)(Y ) = (U2)12Y13 .

Here, Ui denotes the fundamental representation of Ao(Fi).

• (Cu(Ao(F1, F2)), δ1, δ2) is isomorphic with the C∗-algebra Bu andthe actions thereon given by Proposition 3.2.5 and the monoidalequivalence Ao(F1) ∼

monAo(F2) of Corollary 3.4.2.


• The multiplicity of the fundamental representation U1 in the actionδ1 equals n2.

Remark that the condition on the matrices F1 and F2 is not really lessgeneral than the condition in Theorem 3.4.1, but just a normalization: if

c1Tr(F ∗1 F1) = c2

Tr(F ∗2 F2) , we multiply F2 by a scalar and obtain c1 = c2 andTr(F ∗

1F1) = Tr(F ∗2F2).

Proof. Take Fi as in the statement of the theorem and denote by ϕ theunitary fiber functor on G1 := Ao(F1) given by Theorem 3.4.1 and (3.13).We continue to use the identification of Irred(G1) with N. Theorem 3.2.1provides us with a ∗-algebra B generated by the coefficients of unitaryoperators Xn ∈ L(Hn,Hϕ(n)) ⊗ B, n ∈ N. Define X := X1. Since everyelement of Irred(G1) appears in a tensor power of the fundamental repre-sentation and since 1 = 1, it follows that the coefficients of X generate Bas an algebra. Moreover (3.7) precisely says that X = (F2⊗1)X(F−1

1 ⊗1).Indeed, this is a consequence of the observation

X∗13(tF2 ⊗ 1) = (1⊗ F2 ⊗ 1)X(1⊗ F−1

1 ⊗ 1)(tF1 ⊗ 1) .

It follows that Cu(Ao(F1, F2)) 6= 0. Denoting by C the unital ∗-subalgebraof Cu(Ao(F1, F2)) generated by the coefficients of Y , we get a surjective∗-homomorphism ρ : C → B satisfying (id ⊗ ρ)(Y ) = X. It remains toshow that ρ is a ∗-isomorphism.

Denote by U the fundamental representation of Ao(F1) on H = Cn1

and by Un its n-th tensor power, on Hn. As in the proof of The-orem 3.4.1, we denote by Pn ∈ Mor(Un, Un) the projection onto theirreducible representation Un. Denote K = Cn2 and denote by Kn

the n-th tensor power of K. Recall that we constructed a faithful ∗-homomorphism π : (Mor(Um, Un))n,m → (L(Kn,Km))n,m and that ϕis defined by restricting π to the relevant subspaces. The graded C∗-algebra (Mor(Un, Um))n,m is generated by the elements t ∈ Mor(U2, ε)and 1r ⊗ t⊗ 1s.

Put Y n = Y1,n+1 · · ·Yn,n+1 ∈ L(Hn,Kn) ⊗ Cu(Ao(F1, F2)). SinceY 2(t ⊗ 1) = π(t) ⊗ 1, it follows that Y m(S ⊗ 1) = (π(S) ⊗ 1)Y n forall S ∈ Mor(Um, Un). Define a linear map γ : B → C by the formula


(id⊗ γ)(Xn) = Y n(Pn ⊗ 1) = (π(Pn)⊗ 1)Y n. We claim that γ is multi-plicative. Take a, n,m ∈ N and S ∈ Mor(n⊗m,a). Then,

(id⊗ id⊗ γ)(Xn13X

m23)(S ⊗ 1) = (ϕ(S)⊗ 1)(id⊗ γ)(Xa)

= (ϕ(S)⊗ 1)(Pa ⊗ 1)Y a

= (π(S)⊗ 1)Y a = Y n+m(S ⊗ 1)

= (Y n(Pn ⊗ 1))13(Y m(Pm ⊗ 1))23(S ⊗ 1)

= (id⊗ γ)(Xn)13(id⊗ γ)(Xm)23(S ⊗ 1) .

Because (id ⊗ γρ)(Y ) = Y , because γ is multiplicative and because thecoefficients of Y generate C as an algebra, it follows that γ is the inverseof ρ. So, ρ is indeed a ∗-isomorphism.

Remark 3.4.4. A combination of Proposition 6.2.6 in [10] and the resultsin [11] yields an alternative proof for the fact that Cu(Ao(F1, F2)) 6= 0.

A precise parametrization of the unitary fiber functors (and hence, the er-godic actions of full quantum multiplicity) on the quantum groups Ao(F ),amounts to the study of matrices F ∈ GL(n,C) satisfying FF = ±1, upto the equivalence relation (1.10), like we did in 1.5.2.

It was proven that a fundamental domain is given by the matrices 0 D(λ1, . . . , λk) 0D(λ1, . . . , λk)−1 0 0

0 0 1n−2k

with 2k ≤ n and 0 < λ1 ≤ · · · ≤ λk < 1 and the matrices(

0 D(λ1, . . . , λn/2)−D(λ1, . . . , λn/2)−1 0

)with 0 < λ1 ≤ · · · ≤ λn/2 ≤ 1.

Corollary 3.4.5. Let 0 < q ≤ 1. For every even natural number n with2 ≤ n ≤ q + 1

q , the quantum group SUq(2) admits an ergodic action offull quantum multiplicity such that the multiplicity of the fundamentalrepresentation is n. If −1 ≤ q < 0, the same statement holds for everynatural number n with 2 ≤ n ≤

∣∣q + 1q

∣∣.We remark that because the multiplicity is bounded by the quantumdimension, for n >

∣∣q + 1q

∣∣ such an action cannot be constructed.

3.5 Monoidal equivalence for Au(F ) 73

Proof. Suppose 0 < q < 1. Take then

F =(

0 D(λ1, . . . , λn/2)D(λ1, . . . , λn/2)−1 0

)such that λ2

1 + · · ·+λ2n/2 + 1

λ21+ · · ·+ 1

λ2n/2

= q+ 1q . Then then Ao(F ) ∼

mon

SUq(2) and the corresponding ergodic action of full quantum multiplicityof SUq will satisfy the theorem.

The case −1 ≥ q > 0 is analogous.

The following corollary summarizes this section.

Corollary 3.4.6. Let F ∈ GL(n,C) with FF = c1 and c ∈ R. De-note G = Ao(F ). For all F1 ∈ GL(n1,C) satisfying F1F 1 = c1 andTr(F ∗

1F1) = Tr(F ∗F ), we denote by δF1 the action of G on Ao(F, F1)defined in Theorem 3.4.3.

• Up to isomorphism, the δF1 yield all universal ergodic actions of fullquantum multiplicity of G. Moreover, δF1 is isomorphic with δF2 ifand only if n1 = n2 and there exists a unitary v ∈ U(n1) such thatF2 = vF1v

t.

• For all F1 as above with n1 = n, we denote by Ω(F1) the unitary2-cocycle on the dual of G associated with the unitary fiber functorϕF1. The Ω(F1) describe, up to coboundary, all unitary 2-cocycleson the dual of G. Moreover Ω(F1) and Ω(F2) differ by a coboundaryif and only if there exists a unitary v ∈ U(n) such that F2 = vF1v

t.

In the two-dimensional case, we get:

Corollary 3.4.7. Every unitary 2-cocycle on the dual of SUq(2) is acoboundary.

3.5 Monoidal equivalence for Au(F )

In this section, we prove, for the universal unitary quantum groups Au(F )studied by Banica [3], analogous results as for Ao(F ).


Again, we give a complete classification of unitary fiber functors, monoidallyequivalent quantum groups, ergodic actions of full quantum multiplicityand 2-cohomology.

Theorem 3.5.1. Let F ∈ GL(n,C) be normalized such that Tr(F ∗F ) =Tr((F ∗F )−1). Let G = Au(F ).

• If F1 ∈ GL(n1,C) satisfies Tr(F ∗1F1) = Tr((F ∗

1F1)−1) = Tr(F ∗F ),there exists a unitary fiber functor ϕF1 on G, uniquely determinedup to isomorphism, such that

ϕ(tF ) = tF1 and ϕ(sF ) = sF1 .

• Every unitary fiber functor ϕ on G is isomorphic with one of theform ϕF1. Moreover, ϕF1 is isomorphic with ϕF2 if and only ifn1 = n2 and there exist unitary elements v, w ∈ U(n1) such thatF2 = vF1w.

Proof. Let N ? N be the free monoid generated by α and β. Denote by ethe empty word. Elements of N ? N are words in α and β.

Take a parameter c > 0. Let (A(p, q))p,q∈N?N be the universal gradedC∗-algebra generated by elements

Vx(p, q) ∈ A(pq, pxq) for p, q ∈ N ? N, x ∈ αβ, βα

with relations (denoting by 1p the unit of the C∗-algebra A(p) := A(p, p))

Vx(p, q)∗Vx(p, q) = 1pqVαβ(p, αq)∗Vβα(pα, q) = c 1pαqVβα(p, βq)∗Vαβ(pβ, q) = c 1pβqVy(p, qxr)Vx(pq, r) = Vx(pyq, r)Vy(p, qr)

Vx(p, qyr)∗Vy(pxq, r) = Vy(pq, r)Vx(p, qr)∗

Take F ∈ GL(n,C) normalized in such a way that Tr(F ∗F ) = Tr((F ∗F )−1).Put c = Tr(F ∗F ). Consider the quantum group G = Au(F ). Definefor every p ∈ N ? N the unitary representation Up of G inductively byUpq := Up T© U q.

3.5 Monoidal equivalence for Au(F ) 75

Defining t ∈ Mor(Uαβ , ε) and s ∈ Mor(Uβα, ε) by the formulas t =1√

Tr(F ∗F )tF and s = 1√

Tr(F ∗F )sF , we get a natural ∗-homomorphism

π : (A(p, q))p,q∈N?N → (Mor(Up, U q))p,q∈N?N

given by π(Vαβ(p, q)) = 1p ⊗ t ⊗ 1q and π(Vβα(p, q)) = 1p ⊗ s ⊗ 1q. Itfollows from Proposition 4, the proof of Theoreme 1 and Proposition 3 in[3] that π is an isomorphism of C∗-algebras.

Take F1 ∈ GL(n1,C) satisfying Tr(F ∗1F1) = Tr((F ∗

1F1)−1) = Tr(F ∗F ).Write Kα = Kβ = Cn1 and define inductively Kp, for all p ∈ N ? N suchthat Kpq = Kp ⊗Kq. We take Ke = C. From the preceding discussion,we obtain a faithful ∗-homomorphism

π : (Mor(Up, U q))p,q∈N?N → (L(Kp,Kq))p,q∈N?N

satisfying π(t) = 1√Tr(F ∗1 F1)

tF1 , π(s) = 1√Tr(F ∗1 F1)

sF1 and π(1⊗ S ⊗ 1) =

1⊗ π(S)⊗ 1 for all S.

We choose a concrete identification of Irred(G) with N ? N as follows.We define, for p ∈ N ? N, Pp ∈ Mor(Up, Up) as the unique projectionsatisfying PpT = 0 for all r ∈ N ? N with length r < length p and allT ∈ Mor(Up, U r). We define Up as the restriction of Up to the image ofPp. We then identify

Mor(q1 ⊗ · · · ⊗ qk, p1 ⊗ · · · ⊗ pr)

= (Pq1 ⊗ · · · ⊗ Pqk) Mor(Up1···pr , U q1···qk)(Pp1 ⊗ · · · ⊗ Ppr) .

Define Hϕ(p) := π(Pp)Kp and define ϕ(S) by restricting π to Mor(q1 ⊗· · · ⊗ qk, p1 ⊗ · · · ⊗ pr). It is obvious that ϕ is a unitary fiber functor.

The converse statement is proven in exactly the same way as in the proofof Theorem 3.4.1.

The next two results are proven in exactly the same way as the corre-sponding results for Ao(F ).

Corollary 3.5.2. Let F ∈ GL(n,C) with Tr(F ∗F ) = Tr((F ∗F )−1) andconsider G = Au(F ). A compact quantum group G1 is monoidally equi-valent with G if and only if there exists F1 ∈ GL(n1,C) satisfyingTr(F ∗

1F1) = Tr((F ∗1F1)−1) = Tr(F ∗F ) such that G1

∼= Au(F1).


So, we exactly know when the compact quantum groups Au(F1) andAu(F2) are monoidally equivalent. If this is the case, Proposition 3.2.5provides us with a universal C∗-algebra Bu and a pair of ergodic actionsof full quantum multiplicity. It is again possible to give an intrinsic de-scription of this C∗-algebra Bu. The proof is analogous to the proof ofTheorem 3.4.3. Again, the fact that Cu(Au(F1, F2)) 6= 0 can be deducedfrom Proposition 6.2.6 in [10] and the results in [12].

Theorem 3.5.3. Let Fi ∈ GL(ni,C) by such that

Tr(F ∗1F1) = Tr((F ∗

1F1)−1) = Tr(F ∗2F2) = Tr((F ∗

2F2)−1) .

• Denote by Cu(Au(F1, F2)) the universal unital C∗-algebra generatedby the coefficients of

X ∈Mn2,n1(C)⊗ Cu(Au(F1, F2)) with relations

X and (F2 ⊗ 1)X(F−11 ⊗ 1) are unitary .

Then, Cu(Au(F1, F2)) 6= 0 and there exists a unique pair of com-muting universal ergodic actions of full quantum multiplicity, δ1 ofAu(F1) and δ2 of Au(F2), such that

(id⊗ δ1)(X) = X12(U1)13 and (id⊗ δ2)(X) = (U2)12X13 .

Here, Ui denotes the fundamental representation of Au(Fi).

• (Cu(Au(F1, F2)), δ1, δ2) is isomorphic with the C∗-algebra Bu andthe actions thereon given by Proposition 3.2.5 and the monoidalequivalence Au(F1) ∼

monAu(F2) of Corollary 3.5.2.

• The multiplicity of the fundamental representation U1 in the actionδ1 equals n2.

Remark 3.5.4. Exactly as in Corollary 3.4.6, a combination of Theorems3.5.1 and 3.5.3 gives a complete classification of the ergodic actions of fullquantum multiplicity of Au(F ) and of the 2-cohomology of the dual ofAu(F ).

A precise parametrization of the unitary fiber functors on the quantumgroups Au(F ) is easy. If F1, F ∈ GL(n,C), we write

F1 ∼ F ⇔ there exist unitaries v, w ∈ U(n) such that F1 = vFw.

3.6 Monoidal equivalence for Aaut(B,ϕ) 77

We study matrices F ∈ GL(n,C) satisfying Tr(F ∗F ) = Tr((F ∗F )−1)up to the equivalence relation ∼. It is obvious that for any such F ,there exist unique 0 < λ1 ≤ · · · ≤ λn satisfying

∑i λ

2i =

∑i λ

−2i such

that F ∼ D(λ1, . . . , λn). Here D(λ1, . . . , λn) denotes again the diagonalmatrix with the λi along the diagonal.

3.6 Monoidal equivalence for Aaut(B, ϕ)

As a last application, we will examine monoidal equivalence for quantumautomorphism groups.

Theorem 3.6.1. Let B and B1 be finite dimensional C∗-algebras and ϕand ϕ1 respectively a δ-form and a δ1-form on B, respectively B1. ThenAaut(B,ϕ) ∼

monAaut(B1, ϕ1) if and only if δ = δ1.

Proof. Denote by µ, µ1 and η, η1 the multiplication and unital map ofrespectively B and B1.

First suppose that δ = δ1. Take now U , respectively V the fundamentalrepresentation of Aaut(B,ϕ), respectively Aaut(B1, ϕ1) corresponding tothe actions of this quantum groups. Consider the graded C∗-algebras(Mor(Um, Un))n,m and (Mor(V m, V n))n,m. We know from [6] that thereis an isomorphism π : (Mor(Un, Um))n,m → (Mor(V n, V m))n,m whichsatisfies π(µ) = µ1 and π(η) = η1 We now can work analogously to thecase of Ao(F ) and Au(F ).

We now set Irred(G) = N and Pn ∈ Mor(Un, Un) the unique projectionfor which PnT = 0 for all r < n and all T ∈ Mor(U r, Un). We define Unas the restriction of Un to the image of Pn and identify

Mor(n1 ⊗ · · · ⊗ nr,m1 ⊗ · · · ⊗mk)

= (Pm1 ⊗ · · · ⊗ Pmk) Mor(Un1+···+nr , Um1+···+mk)(Pn1 ⊗ · · · ⊗ Pnr) .

Define now Hψ(n) := π(Pn)Bn1 and define for S ∈ Mor(n1⊗· · ·⊗nr,m1⊗

· · ·⊗mk), ψ(S) by the restriction of π to Mor(n1⊗· · ·⊗nr,m1⊗· · ·⊗mk).Then ψ is a unitary fiber functor which gives a monoidal equivalencebetween G and G1.

Conversely, suppose that Aaut(B,ϕ) ∼mon

Aaut(B1, ϕ1). Denote by u1, res-

pectively v1 the irreducible representation with label 1 of Aaut(B,ϕ) and


Aaut(B1, ϕ1). Then dimq(u1)1 = ϕmm∗ϕ∗ = δ21 and because monoidalequivalence preserves the quantum dimension, δ and δ1 must be equal.

Remark 3.6.2. Consider two finite dimensional C∗-algebras (B1, ω1) and(B2, ω2) with δ-forms and their quantum automorphism groupsAaut(B1, ω1) := G1 and Aaut(B2, ω2) := G2. Denote now by H i

1 = BiCand U1

i ∈ L(H i1) ⊗ C(Gi) for i = 1, 2 the representative of the irre-

ducible representation with label 1. Denote by θi ∈ Mor((U1i ), (U

1i )

2),γi ∈ Mor(U0

i , (U1i )

2), Ti ∈ Mor(U1i , U

1i ⊗ U0

i ), Si ∈ Mor(U1i , U

0i ⊗ U1

i ),Ri ∈ Mor(U0, (U0

i )2) the ”components” of the multiplication as in 2.5.

From construction in the theorem above, it follows that there is a monoidalequivalence ϕ between G1 and G2 which sends θ1, γ1, T1, R1 and S1 toθ2, γ2, T2, R2 and S2. If we further below talk about the monoidal equi-valence between Aaut(B1, ω1) and Aaut(B2, ω2), we will always mean thisone.

Two quantum automorphism groups can only be isomorphic if their C∗-algebras together with the invariant states are isomorphic.

Theorem 3.6.3. Consider two finite dimensional C∗-algebras B1 andB2 and δ-forms ω1 and ω2 respectively on B1 and B2. Then G1 =Aaut(B1, ω1) and G2 = Aaut(B2, ω2) are isomorphic if and only if

(B1, ω1) ∼= (B2, ω2) .

Proof. Denote by π : C(G1) → C(G2) the isomorphism of quantum groups.Denote by Irred(G) the set of equivalence classes of irreducible representa-tions of G1 and G2 and by Uxi ∈ L(H i

x)⊗C(Gi) the representative of x forGi. For all x ∈ Irred(G), there has to exist a unitary map vx : H1

x → H2x

with (id⊗ π)(Ux1 ) = (vx ⊗ 1)(Ux2 )(v∗x ⊗ 1). This is true because every ir-reducible representation has a different dimension, which can be deducedfrom the fusion rules. Denote by ϕ the monoidal equivalence implementedby these unitaries.

Denote by θi, γi the components of the multiplication of Bi, i = 1, 2. Be-cause Mor((U1

i ), (U1i )

2) and Mor((U0i ), (U

0i )

2) are one dimensional, ϕ(θ1)and ϕ(R1) must be scalar multiples of respectively θ2 and R2. Moreover,these scalar multiples must be of modulus 1, because

ϕ(θ1)ϕ(θ1)∗ = θ1θ∗1 = δ21 = θ2θ

∗2 and also


ϕ(R1)ϕ(R1)∗ = R1R∗1 = 1 = R2R

∗2 .

Therefore, we can always adapt our unitaries in such a way that ϕ′(θ1) =θ2 and ϕ′(R1) = R2. So in the following we may suppose that it is thecase that v1θ1(v∗1 ⊗ v∗1) = θ2 and v0 = 1. It is now easy to see thatalso the monoidal equivalence sends T1 to T2 and S1 to S2. Indeed,we have that v1T1(v∗1 ⊗ v0)(x ⊗ 1) = v1v

∗1x = x = T2(x ⊗ 1) and the

same argument works for Si. Now θ2, T2, S2 and R2 give rise to a newassociative multiplication • on B2. We prove that this must necessarilybe the original one. The equation v0γ1(v∗1 ⊗ v∗1) = µγ2 means that forx, y ∈ H2

1 , ω2(x • y) = µω2(xy). Take 3 elements x, y, z ∈ H21 . We have

that

(x • y) • z = ((xy − ω2(xy)1) + µω2(xy)) • z= xyz − ω2(xy)z + (µ− 1)ω2(xyz) + µω2(xy)z

and on the other hand

x • (y • z) = x • ((yz − ω2(yz)) + µω2(yz))

= xyz − xω2(yz) + (µ− 1)ω2(xyz) + µxω2(yz) .

So the multiplication can only be associative if

(µ− 1)xω2(yz) = (µ− 1)ω2(xy)z

for all x, y, z ∈ B2 C. Because dim(B2) ≥ 4, this is only possible ifµ = 1 which means that γ1(v∗1 ⊗ v∗1) = γ2.

Define now

ϕ : B1 → B2 : x 7→ v1x if x ∈ H11 and ϕ(1) = 1

We prove that ϕ is a *-homomorphism. Take b = (λ, x), c = (µ, y) ∈B1 = H1

1 ⊕ C. We then have that

ϕ(b)ϕ(c) = (λµ+ γ2(v1x⊗ v1y), θ2(v1x⊗ v1y) + λv1y + µv1x)

= (λµ+ γ1(x⊗ y), v1θ1(x⊗ y) + λv1y + µv1x)

= ϕ(bc) ,

so ϕ is multiplicative.


It also preserves the involution, because

ϕ(b∗) = ϕ(λ, (x∗ ⊗ 1)γ∗1) = (λ, v1((x∗ ⊗ 1)γ∗1)

= (λ, (x∗ ⊗ 1)(1⊗ v1)γ∗1) = (λ, (x∗ ⊗ 1)(v∗1 ⊗ 1)γ∗2)

= (λ, ((v1x)∗ ⊗ 1)γ∗2) = ϕ(b)∗ .

We may now conclude that ϕ indeed is a *-isomorphism between (B1, ω1)and (B2, ω2).

Like we did before for Ao(F ) and Au(F ), we give a concrete descriptionof the link algebra coming from a monoidal equivalence.

Theorem 3.6.4. Consider two finite dimensional C∗-algebras (D1, ω1)and (D2, ω2) with δ-forms.

• Denote by Cu(Aaut((D1, ω1), (D2, ω2))) the universal C∗-algebra gene-rated by the coefficients of a unital *-homomorphism

γ : D1 → D2 ⊗ Cu(Aaut((D1, ω1), (D2, ω2)))

with relations (ω2 ⊗ id)γ(x) = ω1(x)1 for all x ∈ D1 .

Then Cu(Aaut((D1, ω1), (D2, ω2))) 6= 0 and there exists a uniquepair of commuting ergodic actions of full quantum multiplicity δ1 ofAaut(D1, ω1) and δ2 of Aaut(D2, ω2), such that

(id⊗ δ1)γ = (γ ⊗ id)β1 and (id⊗ δ2)γ = (β2 ⊗ id)γ ,

where βi : Di → Di ⊗ Cu(Aaut(Di, ωi)), i = 1, 2, are the canonicalactions of the quantum automorphism groups.

• Cu(Aaut((D1, ω1), (D2, ω2))) is isomorphic with the C∗-algebra Buand the actions thereon given by proposition 3.2.5 and the monoidalequivalence Aaut(D1, ω1) ∼

monAaut(D2, ω2).

Proof. We first remark that if Cu(Aaut((D1, ω1), (D2, ω2))) 6= 0, the ac-tions are given by universality. Indeed,

(γ ⊗ id)β1 : D1 → D2 ⊗ Cu(Aaut((D1, ω1), (D2, ω2)))⊗ Cu(Aaut(D1, ω1))


is a *-homomorphism which satisfies

(ω2 ⊗ id⊗ id)(γ ⊗ id)β1(x) = ω1(x)1

for x ∈ D1. So by universality, there exists a *-homomorphism

δ1 : Cu(Aaut((D1, ω1), (D2, ω2)))

→ Cu(Aaut((D1, ω1), (D2, ω2)))⊗ Cu(Aaut(D1, ω1))

satisfying (id ⊗ δ1)γ = (γ ⊗ id)β1. Because β1 is an action and thecoefficients of γ generate Cu(Aaut((D1, ω1), (D2, ω2)), it follows that δ1 isan action. We define δ2 in an analogous way.

Consider now the C∗-algebra Bu we get from the monoidal equivalence.Denote by θi, γi the components of the multiplication of Di, i = 1, 2.Because of remark 3.6.2, we may suppose that the monoidal equivalencesends θ1 and γ1 respectively to θ2 and γ2. Denote by Ui the irreduciblerepresentation of Aaut(Di, ωi) with label 1. Because every irreduciblerepresentation is a contained in a tensor power of the one with label 1,the matrix coefficients of X1 ∈ L(D1 C, D2 C) ⊗ Bu generate Bu asa C∗-algebra. By identification, X1 provides us with a linear map

Γ : D1 C → (D2 C)⊗Bu

which we can easily extend to D1 by setting Γ(1) = 1. Because

X1(θ1 ⊗ 1) = (θ2 ⊗ 1)X113X

123 and (γ1 ⊗ 1) = (γ2 ⊗ 1)X1

13X123 ,

Γ is multiplicative, obviously unital and ω1(x)1 = (ω2 ⊗ id)Γ(x). It alsopreserves the involution because X23(γ∗1 ⊗ 1) = X∗

13(γ∗2 ⊗ 1) and γ1 and

γ2 implement the involution on respectively B1 and B2. By universalitythere exists now a unital *-homomorphism

ρ : Cu(Aaut((D1, ω1), (D2, ω2))) → Bu

such that Γ = (id⊗ ρ)γ. It is now left to show that ρ is an isomorphism.

Because γ satisfies the equation (ω2 ⊗ id)γ(x) = ω1(x)1, we can look atthe restriction of γ given by

γ : D1 C → (D2 C)⊗ Cu(Aaut((D1, ω1), (D2, ω2)))


Denote by Y ∈ B((D1C), (D2C))⊗Cu(Aaut((D1, ω1), (D2, ω2))) theelement corresponding to this restricted *-homomorphism. This elementsatisfies the equations

Y (θ1 ⊗ 1) = (θ2 ⊗ 1)Y13Y23 and

γ1 ⊗ 1 = (γ2 ⊗ 1)Y13Y23

because γ is a unital homomorphism. Remark that Y is unitary becauseγ also preserves the involution. Because the multiplication and the unitalmap generate all the intertwiners of Aaut(Di, ωi), i = 1, 2 and thereforethis is also true for θi and γi, it holds that

Y ⊗n(Pn ⊗ 1) = (Qn ⊗ 1)Y ⊗n

where Pn and Qn are the unique projections in respectively Mor(Un1 , Un1 )

and Mor(Un2 , Un2 ) on the irreducible representation with label n. Defining

σ such that (id⊗σ)(Xn) = Y ⊗n(Pn⊗1), gives a unital *-homomorphismwith σρ = ρσ = id.

The following theorem says the family of quantum automorphism groupsis closed under monoidal equivalence.

Theorem 3.6.5. Suppose a compact quantum group G is monoidallyequivalent to a quantum automorphism group Aaut(B,ϕ) of a finite di-mensional C∗-algebra B with ϕ a δ-form. Then there exists a finite di-mensional C∗-algebra C and a δ-form ω such that G as a compact quan-tum group is isomorphic Aaut(C,ω).

Proof. Denote by ui ∈ B(Hi)⊗ C(Aaut(B,ϕ)), respectively vi ∈ L(Ki)⊗C(G) the irreducible representations with label i ∈ N of Aaut(B,ϕ),respectively G. Consider the components θ ∈ Mor(u1, u

21) and γ ∈

Mor(u0, u21) of the multiplication on B. Denote by ψ the monoidal equi-

valence. We then have linear isomorphisms

ψ : Mor(u1, u21) → Mor(v1, v2

1) and

ψ : Mor(u0, u21) → Mor(u0, u

21) .

If now a = (λ, x) and b = (µ, y) are elements of K := C ⊕K1, the mapm given by

m(a⊗ b) = (λµ+ ψ(γ)(x⊗ y), λy + µx+ ψ(θ)(x⊗ y))


gives an associative multiplication on K. This makes K into a unitalalgebra with unit (1, 0).

Because γ implements the involution on B, the image ψ(γ) together withS(1) = 1 induces us then an antilinear map S on K1, which we can in thenatural way extend to K. This map turns K into a *-algebra. The onlything left to check is that the adjoint S makes K a C∗-algebra. But K isa Hilbert-space, so we get a trivial representation of the *-algebra K asoperators on K.

It is clear that G acts ergodically on the C∗-algebra K via the representa-tion v1⊕ 1 := v with invariant state ω : K → C : a 7→ 〈(0, 1), a〉. So thereexists an epimorphism π : C(Aaut(K,ω)) → C(G) of quantum groups suchthat (id⊗π)(V ) = v if V is the fundamental representation of Aaut(K,ω).It is left to prove that π is an isomorphism of compact quantum groups. Itsuffices to prove that for each irreducible representation ui of Aaut(K,ω),(id⊗ π)(ui) is an irreducible representation of G.

From the equation (id ⊗ π)(V ) = v, we see that Mor(V n, V m) ⊆Mor(vn, vm). Since Aaut(K,ω) and Aaut(B,ϕ) have the same fusion rules,dim(Mor(V n, V m)) = dim(Mor(Un, Um)), with U the fundamental repre-sentation of Aaut(B,ϕ). By monoidal equivalence, dim(Mor(vn, vm)) =dim(Mor(Un, Um)). We conclude that Mor(V n, V m) = Mor(vn, vm) forall n,m ∈ N. Denote by Qn ∈ Mor(V n, V n) the orthogonal projec-tion onto the irreducible subrepresentation of label n in V n. Denote byPn ∈ Mor(Un, Un) the analogous projection forAaut(B,ϕ). We know thatQn ∈ Mor(V n, V n) is the largest projection satisfying Qn Mor(V n, V r) =0 for all r < n. Hence Qn is the largest projection in Mor(vn, vn)satisfying Qn Mor(vn, vr) = 0 for all r < n. By monoidal equivalenceQn = ψ(Pn). This means that π maps the irreducible representation(Qn⊗1)V n to the irreducible representation (Qn⊗1)vn of G. This provesthat Aaut(K,ω) and G are isomorphic as compact quantum groups.

As the classes of universal orthogonal and unitary compact quantumgroups are closed under having the same fusion rules, the same can behoped for for quantum automorphism groups. We thus end this chapterwith the following natural question. Is every compact quantum groupwith the fusion rules of SO(3) of the form Aaut(B,ϕ) for a finite dimen-sional C∗-algebra B and a δ-form ϕ on B?

Chapter 4

Actions of monoidally

equivalent compact

quantum groups

The content of this and the following chapter mostly coincides with thepreprint [21], written in collaboration with Nikolas Vander Vennet. In theprevious chapter, we introduced and developed the notion of monoidallyequivalent quantum groups.

In the first section of this chapter, we obtain a bijective correspondencebetween (not necessarily ergodic) actions of monoidally equivalent quan-tum groups on unital C∗-algebras. Moreover, the correspondence is ofthat kind that it preserves the spectral subspaces of the actions. Re-stricting this to ergodic actions, this just means that the multiplicitiesof the irreducible representations are preserved through this correspon-dence. It should be emphasized that our approach is not categoric. Thecorrespondence is obtained in a concrete, constructive way.

In a short second section, we apply our construction to the specific casesof a homogeneous space and an invariant subalgebra.

The last section provides concrete examples of this correspondence be-tween actions. In chapter 3, it is proven that the quantum groups SUq(2)and Ao(F ) are monoidally equivalent if Tr(F ∗F ) = |q + 1/q| and FF =−sgn q. We construct, for every F , a natural ergodic action of thecompact quantum group Ao(F ) on an invariant subalgebra, as defined

86 Chapter 4. Actions through monoidally equivalence

by Tomatsu. Using the bijective correspondence of this chapter, thisyields, under some restrictions on q, ergodic actions of SUq(2) which arenot invariant subalgebras and are not of full quantum multiplicity. If2−

√3 < |q| < 1, we even construct continuous (non-conjugate) families

of such ergodic actions of SUq(2). In view of the classification program ofergodic actions of SUq(2), this proves once more that this classification ishighly non-trivial.

4.1 The correspondence between the actions of

monoidally equivalent quantum groups.

In chapter 2, we remarked that for ordinary groups, monoidal equiva-lence implies that the groups are isomorphic ([19]). For quantum groups,this was no longer the case. We can ask ourselves which informationabout the quantum group can be retrieved from its monoidal category.In [39], inspired by the results of chapter 3 (i.e. [14]), Pinzari and Robertsproved that the ergodic actions of a compact quantum group correspondto so-called quasitensor functors on the representation category. As theyproved, the composition of a quasitensor functor and a unitary fiber func-tor is still quasitensor. This yields a bijective correspondence between theergodic actions of two monoidally equivalent compact quantum groups.

The result in this section is more general. We prove, in a very con-crete way, that two monoidally equivalent compact quantum groups have,roughly spoken, the “same actions”.

Remark that we work mostly on an algebraic level. This is not a re-striction because as we stated on page 15 and in proposition 2.1.4, theunderlying Hopf*-algebra action carries all the relevant information.

Consider two monoidally equivalent compact quantum groups G1 and G2

and a C∗-algebra D1. Suppose we have an action α1 : D1 → D1⊗C(G1).As stated above, we work with the underlying Hopf ∗-algebra action α1 :D1 → D1 ⊗alg C(G1). Consider a monoidal equivalence ϕ : G2 → G1.Note that we have exchanged the roles of G1 and G2. This will turn outto be more convenient in what follows. From proposition 3.2.5, we get alink algebra B, unitaries Xx ∈ L(Hx,Hϕ(x)) ⊗alg B and two commutingergodic actions

δ1 : B → C(G1)⊗alg B and δ2 : B → B ⊗alg C(G2) .

4.1 Correspondence between actions 87

given by

(id⊗ δ1)(Xx) = Uϕ(x)12 Xx

13 and (id⊗ δ2)(Xx) = Xx12U

x13 . (4.1)

The following theorem enables us to construct an action of G2 with thesame spectral structure as α1.

Theorem 4.1.1. The restriction of id⊗ δ2 to the *-algebra

D2 := a ∈ D1 ⊗alg B | (α1 ⊗ id)(a) = (id⊗ δ1)(a)

yields a Hopf *-algebra action α2 of G2 on D2 such that

• a 7→ a⊗ 1 is a ∗-isomorphism between the fixed point algebras of α1

and α2.

• The map Tx : Kϕ(x) → Kx : v 7→ v12Xx13 is a bimodular isomor-

phism between the spectral subspaces of α1 and α2. Moreover, T isa unitary element of L(Kϕ(x),Kx) for the inner products 〈·, ·〉l and〈·, ·〉r defined in (2.4) and (2.6).

• The set (Tx)x∈Irred(G2) respects the monoidal structure in the sensethat for x, y, z ∈ Irred(G2) and V ∈ Mor(x⊗ y, z)

Tx(X)13Ty(Y )23(V ⊗ 1) = Tz(X13Y23(ϕ(V )⊗ 1))

for all X ∈ Kϕ(x), Y ∈ Kϕ(y).

• Suppose that α1 is an ergodic action. Then the action α2 as definedabove is also an ergodic action. Moreover for all x ∈ Irred(G2),multq(x) = multq(ϕ(x)).

Proof. From the following easy calculation, one can see that D2 is invari-ant under the action id⊗ δ2. For all a ∈ D2,

(α1 ⊗ id⊗ id)(id⊗ δ2)(a) = (id⊗ id⊗ δ2)(α1 ⊗ 1)(a)

= (id⊗ id⊗ δ2)(id⊗ δ1)(a)

= (id⊗ δ1 ⊗ id)(id⊗ δ2)(a) .

The last step is valid because δ1 and δ2 commute. Hence α2 := (id⊗δ2) |D2

is a well defined action of G2 on D2.


Suppose that α2(a) = a ⊗ 1 for a ∈ D2. This means that (id ⊗ δ2)(a) =a⊗ 1. By ergodicity of δ2 there exists a b ∈ D1 such that a = b⊗ 1. Butbecause (α1 ⊗ id)(a) = (id⊗ δ1)(a), it follows that b ∈ Dα1

1 . So the mapDα1

1 → Dα22 : b 7→ b⊗ 1 gives an *-isomorphism between the C∗-algebras

Dα11 and Dα2

2 .

We now prove that the spectral subspaces of α1 and α2 are isomorphicas Dα1

1 -bimodules. Denote by Kϕ(x) and Kx the spectral subspaces ofrespectively α1 and α2 for the representation ϕ(x), respectively x. Fromsection 2.2 we know that the spectral subspaces have a natural bimodulestructure over the fixed point algebra. We claim that the map

T : Kϕ(x) → Kx : v 7→ v12Xx13

is the bimodule isomorphism we are looking for. If v ∈ Kϕ(x), then

(id⊗ α1 ⊗ id)T (v) = v12Uϕ(x)13 Xx

14 = (id⊗ id⊗ δ1)T (v)

by definition 2.1.2 of the spectral subspace Kϕ(x) and the properties ofXx, so T (v) ∈ Hx ⊗D2. Moreover, it is obvious that

(id⊗ α2)T (v) = (id⊗ id⊗ δ2)T (v) = v12Xx13U

x14 ,

which means T (v) ∈ Kx. The Dα11 -bilinearity of T is clear. Consider

now the spectral subspaces Kx and Kϕ(x) as equipped with the left innerproduct as in (2.4). We show that T is a unitary element of L(Kϕ(x),Kx)for this inner product and obtain in this way that T actually gives anisomorphism between Kϕ(x) and Kx. Consider the map

S : Kx → Hϕ(x) ⊗D1 ⊗ B : w 7→ w(Xx13)

∗ .

If w ∈ Kx, then

(id⊗ id⊗ δ2)S(w) = (id⊗ α2)(w)(id⊗ id⊗ δ2)(Xx13)

∗

= (w ⊗ 1)Ux14(Ux14)

∗(Xx13)

∗

= S(w)⊗ 1 ,

So, by ergodicity of δ2, we may conclude that S(w) ∈ Hϕ(x) ⊗D1 ⊗ C.

Because w has its second leg in D2, we get that

(id⊗ α1 ⊗ id)S(w) = (id⊗ id⊗ δ1)(w)(Xx14)

∗

= (id⊗ id⊗ δ1)(w(Xx13)

∗)Uϕ(x)13

= (id⊗ id⊗ δ1)(S(w))Uϕ(x)13 ,


But we just proved that the third leg of S(w) is scalar, so the last ex-pression is nothing else than (S(w)⊗ 1)Uϕ(x)

13 . Thus, by the definition ofKϕ(x), we get that S : Kx → Kϕ(x) ⊗ C.

For every v ∈ Kϕ(x) and w ∈ Kx, we have that

〈T (v), w〉l = T (v)w∗ = v12Xx13w

∗ = v12S(w)∗ = 〈v, S(w)〉l .

So, S is actually the adjoint T ∗ of T in the sense of Hilbert C∗-modules.Moreover, it is trivial that T ∗T = 1 = TT ∗. Hence T ∈ L(Kϕ(x),Kx) isunitary for 〈·, ·〉l.

Next, we show that T is also a unitary element of L(Kϕ(x),Kx) for theright Hilbert-C∗-module structure given by (2.6). From proposition 3.5 of[33], it suffices to show that T is isometric and surjective. The surjectivityfollows from above. We use the orthogonality relations (1.3) and (3.9) forUx and Xx to prove that T is indeed an isometry.

First notice that the conditional expectation E2 : D2 → Dα22 is nothing

else than the map a 7→ (id⊗ ω)(a)⊗ 1, where ω is the invariant state forδ1 and δ2. Indeed, for a ∈ D2,

E2(a) = (id⊗ h2)α2(a) = (id⊗ id⊗ h2)(id⊗ δ2)(a) = (id⊗ ω)(a)⊗ 1 .

Consider now v ∈ Kϕ(x). On the one hand, we have that

1⊗ 〈T (v), T (v)〉r = (id⊗ E2)((Xx13)

∗v∗12v12Xx13)

= (id⊗ id⊗ ω)((Xx13)

∗v∗12v12Xx13)⊗ 1

=1

dimq(x)(1⊗ (Tr⊗id)((Q−1

ϕ(x) ⊗ 1)v∗v)⊗ 1)

because of the orthogonality relations for Xx.

On the other hand

1⊗ 〈v, v〉r = (id⊗ E1)(v∗v)

= (id⊗ id⊗ h1)(id⊗ α1)(v∗v)

= (id⊗ id⊗ h1)((Ux13)∗v∗12v12U

x13)

=1

dimq(x)(1⊗ (Tr⊗id)((Q−1

ϕ(x) ⊗ 1)v∗v)) ,


where in the last step we used the orthogonality relations for Ux. Con-sidering the map Dα1

1 → Dα22 : a 7→ a ⊗ 1, the calculations above show

that T is indeed isometric and hence unitary.

We now show that (Tx)x∈Irred(G2) preserves the monoidal structure. Takev ∈ Kϕ(x), w ∈ Kϕ(y) and V ∈ Mor(x⊗ y, z). We calculate that

Tx(v)13Ty(w)23(V ⊗ 1) = v13Xx14w23X

y24(V ⊗ 1)

= v13w23(ϕ(V )⊗ 1)Xz13 = Tz(v13w23(ϕ(V )⊗ 1)) ,

which proves the statement.

Finally, we prove the fourth part of the theorem. Recall the operatorsfrom formula (N.2).

Rϕ(x) : v 7→ (ϕ(t)∗ ⊗ 1)(1⊗ v∗) and Lϕ(x) = R∗ϕ(x)Rϕ(x)

with v ∈ Kϕ(x) and

Rx : w 7→ (t∗ ⊗ 1)(1⊗ w∗) and Lx = R∗xRx

with w ∈ Kx. Then

〈v, Lϕ(x)w〉1 = 〈Rϕ(x)v,Rϕ(x)w〉1= 〈(ϕ(t)∗ ⊗ 1)(1⊗ v∗), (ϕ(t)∗ ⊗ 1)(1⊗ w∗)〉= (ϕ(t)∗ ⊗ 1)(1⊗ w∗v)(ϕ(t)⊗ 1)

where v, w ∈ Kϕ(x). Remember the isomorphism Tx : Kϕ(x) → Kx : v 7→v12X

x13. Then:

〈v12Xx13, Lxw12X

x13〉1 = 〈Rxv12Xx

13, Rxw12Xx13〉1

= (t∗ ⊗ 1)(1⊗ (w12Xx13)

∗(v12Xx13))(t⊗ 1)

= (t∗ ⊗ 1⊗ 1)((Xx24)

∗w∗23v23Xx24)(t⊗ 1⊗ 1)

= (ϕ(t)∗ ⊗ 1⊗ 1)(Xx14w

∗23v23(X

x14)

∗)(ϕ(t)⊗ 1⊗ 1)

= (ϕ(t)∗ ⊗ 1⊗ 1)(w∗23v23)(ϕ(t)⊗ 1⊗ 1)

= (ϕ(t)∗ ⊗ 1)(1⊗ w∗v)(ϕ(t)⊗ 1)⊗ 1

In this calculation, we have used that Xx13X

x23(t ⊗ 1) = ϕ(t) ⊗ 1. This

follows from the fact that t ∈ Mor(x⊗ x, ε). Again considering the mapDα1

1 → Dα22 : a 7→ a ⊗ 1, we get that Tx intertwines Lx and Lϕ(x). It

follows trivially from the definition 2.3.2 of quantum multiplicity thatboth quantum multiplicities are the same. This completes the proof ofthe theorem.


We remark that it seems that the statement of the theorem cannot imme-diately be formulated on the C∗-algebraic level. If we define D2 = a ∈D1⊗B | (α1⊗ id)(a) = id⊗ δ1)(a), it is not clear that α2 = id⊗ δ2 goesinto D2 ⊗ C(G2).

However, for von Neumann algebras, there is no problem. Suppose thatα1 : D1 → D1⊗L∞(G) is a von Neumann algebraic action and take thenotations as before, where we now take the von Neumann algebraic link-algebra B = (B, ω)′′. Here it does hold that α2(D2) ⊆ D2⊗L∞(G2), as weonly need to check that (id⊗µ)α2(a) ∈ D2 for all a ∈ D2 and µ ∈ (D2)∗.For C∗-algebras, this argument is not valid.

Claim: The algebra D2 as defined in theorem 4.1.1 is precisely the spec-tral subalgebra of (D2, α2).

Proof. Denote by D2 the spectral subalgebra of (D2, α2). It is clear thatD2 ⊆ D2.

On the other hand,

D2 = 〈(id⊗ h)(α2(a)(1⊗ b)) | a ∈ D2, b ∈ C(G2)x, x ∈ Irred(G)〉 .

Because the elements of D2 of course sit in D2, it is sufficient to provethat D2 ⊆ D1 ⊗alg B.

If a ∈ D2, b ∈ C(G2)x, x ∈ Irred(G2), (id ⊗ h)(α2(a)(1 ⊗ b)) belongs tothe strongly closed linear span of

(id⊗id⊗h)((a⊗δ2(d))(1⊗1⊗b)) | a ∈ D1, d ∈ B, b ∈ C(G2)x ⊆ D1⊗algBx .

Since Bx is finite dimensional, D1 ⊗alg Bx is already strongly closed inD1⊗B. Hence c := (id⊗h)(α2(a)(1⊗ b)) ∈ D1⊗alg Bx for all a ∈ D2 andb ∈ C(G2)x. On the other hand, (α1⊗ id)(c) = (id⊗ δ1)(c), which impliesthat c ∈ (D1)x ⊗alg Bx ⊆ D1 ⊗alg B. This ends the proof.

We can start from the comultiplication on G1 and apply the above con-struction. It is not surprising that we end up with the link algebra andthe action δ2.

Proposition 4.1.2. Consider two monoidally equivalent compact quan-tum groups G1 and G2. In the case that D1 = C(G1) and α1 = ∆1, wehave a ∗-isomorphism between D2 and the link algebra B for the monoidalequivalence ϕ. Moreover, this ∗-isomorphism intertwines the action α2

with the action δ2.


Proof. By definition, D2 := a ∈ C(G1) ⊗alg B | (∆1 ⊗ id)(a) = (id ⊗δ1)(a). We claim that δ1 : B → D2 is the desired ∗-isomorphism. Fromthe definition of δ1, it follows that δ1 : B → C(G1) ⊗alg B is an injective∗-homomorphism. The image of δ1 is contained in D2 because δ1 is anaction. Moreover, if a ∈ D2 and ε1 is the co-unit on C(G1), then

δ1((ε1 ⊗ id)(a)) = (ε1 ⊗ id⊗ id)(∆1 ⊗ id)(a) = a

which means that δ1(B) = D2. So δ1 is also surjective.

Because δ1 and δ2 commute, it is clear that this ∗-isomorphism intertwinesthe actions δ2 and α2.

Now we consider the inverse monoidal equivalence ϕ−1 : G1 → G2. Ac-cording to theorem 3.2.5, we obtain the link algebra B generated by thecoefficients of unitary elements Y x ∈ L(Hϕ(x),Hx)⊗alg B and two commu-ting ergodic actions

γ1 : B → B ⊗alg C(G1) and γ2 : B → C(G2)⊗ B

with

(id⊗ γ1)(Y x) = Y x12U

ϕ(x)13 and (id⊗ γ2)(Y x) = Ux12Y

x13 (4.2)

Denote by ω the invariant state on B. Then we get the following propo-sition.

Proposition 4.1.3. Consider two monoidally equivalent compact quan-tum groups G1 and G2. In the case that D1 = B and α1 = γ1, we obtaina ∗-isomorphism π : C(G2) → D2. This ∗-isomorphism intertwines thecomultiplication ∆2 with the action α2.

Proof. In this case, D2 = a ∈ B ⊗alg B | (γ1⊗ id)(a) = (id⊗ δ1)(a). Wedefine the linear map π : C(G2) → B⊗alg B where (id⊗π)(Ux) = Y x

12Xx13.

Because

(id⊗ γ1 ⊗ id)(Y x12X

x13) = Y x

12Uϕ(x)13 Xx

14 = (id⊗ id⊗ δ1)(Y x12X

x13) ,

the image of π lies in D2.


Consider x, y, z ∈ Irred(G2) and take T ∈ Mor(x⊗ y, z). The multiplica-tivity of π follows from the following calculation:

(id⊗ π)(Ux13Uy23(T ⊗ 1)) = (id⊗ π)((T ⊗ 1)U z) = (T ⊗ 1⊗ 1)(Y z

12Xz13)

= Y x13Y

y23(ϕ(T )⊗ 1⊗ 1)Xz

13

= Y x13Y

y23X

x14X

y24(T ⊗ 1⊗ 1)

= (id⊗ π)(Ux)134(id⊗ π)(Uy)234(T ⊗ 1⊗ 1) .

Take now tx ∈ Mor(x⊗ x, ε). Because

Ux13Ux23(tx ⊗ 1) = tx ⊗ 1 ,

it follows that

(id⊗ id⊗ π)(Ux13∗(tx ⊗ 1)) = (id⊗ id⊗ π)(Ux23(tx ⊗ 1))

= Y x23X

x24(tx ⊗ 1⊗ 1)

= Y x23X

x14∗(ϕ(tx)⊗ 1⊗ 1)

= Xx14∗Y x

13∗(tx ⊗ 1⊗ 1) .

This proves that π also passes trough the involution, so it is a *-homo-morphism. We now show that this map is the desired *-isomorphism.

First we prove the injectivity. It is easy to show that (ω ⊗ ω)π = h,with h the Haar measure of C(G2). Suppose now that for an a ∈ C(G2),π(a) = 0. Then also π(a∗a) = 0, which means that also h(a∗a) = 0. Buth is faithful on C(G2), so a = 0.

To prove the surjectivity of π, we have to take a closer look at the elementsof D2. From definition 2.1.3, we get that

B =⊕

x∈Irred(G1)

Bx and B =⊕

x∈Irred(G1)

Bx

where δ1(Bx) ⊆ C(G1)x ⊗alg Bx and γ1(Bx) ⊆ Bx ⊗alg C(G1)x. Sup-pose b ∈ B ⊗alg B and (γ1 ⊗ id)(b) = (id ⊗ δ1)(b). We claim thatb ∈

⊕x∈Irred(G1) Bx ⊗alg Bx. First notice that

B ⊗alg B =⊕

x,y∈Irred(G1)

Bx ⊗alg By .


So b =∑bxy with bxy ∈ Bx ⊗alg By. Since δ1(By) ⊆ C(G1)x ⊗alg By and

γ1(Bx) ⊆ Bx ⊗alg C(G1)x, it follows that bxy = 0 if x 6= y.

So we only need to prove that π(C(G2)x) = (D2)x, with

(D2)x := a ∈ Bx ⊗alg Bx | (γ1 ⊗ id)(a) = (id⊗ δ1)(a) .

Therefore, remember the formulas (id ⊗ γ1)(Y x) = Y x12U

ϕ(x)13 and

(id ⊗ δ1)(Xx) = Uϕ(x)12 Xx

13 where Y x ∈ L(Hϕ(x),Hx) ⊗alg Bx and Xx ∈L(Hx,Hϕ(x))⊗alg Bx.

We know that a basis of Bx (resp. Bx) is given respectively by elementsof the form (ω

gekx ,elx⊗ id)(Xx) and (ωelx ,gekx

⊗ id)(Y x) with ekx , kx ∈1, . . . ,dim(ϕ(x)) an orthonormal basis in Hϕ(x) and elx , lx ∈1, . . . ,dim(x) an orthonormal basis inHx. Denote (ω

gekx ,elx⊗id)(Xx) :=

bkx,lx and (ωelx ,gekx⊗ id)(Y x) = blx,kx . We also have a basis for C(G2)x

given by (ωekx ,elx⊗ id)(Ux), again with ekx , kx ∈ 1, . . . ,dim(x) an or-

thonormal basis in Hx. Denote by (ωekx ,elx⊗ id)(Ux) := ukx,lx . In the

following, we drop the subscript x. With these notations, we get that

γ1(bkl) =dim(ϕ(x))∑

p=1

bkp ⊗ upl and δ1(bij) =dim(ϕ(x))∑

q=1

uiq ⊗ xqj

and

π(ukj) =dim(ϕ(x))∑

l=1

bkl ⊗ blj .

An arbitrary element of Bx ⊗ Bx has the form

a =∑klij

λkjli bkl ⊗ bij .

We get that(γ1 ⊗ id)(a) =

∑klijp

λkjli bkp ⊗ upl ⊗ bij

equals(id⊗ δ1)(a) =

∑klijq

λkjli bkl ⊗ uiq ⊗ bqj .


From this equality, we immediately see that λkjli = 0 if l 6= i. We havealso that λkjll = λkjpp for all k, l, j, p. Indeed, the above equality providesus with the following equalities:∑

lq

λkjll bkl ⊗ ulq ⊗ bqj =∑lp

λkjll bkp ⊗ upl ⊗ blj

for every k, j. This can only happen when λkjll = λkj for every l ∈1, . . .dim(Hϕ(x)). So a ∈ (D2)x has the form

a =∑kj

λkj

dim(ϕ(x))∑l=1

bkl ⊗ blj

,

which is a linear combination of the π(ukj). This proves the surjectivityof π.

Moreover

(id⊗ π ⊗ id)(id⊗∆2)(Ux) = (id⊗ π ⊗ id)(Ux12Ux13)

= Y x12X

x13U

x14

= (id⊗ α2π)(Ux) ,

so the action α2 indeed corresponds to the comultiplication on G2.

Combination of the two previous propositions now enables us to provethe reversibility of our construction.

Proposition 4.1.4. The construction in theorem 4.1.1 applied to theinverse monoidal equivalence ϕ−1 : G1 → G2 and the action α2 : D2 →D2⊗algC(G2) gives back, up to isomorphism, the action α1 : D1 → D1⊗alg

C(G1).

Proof. Denote by D′1 and α′1 the *-algebra and action we obtain after

applying proposition 4.1.1 on D2 and α2. Then

D′1 = a ∈ D1 ⊗alg B ⊗alg B | (id⊗ δ2 ⊗ id)(a) = (id⊗ id⊗ γ2)(a)

and (α1 ⊗ id⊗ id)(a) = (id⊗ δ1 ⊗ id)(a)

and α′1 = (id ⊗ id ⊗ γ1) |D′1. In exactly the same way as in proposition

4.1.3, we can prove that C(G1) is ∗-isomorphic to the ∗-algebra D := a ∈


B ⊗alg B | (δ2 ⊗ id)(a) = (id⊗alg γ2)(a). In this case, the ∗-isomorphismis given by π : C(G1) → D where (id ⊗ π)(Uϕ(x)) = Xx

12Yx13. Also in the

same way, we can prove that π intertwines the actions δ1⊗ id |D and ∆1.From this, we get that D′

1 is isomorphic to

D′′1 := a ∈ D1 ⊗alg C(G1) | (α1 ⊗ id)(a) = (id⊗∆1)(a) .

From the calculation

(id⊗ π ⊗ id)(id⊗∆1)(Uϕ(x)) = (id⊗ π ⊗ id)(Uϕ(x)12 U

ϕ(x)13 )

= Xx12Y

x13U

ϕ(x)14

= (id⊗ id⊗ γ1)(id⊗ π)(Uϕ(x)) ,

it follows that (id⊗ id⊗ γ1) |D′1

is equivalent with id⊗∆1 |D′′1

under the∗-isomorphism id⊗ π.

In the same way as in proposition 4.1.2, we see that α1 : D1 → D′′1 is

a ∗-isomorphism. It is obvious that this ∗-isomorphism intertwines theactions id⊗∆1 |D′′

1and α1. This concludes the proof.

The results of this section now provide us with a natural correspondencebetween the actions of two monoidally equivalent quantum groups G1

and G2. Indeed, suppose we have an action α1 : D1 → D1 ⊗ C(G1).As we remarked, we can also work with the underlying Hopf ∗-algebraaction α1 : D1 → D1 ⊗alg C(G1). Then theorem 4.1.1 provides us with a∗-algebra D2 and an action α2 : D2 → D2 ⊗alg C(G2) such that the fixedpoint algebras of α1 and α2 are ∗-isomorphic and such that there exista bimodular isomorphism between the spectral subspaces of α1 and α2.Moreover, because of the preceding proposition 4.1.4, this constructiongives, up to isomorphism, a one-to-one correspondence between actionsof G1 and G2. In the von Neumann-algebraic case, this algebraic detouris not necessary. We can then immediately do everything on the level ofvon Neumann algebras.

4.2 Homogeneous spaces through monoidal equivalence 97

4.2 Homogeneous spaces through monoidal equi-

valence

In this section, we apply the construction of theorem 4.1.1 to the specialcase of a homogeneous space. This gives us a generalization of proposition4.1.2.

Suppose we are given two monoidally equivalent compact quantum groupsG1 and G2. Denote by δ1 and δ2 the ergodic actions of full quantummultiplicity on the corresponding link-algebra B. Consider a quantumsubgroup H of G1 and the corresponding action

∆H\G1: C(H\G1) → C(H\G1)⊗alg C(G1)

on the homogeneous space.

To a homogeneous space of a compact quantum group, naturally therecorresponds a homogeneous space of the link algebra.

Definition 4.2.1. We define the homogeneous space BH by

BH := a ∈ B | (rH ⊗ id)δ1(a) = 1⊗ a .

Now δ2 is an action on BH because δ1 and δ2 commute.

Theorem 4.2.2. The construction of theorem 4.1.1 applied to D1 =C(H\G1) and α1 = ∆H\G1

gives us a *-algebra which is isomorphic withBH. Moreover, this *-isomorphism intertwines the action α2 with therestriction of δ2 to BH.

Proof. By definition,

D2 := a ∈ C(H\G1)⊗alg B | (∆1 ⊗ id)(a) = (id⊗ δ1)(a) .

We prove that as in proposition 4.1.2, δ1 : BH1 → D2 is a *-isomorphism.Because

(rH1 ⊗ id)(∆1 ⊗ id)δ1(a) = (rH1 ⊗ id)(id⊗ δ1)δ1(a)

= (id⊗ δ1)(1⊗ a) = 1⊗ δ1(a) ,

we get that δ1(BH1) ⊆ D2. The injectivity of δ1 is clear. The surjectivityfollows from the fact that for a ∈ D2,

δ1(ε⊗id)(a) = (ε⊗id⊗id)(id⊗δ1)(a) = (ε⊗id⊗id)(∆1⊗id)(a) = a (4.3)


and(rH ⊗ id)δ1(ε⊗ id)(a) = (rH ⊗ id)(a) = (ε⊗ id)(a),

where in the last step we used proposition 2.4.3. Because δ1 and δ2commute, δ1 intertwines α2 with the restriction of δ2 to BH. This endsthe proof.

Remark that we can generalize this theorem to right invariant subalgebrasof a compact quantum group. If D1 ⊆ C(G1) is an invariant subalgebra,ε⊗id embeds D2 as an invariant subalgebra of B, which in turn is mappedonto D2 by δ1. Indeed, injectivity is clear and surjectivity again followsfrom (4.3). Right invariant subalgebras of G1 thus correspond to rightinvariant subalgebras of the link-algebra. The converse statement is alsotrue. We denote again by π : C(G2) → B⊗alg B the injective *-morphismfrom proposition 4.1.3.

Proposition 4.2.3. Consider two monoidally equivalent compact quan-tum groups G1 and G2. If D1 ⊆ B is a right invariant subalgebra, andα1 = γ1, π−1(D2) gives a right invariant subalgebra of C(G2).

Proof. In this case, D2 = a ∈ B ⊗alg D1 | (γ1 ⊗ id)(a) = (id ⊗ δ1)(a).Take now b ∈ π−1(D2). It is sufficient to prove that (π ⊗ id)∆2(b) ∈D2 ⊗alg C(G2). Now it holds that

(γ1 ⊗ id⊗ id)(π ⊗ id)∆2(b) = (γ1 ⊗ id⊗ id)α2π(b)

= (id⊗ id⊗ δ2)(γ1 ⊗ id)π(b)

= (id⊗ id⊗ δ2)(id⊗ δ1)π(b)

= (id⊗ δ1 ⊗ id)(id⊗ δ2)π(b)

= (id⊗ δ1 ⊗ id)(π ⊗ id)∆2(b) .

This proves the statement.

4.3 Application: Invariant subalgebras of Ao(F )

In this section, we apply the above construction to the quantum groupsAo(F ).

In section 3.4 of chapter 3, it was proven that for each F ∈ GLn(C)with FF = ±1, there exists a q ∈ ]−1, 1[ \0 such that SUq(2) ∼

mon

4.3 Application: Invariant subalgebras of Ao(F ) 99

Ao(F ). Theorem 4.1.1 gives, for every action of Ao(F ), an action ofthe corresponding SUq(2). We use this to construct examples of actionsof SUq(2), under some restrictions on q, which are different from theones classified by Tomatsu in [44] and also different from the actions offull quantum multiplicity in chapter 3. For 2 −

√3 < |q| < 1, we get

a continuous family of non conjugate actions of SUq(2), which are notinvariant subalgebras and not of full quantum multiplicity.

To achieve this, we look at the following kind of actions of Ao(F ). FixF ∈ GLn(C) with FF = ±1. Consider a unitary matrix v ∈ Mn(C) forwhich

FvF−1 = v . (4.4)

Then, with U ∈ L(H ⊗ C(Ao(F )) the fundamental representation ofAo(F ),

(id⊗ αv)(U) = (v ⊗ 1)U

defines an automorphism of C(Ao(F )). Indeed, because v satisfies thesame equation as the fundamental representation, there is a *-homo-morphism ϕv : C(Ao(F )) → C such that (id ⊗ ϕv)(U) = v. Thusαv = (ϕv ⊗ id)∆. Also, for all T ∈ Mor(Un, Um), T intertwines alsovn and vm. Because

(id⊗∆αv)(U) = (id⊗∆)(v ⊗ 1)(U) = (v ⊗ 1⊗ 1)(U12U13)

= (id⊗ αv ⊗ id)(id⊗∆)(U) ,

the fixed point algebra Cv := C(Ao(F ))αv is a right invariant subalgebra,i.e. ∆(Cv) ⊂ Cv ⊗ C(Ao(F )). Hence, by restricting the comultiplicationto Cv, we get an ergodic action of Ao(F ).

Take now q ∈] − 1, 1[\0 such that SUq(2) and Ao(F ) are monoidallyequivalent. Through the above construction, we get an action αF ofSUq(2) on an algebra CF with the same multiplicities as ∆|Cv . We canalso describe this action in a direct way. We do this now, although furtheron, we will not use the explicit description of this action.

So consider the monoidal equivalence ϕ : SUq(2) → Ao(F ) and the cor-responding link algebra B := C(Ao(Fq, F )) as in theorem 3.4.3 which isgenerated by the coefficients of a unitary Y . In the same way as was donefor Ao(F ), we can define an automorphism of B by

(id⊗ βv)(Y ) := (v ⊗ 1)(Y ) .


It is now easy to check that for a ∈ B, (βv ⊗ id)δ2(a) = δ2(βv(a)), soδ2(Bv) ⊂ Bv ⊗ C(SUq(2)). We claim that the actions αF and δ2|Bv areisomorphic.

By Theorem 4.1.1, CF = a ∈ Cv⊗algB | (∆⊗ id)(a) = (id⊗δ1)(a). Nowfrom the previous section we now that δ1 gives an isomorphism between(ε ⊗ id)CF and CF , where ε is the co-unit on Ao(F ). We show that(ε ⊗ id)CF is precisely Bv. First of all it is clear that for x ∈ Bv, δ1(x)satisfies (∆⊗ id)δ1(x) = (id⊗ δ1)δ1(x) because δ1 is a action. Moreover,because (αv ⊗ id)δ1 = δ1βv, δ1(Bv) ⊂ CF . It holds that (ε⊗ id)δ1(a) = a

for all a ∈ B. Hence Bv ⊆ (ε⊗ id)CF .

On the other hand, for a ∈ CF , (ε⊗ id)(a) ∈ Bv. Indeed,

a = (αv ⊗ id)(a) = (αv ⊗ id)(ε⊗ id⊗ id)(∆⊗ id)(a)

= (αv ⊗ id)(ε⊗ δ1)(a) = (αv ⊗ id)δ1(ε⊗ id)(a)

= δ1βv(ε⊗ id)(a)

and so (ε⊗ id)(a) = βv((ε⊗ id)(a)) which proves the statement. We mayconclude that δ1 : Bv → CF is an isomorphism. It is also clear that δ1intertwines δ2 and αF .

We will need to calculate the multiplicities of the irreducible represen-tations of this action αF . Because of theorem 4.1.1, these coincide withthe multiplicities of the action ∆|Cv . Let ul be the subrepresentation ofU⊗l which is equivalent with the irreducible representation of label l. Weknow that mult(l), the multiplicity of ul is exactly the dimension of Cvldivided by dim(ul). So it is enough to calculate this dimension.

We first calculate the multiplicity of the fundamental representation U ,which is the irreducible representation with label 1. Choose an orthonor-mal basis ei, i = 1 . . . n for H. Take a ∈ Cv1 . Certainly, a must be inC(Ao(F ))1. We can decompose a in a unique way as

a =n∑i=1

(ωξi,ei⊗ id)(U) ,


where ξi are vectors in H. Because αv(a) = a,

n∑i=1

(ωξi,ei⊗ id)(U) =

n∑i=1

(ωξi,ei⊗ id)(id⊗ αv)(U)

=n∑i=1

(ωξi,ei⊗ id)((v ⊗ 1)(U))

=n∑i=1

(ωv∗ξi,ei⊗ id)(U) ,

which can only happen if vξi = ξi for all i = 1, . . . , n as U is irreducible.Hence mult(1) equals the dimension of the eigenspace of v at eigenvalue1.

Suppose ul = (Pl ⊗ 1)U⊗l, which means Pl ∈ Mor(U l, U l) is the uniqueprojection of U⊗l on ul. Take a ∈ Cvl . Then again, a must be inC(Ao(F ))l. Choose an orthonormal basis fi, i = 1 . . .dim(l) for Hl :=Pl(H l). We can decompose a in a unique way as

a =dim(l)∑i=1

(ωξi,fi⊗ id)(ul) ,

where ξi are vectors in Hl. If a ∈ Cv, αv(a) = a, so

dim(l)∑i=1

(ωξi,fi⊗ id)(ul) =

dim(l)∑i=1

(ωξi,fi⊗ id)(id⊗ αv)((Pl ⊗ 1)(U l))

=dim(l)∑i=1

(ωξi,fi⊗ id)((v⊗l ⊗ 1)ul)

=dim(l)∑i=1

(ω(v∗)⊗l(ξi),fi⊗ id)(ul) ,

which, because ul is irreducible, can only happen if v⊗l(ξi) = ξi forξi ∈ Hl. Moreover, the eigenspace E of v⊗l at eigenvalue 1 is invari-ant under Pl because v⊗l and Pl commute. These last two results implythat the calculation of the multiplicities of the irreducible representationsul of the above action ∆|Cv can be done just by examining the invariantvectors of the operators v⊗n for every n ∈ N. Remember that this gives


also the multiplicities of the action αF of SUq(2). If the CF is an invari-ant subalgebra of SUq(2), by proposition 2.4.6 the multiplicity of everyirreducible representation does not exceed the dimension of the represen-tation. So, if we find an irreducible representation x ∈ Irred(SUq(2)) forwhich dim(x) < mult(αF , x), then CF is not an invariant subalgebra.

We will also need to calculate the quantum multiplicity of αF which is,according to theorem 4.1.1 equal to the quantum multiplicity of ∆|Cv .Therefore, we have to calculate Tr(Ln) (recall that n = n for Ao(F )). Werecall the construction of Ln : Kn → Kn. From the definition of spectralsubspaces, we get

Kn = X ∈ Hn ⊗ Cv | (id⊗∆)(X) = X12Un13 .

Like in proposition 2.4.6

Kn ⊂ Kn := X ∈ Hn ⊗ C(Ao(F )) | (id⊗∆)(X) = X12Un13 ,

with Kn the spectral subspace of the comultiplication ∆. Then we getthat

Kn = X ∈ Kn | (id⊗ αv)(X) = X .

Here we used that ∆ αv = (αv ⊗ id) ∆ to show that if X ∈ Kn thenalso (id⊗ αv)(X) ∈ Kn. As

Kn = (ξ∗ ⊗ 1)Un | ξ ∈ Hn ,

it holds that

Kn = (ξ∗ ⊗ 1)Un | ξ ∈ Hn and v⊗nξ = v .

Now take ξ, η ∈ Hn and such that v⊗nξ = ξ and vη = η. Then wecalculate

〈(ξ∗ ⊗ 1)Un, Ln(η∗ ⊗ 1)Un〉= 〈Rn(ξ∗ ⊗ 1)Un, Rn(η∗ ⊗ 1)Un〉= 〈(t∗ ⊗ 1)(1⊗ (Un)∗(ξ ⊗ 1)), (t∗ ⊗ 1)(1⊗ (Un)∗(η ⊗ 1))〉= h ((t∗ ⊗ 1) (1⊗ (Un)∗(ηξ∗ ⊗ 1)Un) (t⊗ 1))

= (t∗ ⊗ 1) (id⊗ id⊗ h)(1⊗ (Un)∗(ηξ∗ ⊗ 1)Un) (t⊗ 1)

=1

dimq(x)(t∗ ⊗ 1)(1⊗ 1〈ξ,Q−1

n η〉)(t⊗ 1)

=1

dimq(x)t∗t〈ξ,Q−1

n η〉 = 〈ξ,Q−1n η〉 .


This means that Ln is the restriction of Q−1n to the space of invariant

vectors of v⊗n. This gives an easy way to calculate the quantum mul-tiplicity. Remember that dimq(n) = Tr(Qn) = Tr(Q−1

n ). So the action∆|Cv is of full quantum multiplicity if and only if v = 1.

The above construction also works for a one parameter group of unitarymatrices vt | t ∈ R, satisfying equation 4.4. These give rise to anaction of the real numbers on Ao(F ), for which the fixed points algebrawill again be an invariant subalgebra. The multiplicities and quantummultiplicities of this invariant subalgebra (and hence of the correspondingaction of SUq(2)), can analogously as above be calculated by consideringthe intersection of the eigenspaces of (vt)⊗n, t ∈ R, n ∈ N.

We now give concrete examples of matrices v that satisfy equation (4.4)to obtain examples of actions of SUq(2) which are not one of the in-variant subalgebras Tomatsu computed in [44] and not of full quantummultiplicity.

Example 1

As a first example, take dim(F ) = 3. We know that, up to isomorphism,F is of the form

F =

0 λ 0λ−1 0 00 0 1

.

Then take the matrix v = diag(1, 1,−1). It is clear that v satisfies equa-tion (4.4). We have that FF = 1 and F qFq = 1, if q ∈]− 1, 0[ and −1 ifq ∈]0, 1[, where Ao(Fq) ∼= SUq(2) (see formula (N.1)). Also, we get thatTr(F ∗F ) = 1 + λ2 + 1

λ2 and Tr(F ∗q Fq) = |q + 1

q |. Remember from corol-lary 3.4.2 that Ao(F ) ∼

monSUq(2) if and only if Tr(F ∗

q Fq) = Tr(F ∗F ) and

sgn(F qFq) = sgn(FF ). This can only happen when q satisfies |q+1/q| ≥ 3and q < 0. From this it follows that, if (

√5 − 3)/2 < q < 0, there is a

unique 3 by 3 matrix F such that SUq(2) is monoidally equivalent withAo(F ). So, for these values of q, we obtain an ergodic action αF of SUq(2)with F of the above form. First we show that this action is not a invari-ant subalgebra from Tomatsu. Take u2, which is the second irreduciblerepresentation of SUq(2). Then, dim(u2) = 3. From the fusion rules ofSUq(2), we get u1 ⊗ u1 = u2 ⊕ u0. In order to calculate the multiplicityof u2, we have to study the dimension of the invariant subspaces of v⊗ v.


This space is 5-dimensional, but one invariant vector comes from u0, sothe multiplicity of u2 equals 4 which is strictly greater than dim(u2) = 3.So, the action obtained is not a invariant subalgebra of Tomatsu.

Now we show that the action is not of full quantum multiplicity. Thisfollows immediately from the fact that v 6= 1. This can also be seen bycalculating Tr(LF ). But, as we know, Tr(LF ) = Tr(Q−1

F ) restricted to theinvariant vectors from v. ButQF = F tF = and soQ−1

F = diag(λ2, λ−2, 1).So Tr(LF ) = λ2 + λ−2 = |q+ 1/q| − 1. But the quantum dimension of u1

equals |q + 1/q|.

Example 2

In the same way, with the same F , we can take the one parameter groupvt = diag(λit, λ−it, 1). This again gives an action. From the fusion rules ofSUq(2), we get that u⊗3

1 = u3⊕ 2u1. So we need the know the dimensionof the subspace of vectors invariant under v⊗3

t . This dimension equals7. But u⊗3 contains the fundamental representation u1 two times. Thedimension of the invariant subspace under vt equals 1. So the multiplicityof u3 is 5. But the dimension of u3 is 4. So this action is not a invariantsubalgebra. Again, it is not of full quantum multiplicity either becausenot all the matrices vt are equal to the identity.

Example 3

Finally, we show that for certain values of q, we can construct in thisway a continuous family of actions of SUq(2) which are not invariantsubalgebras and not of full quantum multiplicity. Therefore, look at thefollowing 4 by 4 matrix

F =(Fr 00 Fs

).

with both r, s ∈]0, 1[ or r, s ∈]− 1, 0[ to ensure that FF = ∓1. Then allmatrices

vt =(Qitr 00 1

).

satisfy formula (4.4) because Qr = F trFr. So we can take this one param-eter group vt = diag(|r|−it, |r|it, 1, 1|). We have that Tr(F ∗

q Fq) = |q+1/q|


and FqFq = 1 if q ∈] − 1, 0[ and −1 if q ∈]0, 1[. Also Tr(F ∗F ) =|r + 1/r + s + 1/s| and FF = 1 if r, s ∈] − 1, 0[ and −1 if r, s ∈]0, 1[.Because Ao(F ) ∼

monSUq(2) if and only if Tr(F ∗

q Fq) = Tr(F ∗F ) and

sgn(F qFq) = sgn(FF ), this can only be the case when |q + 1/q| ≥ 4.So, if 2−

√3 < |q| < 1, then there exists a matrix F as above such that

Ao(F ) is monoidally equivalent with SUq(2). Again, we show that theaction αF obtained in this way is not a invariant subalgebra. Therefore,notice that u1 ⊗ u1 = u2 ⊕ u0. But the dimension of the eigenspace witheigenvalue one for v ⊗ v equals 6. This means that the multiplicity ofu2 is 5, but dim(u2) is 3. So, this action is not a invariant subalgebra.To show that it is not of full quantum multiplicity follows from the factthat not all the vt equal the identity matrix. We immediately see thatLF = diag(|s|−1, |s|). So, for different s, these actions are all non isomor-phic, otherwise, the LF -matrices should have the same eigenvalues..

Final remark

As we remarked in the previous section, we can generalize the above con-struction for all invariant subalgebras of Ao(F ). Suppose C ⊂ C(Ao(F ))is a left invariant subalgebra. This gives rise to an ergodic action of Ao(F )on C by restricting the comultiplication. If q is so that SUq(2) ∼

monAo(F ),

via the construction of this chapter we get a *-algebra CF and an actionαF : CF → CF ⊗ C(SUq(2)) with the same spectral subspaces as theinvariant subalgebra.

Now CF equipped with αF is isomorphic to a subalgebra of the link-algebra B equipped with the restriction of δ2 in the following way. DefineBF := (ε⊗ id)(CF ) with ε the counit. We showed that δ1 : BF → CF is a*-isomorphism.

The invariant subalgebras of Ao(F ) hence correspond to subalgebras ofthe link-algebras B with SUq(2) thereon acting by δ2 and vice versa.Considering the fact that for the classical group SU(2) every ergodicaction comes from a subgroup, it would be natural and interesting toexamine if every ergodic action of SUq(2) is coming from an invariantsubalgebra of some Ao(F ) via the above construction.

Chapter 5

Poisson boundaries through

monoidal equivalence

We apply the results of the previous chapter to compute the Poissonboundaries for certain non-amenable discrete quantum groups. The re-sults in this chapter were found in collaboration with Nikolas VanderVennet. They coincide with the results in the second part of preprint[21].

In the first section, we give an overview of the theory of Poisson boun-daries. Next, we prove that the Poisson boundaries of the duals of twomonoidally equivalent compact quantum groups, correspond through theconstruction of chapter 4.

Recently, Tomatsu ([45]) managed to identify the Poisson boundaries ofall amenable discrete quantum groups having commutative fusion ruleswith the homogeneous space of the maximal quantum subgroup of Kactype. Combining Tomatsu’s result with ours, we identify in the thirdsection, the Poisson boundary for quantum groups with commutativefusion rules that are monoidally equivalent with coamenable quantumgroups.

In the last section we apply our results to the class of quantum auto-morphism groups Aaut(B,ϕ). These are coamenable if and only if thedimension of the C∗-algebra B equals 4. Since every quantum automor-phism group is monoidally equivalent with a coamenable one, we canidentify the Poisson boundary of the dual of all quantum automorphism

108 Chapter 5. Poisson boundaries

groups.

5.1 The Poisson boundary of a discrete quantum

group

We give a brief survey of Izumi’s theory of Poisson boundaries for discretequantum groups.

Fix a discrete quantum group G.

Notation 5.1.1. For every normal state φ ∈ `∞(G)∗, we define theconvolution operator

Pφ : `∞(G) → `∞(G) : Pφ(a) = (id⊗ φ)∆(a) .

We are only interested in special states φ ∈ `∞(G), motivated by thefollowing straightforward proposition. For every probability measure µon Irred(G), we set

ψµ =∑

x∈Irred(G)

µ(x)ψx and Pµ := Pψµ .

Recall that the states ψx are defined in 1.4.5. Note that we have a con-volution product µ ∗ ν on the measures on Irred(G), such that ψµ∗ν =(ψµ ⊗ ψν)∆.

Proposition 5.1.2. Let φ be a normal state on `∞(G). Then the follow-ing conditions are equivalent.

• φ has the form ψµ for some probability measure µ on Irred(G).

• The Markov operator Pφ preserves the center of `∞(G).

• φ is invariant under the adjoint action of G on `∞(G)

αG : `∞(G) → `∞(G)⊗L∞(G) : a 7→ V(a⊗ 1)V∗ .

Definition 5.1.3 ([30], Section 2.5). Let µ be a probability measure onIrred(G). Set

H∞(G, µ) = a ∈ `∞(G) | Pµ(a) = a .

5.1 The Poisson boundary of a discrete quantum group 109

Equipped with the product defined by

a · b := w∗- limn→∞

1n

n∑k=1

P kµ (ab) , (5.1)

and the involution, norm and σ-weak topology inherited from `∞(G), thespace H∞(G, µ) becomes a von Neumann algebra that we call the Poissonboundary of Irred(G) with respect to µ.

Terminology 5.1.4. A probability measure µ on Irred(G) is called ge-nerating if there exists, for every x ∈ Irred(G), an n ≥ 1 such thatµ∗n(x) 6= 0.

Remark 5.1.5. The restriction of the co-unit ε yields a state onH∞(G, µ),called the harmonic state. This state is faithful when µ is generating. Inwhat follows, we always assume that µ is generating.

Definition 5.1.6. Let µ be a generating measure on Irred(G). The Pois-son boundary H∞(Irred(G), µ) comes equipped with two natural actions,one of G and one of G:

αG : H∞(G, µ) → H∞(G, µ)⊗L∞(G) : αG(a) = V(a⊗ 1)V∗ ,

αbG : H∞(G, µ) → `∞(G)⊗H∞(G, µ) : α

bG(a) = ∆(a) .

Note that αG is the restriction of the adjoint action of G on `∞(G), whileαbG is nothing else than the restriction of the comultiplication. The maps

αG and αbG are well defined because of the following equivariance formulae:

(id⊗ Pµ)(∆(a)) = ∆(Pµ(a)) and (Pµ ⊗ id)(αG(a)) = αG(Pµ(a)) .(5.2)

Remark 5.1.7. With the product defined by formula (5.1), the mappingαG : H∞(G, µ) → H∞(G, µ)⊗L∞(G) is multiplicative. This follows fromthe second equivariance formula from (5.2). Hence αG is an action of Gon H∞(G, µ). Because

(ε⊗ id)αG(a) = (ε⊗ id)(V(a⊗ 1)V∗) = ε(a)1

we see that ε is an invariant state for the action αG : H∞(G, µ) →H∞(G, µ)⊗ L∞(G).


As all our concrete computations apply to quantum groups G with com-mutative fusion rules, we include the following important result.

Proposition 5.1.8. Suppose that the fusion algebra of G is commutative(i.e. mult(y ⊗ z, x) = mult(z ⊗ y, x) for every x, y, z ∈ Irred(G)) and letµ be a generating probability measure on Irred(G).

• (Cor. 3.5 in [26] and Cor. 3.2 in [31]) There are no non-trivialcentral harmonic elements, i.e.

Z(`∞(G)) ∩H∞(G, µ) = C1 .

• (Prop. 1.1 in [31]) The Poisson boundary does not depend on thechoice of generating measure:

H∞(G, µ) = a ∈ `∞(G) | Px(a) = a for all x ∈ Irred(G) .

• (Cor. 3.5 in [31]) Using the notation of 2.1.2, we have

mult(x, αG) ≤ supmult(y ⊗ y, x) | y ∈ Irred(G) ,

for all x ∈ Irred(G).

We remark that in this chapter, we mostly work with von Neumannalgebras. So all square letters are von Neumann algebras, while the curvedones are the underlying dense *-algebras.

5.2 Poisson boundaries of monoidally equivalent

quantum groups

In this section we prove that the Poisson boundaries of two monoidallyequivalent quantum groups correspond with each other trough the con-struction of Theorem 4.1.1. Recall (p. 91) that we may do all computa-tions immediately on the von Neumann algebraic level.

Consider two monoidally equivalent compact quantum groups G1 and G2

where the monoidal equivalence is given by ϕ : G2 → G1.

5.2 Poisson boundaries and monoidal equivalence 111

Notation 5.2.1. From now on, we denote respectively V1 :=⊕

x∈Irred(G1) Ux

and V2 :=⊕

x∈Irred(G2) Uy. We also denote by B = B′′ the von Neu-

mann algebraic link algebra of the monoidal equivalence and by X :=⊕x∈Irred(G2)X

x. We denote the states ϕ1µ and ψ1

µ, respectively ϕ2µ and

ψ2µ on `∞(G1), respectively `∞(G2).

Let µ be a generating probability measure on Irred(G1). Consider thePoisson boundary H∞(G1, µ) of G1 with adjoint action

αG1 : H∞(G1, µ) → H∞(G1, µ)⊗L∞(G1) : αG1(a) = V1(a⊗ 1)V∗1 .

It turns out that the Poisson boundary of G2 with adjoint action αG2 isobtained through the construction of theorem 4.1.1. We have to provethat

D2 := a ∈ H∞(G1, µ)⊗B | (αG1 ⊗ id)(a) = (id⊗ δ1)(a)

is isomorphic to H∞(G2, µ) and that the action id⊗δ2 on D2 correspondsto the adjoint action αG2 on H∞(G2, µ). We formulate the main resultof this section in the following theorem.

Theorem 5.2.2. Consider two monoidally equivalent compact quantumgroups G1 and G2. Let µ be a generating probability measure on Irred(G1).The Poisson boundary H∞(G2, µ) is ∗-isomorphic to

D2 := a ∈ H∞(G1, µ)⊗B | (αG1 ⊗ id)(a) = (id⊗ δ1)(a) .

Moreover, the ∗-isomorphism intertwines the action α2 with the conjugateaction αG2.

The proof consists of several steps.

The generalized Izumi operator

Izumi proved the following general result:

Proposition 5.2.3 (Lemma 3.8 in [30]). Let G be a compact quantumgroup. Defining

Φ : L∞(G) → `∞(G) : Θ(a) = (id⊗ h)(V∗(1⊗ a)V) ,

the image of Φ is contained in H∞(G, µ) for any probability measure µon Irred(G).


We call Φ the Izumi operator.

Inspired by this, we define a generalized Izumi operator

Θ : D2 → `∞(G2) : Θ(a) = (id⊗ ω)(X∗aX) . (5.3)

We prove that this Izumi operator produces an isomorphism between D2

and H∞(G2, µ).

Proposition 5.2.4. Let Θ be as in (5.3), and let µ be a generating pro-bability measure on Irred(G1). Then, Θ(a) ∈ H∞(G2, µ) for all a ∈ D2.

Proof. Let a ∈ D2. We claim that for x, y ∈ Irred(G2),

(px ⊗ py)∆2(Θ(a)) = (id⊗ id⊗ ω)((Xy

23)∗(Xx

13)∗((∆1 ⊗ id)(a)

)Xx

13Xy23

).

Take now z ∈ Irred(G) and T ∈ Mor(x⊗ y, z). Then

(px ⊗ py)∆(Θ(a))T = TΘ(a) = (id⊗ id⊗ ω)((T ⊗ 1)(Xz)∗aXz

)= (id⊗ id⊗ ω)

((Xy

23)∗(Xx

13)∗(ϕ(T )⊗ 1)aXz

)(5.4)

= (id⊗ id⊗ ω)((Xy

23)∗(Xx

13)∗(∆1 ⊗ id)(a)(ϕ(T )⊗ 1)Xz

)= (id⊗ id⊗ ω)

((Xy

23)∗(Xx

13)∗((∆1 ⊗ id)(a)

)Xx

13Xy23

)(T ⊗ 1)

where (5.4) is valid because

(T ⊗ 1)(Xz)∗ = (Xy)∗23(Xx)∗13(ϕ(T )⊗ 1) .

Then, we get that

px(id⊗ ψ2y)∆2(Θ(a))

= (id⊗ ψ2y ⊗ ω)

((Xy

23)∗(Xx

13)∗((∆1 ⊗ id)(a)

)Xx

13Xy23

)= (id⊗ ω)

((Xx)∗

(((id⊗ ϕ1

y)∆1 ⊗ id)(a))Xx

)= Θ(a)px

The equality follows from the fact that (ψ2y⊗ω)

((Xy)∗rXy

)= (ϕ1

y⊗ω)(r)for r ∈ `∞(G1)⊗B. Indeed, the KMS-property of the state ω (see (2.3.4))gives

(ψ2y ⊗ω)

((Xy)∗rXy

)= (ψ1

y ⊗ω)(((Q2)−2

y ⊗ 1)Xy(Xy)∗r)

= (ϕ1y ⊗ω)(r) .

So, we obtain that

(id⊗ ψ2µ)∆2(Θ(a)) = (id⊗ ω)

(X∗(((id⊗ ϕ2

µ)∆1 ⊗ id)(a))X

)(5.5)


whenever a ∈ D2.

Because of the fact that ω is invariant under δ1 and using formula (4.1),expression (5.5) equals

(id⊗ h1 ⊗ ω)(X∗

13(V1)∗12

(((id⊗ ϕ1

µ)∆1 ⊗ δ1)(a))(V1)12X13

)= (id⊗ h1 ⊗ ω)

(X∗

13((V1)∗12

(((id⊗ ϕ1

µ)∆1 ⊗ id⊗ id)(αG1 ⊗ id)(a))(V1)12X13

)= (id⊗ ϕ1

µ ⊗ h1 ⊗ ω)(X∗

14((V1)∗13(∆1 ⊗ id⊗ id)((V1)12a13(V1)∗12)(V1)13X14

)= (id⊗ ϕ1

µ ⊗ h1 ⊗ ω)(X∗

14(V1)23(∆1 ⊗ id)(a)124(V1)∗23X14

)= (id⊗ ψ1

µ ⊗ h1 ⊗ ω)(X∗

14(∆1 ⊗ id)(a)124X14

)(5.6)

= (id⊗ ω)(X∗(Pµ ⊗ id)(a)X) = Θ(a) .

Step 5.6 follows from the KMS-property of h1 (see 1.4). So, we get that

(ϕy ⊗ h1)(V1sV∗1) = (ϕ1

y ⊗ h1)(sV∗1V1((Q1)2y ⊗ 1))

= Tr((Q1)−1y ⊗ 1)s((Q1)2y ⊗ 1)) = (ψ1

y ⊗ h1)(s)

for every s ∈ `∞(G1)⊗L∞(G1).

From the above computation, it follows that Θ(a) ∈ H∞(G2, µ), whenevera ∈ D2. This completes the proof.

By the construction of D2 we get a natural action α2 on D2 which is givenby id⊗ δ2|D2 .

Proposition 5.2.5. The generalized Izumi operator Θ defined by (5.3)intertwines the action α2 with the conjugate action αG2 defined in 5.1.6.

Proof. This follows from the calculation below. For all a ∈ D2, it holdsthat

(Θ⊗ id)α2(a) = (id⊗ ω ⊗ id)(X∗12(α2)13(a)X12)

= (id⊗ ω ⊗ id)(X∗12(id⊗ δ2)(a)X12)

= (id⊗ ω ⊗ id)((V2)13(id⊗ δ2)

(X∗aX

)(V2)∗13

)= V2

((id⊗ ω ⊗ id)(id⊗ δ2)

(X∗aX

))V∗

2

= V2

((id⊗ ω)

(X∗aX

)⊗ 1

)V∗

2 = αG2(Θ(a)) .

Remark 5.2.6. The operator Θ : `∞(G1)⊗B → H∞(G2, µ) is clearlynormal and completely positive.


Multiplicativity of the generalized Izumi operator

To prove that Θ is multiplicative, we prove that it is a complete orderisomorphism, i.e. that Θ has a completely positive inverse.

We consider the inverse monoidal equivalence ϕ−1 : G1 → G2. This pro-vides us with a link algebra B, two commuting actions γ1 and γ2 and uni-taries Y x ∈ L(Hϕ(x),Hx)⊗alg B as in (4.2). Denote Y =

⊕x∈Irred(G1) Y

x.

In that way, we obtain another generalized Izumi operator Θ′ : D1 →H∞(G1, µ) which is now given by

Θ′(a) = (id⊗ ω)(Y∗aY) (5.7)

with

D1 := a ∈ H∞(G2, µ)⊗B | (α2 ⊗ id)(a) = (id⊗ γ2)(a)

Again, Θ′ : D1 → H∞(G1, µ) intertwines (id⊗ γ1)|fD1and αG1 .

We calculate the compositions of Θ′ and Θ⊗id. To make the compositionsmeaningful, we need the canonical isomorphism T between H∞(G1, µ)and

D′1 = a ∈ H∞(G1, µ)⊗B⊗B | (id⊗ δ2 ⊗ id)(a) = (id⊗ id⊗ γ2)(a)

and (αG1 ⊗ id⊗ id)(a) = (id⊗ δ1 ⊗ id)(a)

as in Proposition 4.1.4. This isomorphism is explicitly given by T =(id⊗ π) αG1 , with (id⊗ π)(V1) = X12Y13. This means that

T : H∞(G2, µ) → H∞(G2, µ)⊗B⊗B : a 7→ X12Y13(a⊗ 1⊗ 1)Y∗13X∗

12 .

Proposition 5.2.7. Both mappings

Θ′ (Θ⊗ id) T : H∞(G1, µ) → H∞(G1, µ)

and(Θ⊗ id) T Θ′ : D1 → D1

are identity mappings. Hence

S1 := Θ′ : D1 → H∞(G1, µ) and S2 := (Θ⊗ id)T : H∞(G1, µ) → D1

are bijective operators.


Proof. Let a ∈ H∞(G1, µ), we get that

Θ′(Θ⊗ id)T (a) = Θ′(id⊗ ω ⊗ id)(X∗12T (a)X12)

= (id⊗ ω)(Y∗(id⊗ ω ⊗ id)(X∗12T (a)X12)Y)

= (id⊗ ω ⊗ ω)(Y∗13X∗

12T (a)X12Y13) = a .

We now compute the converse composition.

Therefore, take a ∈ D1, meaning that a ∈ H∞(G2, µ)⊗B and (αG2 ⊗id)(a) = (id⊗ γ2)(a).

It holds that

TΘ′(a) = X12Y13(Θ′(a)⊗ 1⊗ 1)Y∗13X∗

12

and hence

(Θ⊗ id)TΘ′(a) = (id⊗ ω ⊗ id)(X∗12X12Y13(Θ′(a)⊗ 1⊗ 1)Y∗

13X∗12X12)

= Y(Θ′(a)⊗ 1)Y∗ .

Because ω(b)1 = (h2 ⊗ id)γ2(b) for b ∈ B, and the fact that a ∈ D1, wefind that

Θ′(a)⊗ 1 = (id⊗ h2 ⊗ id)(id⊗ γ2)(Y∗aY)

= (id⊗ h2 ⊗ id)(Y∗13(V2)∗12(α2 ⊗ id)(a)(V2)12Y13)

= Y∗(id⊗ h2 ⊗ id)((V2)∗12(V2)12a13(V2)∗12(V2)12 = Y∗aY .

This gives us that (Θ⊗ id)TΘ′(a) = a for a ∈ D1.

Corollary 5.2.8. The mapping Θ : D2 → H∞(G2, µ) is a ∗-isomorphism.

Proof. Because T is a ∗-homomorphism, S1 and S2 are completely positiveunital mappings. By the Schwarz inequality for contractive completelypositive mapping (5.2.2 in [23]), it follows that S1(a∗)S1(a) ≤ S1(a∗a)and S2(b∗)S2(b) ≤ S2(b∗b) for every a ∈ D1 and b ∈ H∞(G1, µ). BecauseS1 and S2 are each others inverses, it holds that

a∗a = (S2S1(a))∗S2S1(a) ≤ S2(S1(a)∗S1(a)) ≤ a∗a .

Hence a∗a = S2(S1(a)∗S1(a)), so S1(a∗a) = S1(a)∗S1(a). Again by corol-lary 5.2.2 in [23], this implies that S1, and hence also S2 are multiplicative.


From this and the bijectivity of S1 and S2, it follows that S1 and S2

are ∗-isomorphisms. Thus, we obtain that Θ′ : D1 → H∞(G1, µ) is a∗-isomorphism. By symmetry, we get that Θ : D2 → H∞(G2, µ) is a∗-isomorphism.

5.3 Concrete computations of certain Poisson

boundaries using Tomatsu’s results.

In [45], Tomatsu has proven that the Poisson boundary of the dual ofa coamenable compact quantum group with commutative fusion rulescan be identified with the homogeneous space coming from its canoni-cal Kac subgroup (see definition 2.4.4). This gives us immediately thePoisson boundary of a whole class of quantum groups. For the duals ofq-deformations of classical Lie-groups, he moreover concretely calculatesthe canonical Kac quotient.

Using the previous section, we find the Poisson boundary of the dualof every compact quantum group with commutative fusion rules whichis monoidally equivalent to a coamenable one. Moreover, we obtain aconcrete description of the Poisson boundary as a homogeneous space ofthe link algebra.

So, consider a compact quantum group (G1,∆1) which is co-amenableand has commutative fusion rules. Let (G2,∆2) be a compact quantumgroup monoidally equivalent with G1 with monoidal equivalence given byϕ : G2 → G1. This gives us a link-algebra B, and two commuting actionsδ1 and δ2 as before. Denote by H1 the canonical Kac quotient of G1 andby rH1 : C(G1) → C(H1) be the corresponding restriction map.

In [45], the following is proven:

Theorem 5.3.1. Let G be a coamenable compact quantum group withcommutative fusion rules and H its canonical Kac quotient. Let µ be agenerating measure on Irred(G). The Izumi operator

Θ : L∞(H\G) → H∞(G, µ) : a 7→ (id⊗ h)(V∗(1⊗ a)V)

is a *-isomorphism and intertwines the adjoint action αG with the action∆H\G as defined in remark 2.4.

5.3 Concrete computations 117

In definition 4.2.1, we associated to a homogeneous space of a quantumgroup, a homogeneous space of the link-algebra.

Tomatsu’s result combined with theorem 4.2.2 and 5.2.2, immediatelygives us the following theorem.

Theorem 5.3.2. Consider a coamenable compact quantum group G1 withcommutative fusion rules. Let G2 be a compact quantum groups that ismonoidally equivalent to G1. Denote by B be the link algebra associatedto the monoidal equivalence. Let H1 be the canonical Kac subgroup of G1.Consider a generating measure µ on Irred(G1). Then (H∞(G2, µ), αG2)is isomorphic to (BH1 , δ2). The isomorphism is given by the followinggeneralized Izumi operator

Θ : BH1 → H∞(G2, µ) : a 7→ (id⊗ ω)(X∗(1⊗ a)X) (5.8)

This ∗-isomorphism intertwines the adjoint action αG2 and the action δ2.

Notation 5.3.3. As above, D2 := a ∈ H∞(G1, µ)⊗B | (αG1 ⊗ id)(a) =(id⊗ δ1)(a). Denote by Θ2 : D2 → H∞(G2, µ) : a 7→ (id⊗ω)(X∗aX) thegeneralized Izumi operator as defined in (5.3).

Proof. By Tomatsu ([45]), Θ : L∞(H1\G1) → H∞(G1, µ) is a ∗-isomorphism.Because Θ intertwines the actions ∆H1\G1

and αG1 ,

Θ⊗ id : D2 := a ∈ L∞(H1\G1)⊗B | (∆1 ⊗ id)(a) = (id⊗ δ1)(a) → D2

is also an isomorphism. Combining this with Theorem 5.2.2, it followsthat D2 and H∞(G2, µ) are isomorphic through Θ2 (Θ⊗ id).

It follows from 4.2.2 that δ1 : BH1 → D2 is an isomorphism. Hence

Θ2(Θ⊗ id)δ1 : BH1 → H∞(G2, µ)

is a *-isomorphism.

Now we just need to prove that Θ2(Θ ⊗ id)δ1 = Θ, which follows fromthe next obvious calculation.

Θ2(Θ⊗ id)δ1(a) = (id⊗ ω)(X∗(id⊗ h1 ⊗ id)((V1)∗12(δ1(a))23(V1)12)X)

= (id⊗ ω)(id⊗ h1 ⊗ id)(id⊗ δ1)(X∗(1⊗ a)X)

= (id⊗ ω)(((id⊗ ω)(X∗(1⊗ a)X))⊗ 1

)= Θ(a)


for all a ∈ BH1 .

Moreover, Θ intertwines αG2 and δ2 because for all a ∈ BH1 ,

(Θ⊗ id)δ2(a) = (id⊗ ω ⊗ id)(X∗12δ2(a)23X12)

= (id⊗ ω ⊗ id)((V2)13(id⊗ δ2)(X∗(1⊗ a)X)(V2)∗13)

= V2((id⊗ ω)(X∗(1⊗ a)X)⊗ 1)(V2)∗ = αG2(Θ(a)) .

This completes the proof.

This result enables us to calculate the Poisson boundaries of a lot ofquantum groups. Tomatsu already concretely obtained the canonical Kacsubgroup of q-deformed classical compact Lie groups. The theorem abovegives us the Poisson boundary of all duals of compact quantum groupsthat are monoidally equivalent to such q-deformation.

One class of quantum groups that satisfy this are the Ao(F )’s. The Pois-son boundary of their duals had already be obtained in a different way(but also using monoidal equivalence) in [52]. However, as every Ao(F )is monoidally equivalent to some SUq(2), we can identify the Poissonboundary of Ao(F ) also using theorem 5.3.2.

Another class, namely the class of quantum automorphism groups is ex-plored in the next section. It would be interesting to examine how manyquantum groups satisfy the requirements of theorem 5.3.2. The questionarises if there are compact quantum groups with commutative fusion ruleswhich are not monoidally equivalent to some coamenable quantum group.

5.4 Example: Quantum automorphism groups

In section 2.5, we recalled the notion of a quantum automorphism group.In this section we identify the Poisson boundary for Aaut(B,ϕ) with B aC∗-algebra of finite dimension bigger than 4 and ϕ a δ-state on B. To dothis, we make use of the previous section 5.3.

In 3.6, we proved that every quantum automorphism group of this typeis monoidally equivalent with one of the form Aaut(M2(C),Tr(·F )), where

5.4 Example: Quantum automorphism groups 119

Tr(F−1) = δ2. Because of the quantum Kesten result (see [8]),Aaut(M2(C),Tr(·F )) is coamenable. Moreover, it has the fusion rulesof SO(3) and those are commutative. Hence, we can apply Tomatsu’sresult 5.3.1.

5.4.1 The canonical Kac subgroup of Aaut(M2(C), Tr(·D))

Denote by G the compact quantum group Aaut(M2(C),Tr(·D)) and byH its canonical Kac subgroup. Observe that we can take D a diagonalmatrix. We here consider only non-trivial D. If D equals the identitymatrix, we just get the compact group SO(3) which is already Kac. HenceSO(3) has trivial Poisson boundary. Denote by π : C(G) → C(H) thecanonical projection map. We denote by U the fundamental irreduciblerepresentation with label 1 and by Q the matrix corresponding to U asdefined in 1.2, normalized such that Tr(Q) = Tr(Q−1). The eigenvaluesof Q are of the form 1, q, q−1.

Now V := (id ⊗ π)(U) is a representation of H and because H is Kac,V = (id ⊗ π)(U) must be unitary. The matrix F =

√QT , unitarizes U ,

what in this case means (F ⊗1)U(F−1⊗1) = U . We claim that V breaksup in 3 one-dimensional representations. As every representation of Happears in a repeated tensor power of V , it follows that all irreduciblerepresentations of H have dimension one.

Proof of claim: As F ∗F has 3 different eigenvalues, it suffices to provethat F ∗F and V commute. It holds that V = FV F−1, so

V F = FV and F ∗V ∗ = V∗F ∗ .

As V is unitary, it follows that V FF ∗V ∗ = FF ∗, which means thatF ∗F ∈ End(V ).

Since all irreducible representations of H have dimension 1, we concludethat H is the dual of a discrete group Γ. Denote by ug the irreduciblerepresentation of Γ corresponding to g ∈ Γ. Since V ∼= V , there existg, h ∈ Γ such that

V ∼= ug ⊕ uh ⊕ ug−1 .

Observe that Γ is generated by g and h. We claim that Γ is abelian. SinceU is a subrepresentation of U⊗2, V is a subrepresentation of V ⊗2. But

V ⊗2 = ug2 ⊕ uh2 ⊕ ug−2 ⊕ 2ue ⊕ uhg ⊕ ugh ⊕ uhg−1 ⊕ ug−1h ,


implying that h ∈ g2, h2, g−2, e, hg, gh, hg−1, g−1h. Any of these possi-bilities for h imply that the group generated by g and h is abelian.

As Γ is commutative, H is just a commutative compact group. Hence themaximal quantum subgroup of Kac type of G is the maximal compactsubgroup of G.

Suppose χ : C(G) → C is a character and α : M2(C) → M2(C) ⊗ C(G)the canonical action of G on M2(C) coming from U . Now (id⊗χ)α is anautomorphism of M2(C), and hence implemented by a unitary matrix A.Moreover, as (id⊗ χ)α is invariant under Tr(·D),

Tr(DAxA∗) = Tr(Dx) for all x ∈M2(C) .

Hence A is a diagonal matrix. But then Ad(A) = Ad(diag(z, z)) for somez ∈ T.

On the other hand, T acts on M2(C) in the way described above. Thisaction δ is Tr(·D)-invariant, so because of the universality of G, thereexists a morphism of quantum groups π : C(G) → C(T) such that (id ⊗π)α = δ.

We may conclude that the maximal Kac subgroup of G is the torus T.

5.4.2 The Poisson boundary of Aaut(B, ϕ)

We now apply theorem 5.3.2 and obtain the following result:

Theorem 5.4.1. Consider Aaut(B,ϕ) with B a C∗-algebra of finite di-mension strictly bigger than 4 and ϕ a δ-state on B. Take F ∈ M2(C)such that Tr(F−1) = δ2. Then the Poisson boundary of Aaut(B,ϕ) isgiven by L∞(Aaut((B,ϕ), (M2(C),Tr(·F ))))T.

The 4-dimensional case was considered above, except for the caseAaut(C4).But this compact quantum group is coamenable by the quantum Kestenresult and moreover Kac, so Aaut(C4) has trivial Poisson boundary. Thiscompletes the identification of Poisson boundaries of the duals of quantumautomorphism groups.

Nederlandse samenvatting

In deze thesis pakken we twee soorten problemen omtrent compacte endiscrete kwantumgroepen aan. Enerzijds geven we de eerste voorbeeldenvan ergodische acties van compacte kwantumgroepen met grote multi-pliciteit. Anderszijds berekenen we de Poissonranden voor bepaalde niet-amenable discrete kwantumgroepen met commutatieve fusieregels. Hoeweldit op zich verschillende problemen zijn, gebruiken we toch hetzelfde werk-tuig om ze op te lossen, namelijk monoidale equivalentie. Dit begrip wordtuitvoerig bestudeerd voor compacte kwantumgroepen om zo tot een ant-woord op de hierboven gestelde vragen te komen.

N.1 Compacte en discrete kwantumgroepen

In dit deel herhalen we de basisdefinities en resultaten omtrent compacteen discrete kwantumgroepen.

N.1.1 Notaties

Zij S een deelverzameling van een C∗-algebra. We noteren de lineaire spanvan S door 〈S〉 en de gesloten lineaire span van S door [S]. We gebruikende notatie ωξ,η(a) = 〈ξ, aη〉 en we beschouwen inproducten die lineair zijnin de tweede variabele. We stellen ξ∗ : H → C : η 7→ 〈ξ, η〉 en we noterendoor H de duale Hilbertruimte van H, d.w.z. H := ξ∗ | ξ ∈ H.

Het symbool ⊗ gebruiken we voor tensorproducten van Hilbertruimten enminimale tensorproducten van C∗-algebra’s. We gebruiken het symbool⊗alg voor algebraische tensorproducten van *-algebra’s en ⊗ voor het ten-sorproduct van von Neumannalgebra’s. We maken ook vaak gebruik van

124 Nederlandse samenvatting

de leg-numbering notatie in meervoudige tensorproducten: als a ∈ A⊗Adan zijn a12, a13 en a23 de logische elementen in A⊗A⊗A, bijvoorbeelda12 = a⊗ 1.

De toevoegbare operatoren tussen C∗-modules of begrensde operatorentussen Hilbertruimten H en K worden genoteerd als L(H,K). We schrij-ven ook L(K,K) = L(K).

Zij B een unitale *-algebra. Dan noemen we een lineaire afbeelding ω :B → C zodat ω(1) = 1 een trouwe toestand als ω(a∗a) ≤ 0 voor alle a ∈ Ben ω(a∗a) = 0 als en slechts als a = 0.

N.1.2 Compacte kwantumgroepen

Reeds in de jaren ’80 introduceerde Woronowicz de notie van een com-pacte kwantumgroep en veralgemeende hij de klassieke Peter-Weyl theo-rie. In de loop der tijd zijn er veel interessante voorbeelden van compactekwantumgroepen bestudeerd. Zo voerden Drinfel’d and Jimbo [20, 32] deq-deformaties van compacte semi-simpele Lie groepen waarna Rosso [40]aantoonde dat deze in het kader van Woronowicz passen. In [50] voer-den Van Daele en Wang de universele orthogonale en unitaire kwantumgroepen. Deze werden in detail bestudeerd door Banica in [3, 2].

Definitie N.1.1. Een compacte kwantumgroep G is een paar (C(G),∆),zodat

• C(G) een unitale C∗-algebra is;

• ∆ : C(G) → C(G)⊗C(G) een unitaal en co-associatief ∗-homomorfismeis, d.w.z.

(∆⊗ id)∆ = (id⊗∆)∆ ;

• G voldoet aan de linkse en rechtse schrappingswet d.w.z.

[∆(C(G))(1⊗ C(G))] = [∆(C(G))(C(G)⊗ 1)] = C(G)⊗ C(G) .

De afbeelding ∆ noemen we de co-vermenigvuldiging.

Deze definitie en notatie komen natuurlijk niet uit het niets. Het ba-sisvoorbeeld wordt namelijk gegeven door de continue functies op eencompacte groep G. Inderdaad, als G een compacte groep is, is C(G) een

N.1 Compacte en discrete kwantumgroepen 125

unitale C∗-algebra. Hierop kunnen we met behulp van de vermenigvuldig-ing op G makkelijk een co-vermenigvuldiging leggen door

∆ : C(G) → C(G)⊗ C(G) : ∆(f)(s, t) = f(st)

te definieren.

Het analogon voor de Haarmaat op compacte groepen is de Haartoestandop een compacte kwantumgroep.

Stelling N.1.2 (Woronowicz, [60]). Zij G een compacte kwantumgroep.Er bestaat een unieke toestand h op C(G) waarvoor

(id⊗ h)∆ = h(a)1 = (h⊗ id)∆(a)

voor alle a ∈ C(G). Deze toestand h noemen we de Haartoestand van G.

De Peter-Weyl representatietheorie voor compacte groepen, kan bijnavolledig worden veralgemeend naar compacte kwantumgroepen.

Definitie N.1.3. Een unitaire representatie U van een compacte kwan-tumgroep G op een Hilbertruimte H is een unitair element U ∈ L(H ⊗C(G)) dat voldoet aan

(id⊗∆)(U) = U12U13 .

Als U1 en U2 unitaire representaties van G zijn op respectievelijk deHilbertruimten H1 en H2, definieren we de intertwiners tussen U1 en U2

als

Mor(U1, U2) := T ∈ L(H2,H1) | U1(T ⊗ 1) = (T ⊗ 1)U2 .

We gebruiken de notatie End(U) := Mor(U,U). Een unitaire represen-tatie U wordt irreducibel genoemd als End(U) = C1. Als Mor(U1, U2)een unitaire operator bevat, noemen we U1 en U2 unitair equivalent.

Stelling N.1.4. Elke irreducibele representatie van een compacte kwan-tumgroep is eindigdimensionaal. Elke unitaire representatie is unitairequivalent met een directe som van irreducibele representaties.

Dit resultaat zorgt ervoor dat we bijna alleen werken met eindigdimen-sionale representaties.


Tevens kunnen de directe som en tensor product van representatiesgedefinieerd worden. Hierbij merken we op dat in tegenstelling tot hetklassieke geval, het tensorproduct van representaties voor kwantumgroepenniet commutatief is. De manier waarop het tensorproduct van irreducibe-len opbreekt in irreducibele representaties, noemen we de fusieregels.

N.1.3 Discrete kwantumgroepen

Discrete kwantumgroepen doken het eerst op als dualen van compactekwantumgroepen. Ze kunnen echter intrinsiek gekarakteriseerd engedefinieerd worden. Dit werd ontdekt door Van Daele in [48] en Effrosen Ruan in [22]. In [48], definieert Van Daele een discrete kwantumgroepals een multiplier Hopf *-algebra waarvoor de onderliggende *-algebra eendirecte som van matrixalgebra’s is. De duale van een compacte kwantum-groep is zo ’n discrete kwantumgroep en wordt als volgt gedefinieerd.

Definitie N.1.5. Zij G een compacte kwantumgroep. We definieren deduale (discrete) kwantumgroep G als volgt.

c0(G) =⊕

x∈Irred(G)

L(Hx) , `∞(G) =∏

x∈Irred(G)

L(Hx) .

Deze notatie komt voort uit het basisvoorbeeld waar G de duale van eendiscrete groep Γ is. Dan is `∞(G) = `∞(Γ) en co(G) = co(Γ).

We hebben een natuurlijk unitair element V ∈ M(c0(G)⊗C(G)), gegevendoor

V =⊕

x∈Irred(G)

Ux .

Deze unitaire V draagt de dualiteit tussen G en G in zich. We hebbeneen natuurlijke covermenigvuldiging

∆ : `∞(G) → `∞(G)⊗`∞(G) zodat (∆⊗ id)(V) = V13V23 ,

dewelke ook op volgende analoge manier kan gedefinieerd worden

∆(a)S = Sa voor alle a ∈ `∞(G), S ∈ Mor(y ⊗ z, x) .

N.2 Acties van kwantumgroepen en spectrale deelruimten 127

N.1.4 De universele orthogonale compacte kwantumgroepen

De universele orthogonale compacte kwantumgroepen werden ingevoerddoor Wang en Van Daele in [50]. Deze compacte kwantumgroepen kun-nen in het algemeen niet bekomen worden als deformaties van klassiekeobjecten.

Definitie N.1.6. Zij F ∈ GL(n,C) zodat FF = ±1. We definieren decompacte kwantumgroep Ao(F ) als volgt.

• Cu(G) is de universele C∗-algebra met generatoren (Uij) and relatieszodat U = (Uij) een unitair element is in Mn(C) ⊗ Cu(G) en U =FUF−1, waarbij (U)ij = (Uij)∗.

• ∆(Uij) =∑

k Uik ⊗ Ukj .

De matrix U is met deze definitie een irreducibele representatie, die wede fundamentele representatie noemen.

Als F ∈ GL(2,C), krijgen we op equivalentie na de matrices

Fq =(

0 |q|1/2−sgn(q)|q|−1/2 0

)(N.1)

voor q ∈ [−1, 1], q 6= 0, met bijhorende kwantumgroepen Ao(Fq) ∼=SUq(2) ( [59]).

Banica heeft in [2] bewezen dat de Ao(F ) precies die kwantumgroepenzijn met dezelfde fusieregels als de groep SU(2).

Stelling N.1.7. Zij F ∈ GL(n,C) en FF = ±1. Zij G = Ao(F ). Dankan Irred(G) geidentificeerd worden met N zodat

x⊗ y ∼= |x− y| ⊕ (|x− y|+ 2)⊕ · · · ⊕ (x+ y) ,

voor alle x, y ∈ N.

N.2 Acties van kwantumgroepen en spectrale deel-

ruimten

De resultaten in dit deel zijn niet nieuw, hoewel we een persoonlijkevoorstelling geven van de theorie van acties van compacte kwantum-groepen.


Definitie N.2.1. Zij B een unitale C∗-algebra. Een (rechtse) actie vanG op B is een unitaal *-homomorfisme δ : B → B ⊗ C(G) dat voldoetaan de volgende eigenschappen:

(δ ⊗ id)δ = (id⊗∆)δ en [δ(B)(1⊗ C(G))] = B ⊗ C(G) .

We noemen de actie δ ergodisch als de vastepuntsalgebra Bδ := a ∈ B |δ(a) = a ⊗ 1 gelijk is aan C1. In dat geval bestaat er op B een uniekeinvariante toestand ω, gegeven door ω(a)1 = (id⊗ h)δ(a).

Definitie N.2.2. Zij δ : B → B ⊗ C(G) een actie van de compactekwantumgroep G op de unitale C∗-algebra B. Voor elke x ∈ Irred(G)definieren we de spectrale deelruimte geassocieerd met x als

Kx = X ∈ Hx ⊗B | (id⊗ δ)(X) = X12Ux13 .

Als we Mor(δ, x) := S : Hx → B | S lineair en δ(Sξ) = (S ⊗id)(Ux(ξ ⊗ 1)) definieren, dan geldt dat Kx

∼= Mor(δ, x) waarbij deidentificatie gebeurt door aan elke X ∈ Kx de operator SX : Hx → B :ξ 7→ X(ξ ⊗ 1) te associeren.

We merken nog op dat er op de spectrale deelruimten Kx een natuurlijkelinkse-en rechtse Hilbert C∗-module structuur ligt. In het ergodische gevalzijn de spectrale deelruimten Hilbertruimten.

Definitie N.2.3. We definieren B als de deelruimte van B, opgespannendoor de spectrale deelruimten, d.w.z.

B := 〈X(ξ ⊗ 1) | x ∈ Irred(G), X ∈ Kx, ξ ∈ Hx〉 .

Tevens definieren we

Bx := 〈X(ξ ⊗ 1) | X ∈ Kx, ξ ∈ Hx〉 .

We merken op dat δ : Bx → Bx ⊗alg C(G)x en dat B∗x = Bx. Het is danook eenvoudig in te zien dat B een dichte unitale *-deelalgebra is van B

en dat de beperking δ : B → B⊗alg C(G) een actie definieert van de Hopf*-algebra C(G) op B. We noemen B de spectrale deelalgebra van δ.

Opmerking N.2.4. Merk op dat als vectorruimte Bx ' Hx⊗Kx. Als δergodisch is, is Bx eindigdimensionaal en noteren we mult(δ, x) = dimKx.We noemen mult(δ, x) de multipliciteit van x in δ.

N.2 Acties van kwantumgroepen en spectrale deelruimten 129

Stel nu dat δ : B → B ⊗ C(G) een ergodische actie is. Beschouw x ∈Irred(G). Zij t ∈ Mor(x ⊗ x, ε), genormalizeerd zodat t∗t = dimq(x).Definieer de antilineaire afbeelding

Rx : Kx → Kx : Rx(v) = (t∗ ⊗ 1)(1⊗ v∗) . (N.2)

Omdat t vast is op een getal van modulus 1 na, is Lx := R∗xRx een goed

gedefinieerde positieve operator in L(Kx).

Definitie N.2.5. We stellen multq(x) :=√

Tr(Lx) Tr(Lx) en we noemenmultq(x) de kwantummultipliciteit van x in δ.

Er geldt dat multq(x) ≤ dimq(x) voor alle x ∈ Irred(G). Als gelijkheidbereikt wordt voor alle x ∈ Irred(G), zeggen we dat δ van volle kwantum-multipliciteit is.

Terminologie N.2.6. We noemen een actie δ : B → B ⊗ C(G) van Gop B universeel als B de universeel omhullende C∗-algebra van B is. Wenoemen de actie gereduceerd als de conditionele verwachting (id⊗h)δ vanB naar de vaste puntsalgebra Bδ trouw is.

Opmerking N.2.7. Een compacte kwantumgroep (C(G),∆) heeft veleC∗-versies, terwijl de onderliggende Hopf∗-algebra hetzelfde blijft. Vooracties geldt hetzelfde. We beschouwen twee acties dan ook enkel als ver-schillend als de onderliggende Hopf ∗-algebra acties verschillend zijn. Ditwordt verder veelvuldig gebruikt.

Acties op Neumann algebras worden als volgt gedefinieerd.

Definitie N.2.8. Een rechtse actie van een compacte (resp. discrete)kwantumgroep G (resp. G) op een von Neumann algebraN is een injectiefnormaal unitaal ∗-homomorfisme

δ : N → N⊗L∞(G) resp. δ : N → N⊗`∞(G)

dat voldoet aan (δ ⊗ id)δ = (id⊗∆)δ, resp. (δ ⊗ id)δ = (id⊗ ∆)δ.

Merk op dat we geen dichtheidseis opleggen zoals voor C∗-algebraische ac-ties. Voor von Neumann algebras is hieraan immers automatisch voldaan(zie bijvoorbeeld [51] voor een bewijs. Dit impliceert dat de spectraledeelalgebra ook dicht is in N , hetgeen ook rechtstreeks bewezen werd indeze thesis.


N.3 Ergodische acties met grote multipliciteit

Høegh-Krohn, Landstad and Størmer bewezen in [29] dat compacte groepenalleen ergodisch kunnen werken op C∗-algebras met een spoor. De uniekeinvariante toestand is in dat geval immers een spoor. Bovendien toon-den ze aan dat de multipliciteit van een irreducibele representatie in eenergodische actie noodzakelijk naar boven begrensd is door haar dimensie.

In zijn artikels [56, 57, 58] voerde Wassermann een doorgedreven studieuit van ergodische acties van compacte kwantumgroepen. Dit leidde tothet belangrijke resultaat dat de compacte groep SU(2) alleen ergodischeacties toelaat op von Neumann algebras van eindig type I. Het is nog eenopen probleem of dit voor alle compacte groepen geldt. Het probleem iszelfs nog niet opgelost voor SU(n) met n ≥ 3.

N.3.1 Monoidale equivalentie van kwantumgroepen

Men kan naar de representatiecategorie van een compacte kwantumgroepkijken op verschillende manieren, de ene al meer in detail dan de andere.De minst precieze manier is kijken naar de fusieregels. Deze bepalen verrevan de kwantumgroep. Zelfs op het klassieke groepsniveau zijn er groependie dezelfde fusieregels hebben en toch niet isomorf zijn.

Als we de representatiecategorie bekijken als een concrete, monoidale ten-sor C∗-categorie, dan bepaalt dit de kwantumgroep volledig. Dit is deinhoud van de Tannaka-Krein reconstructiestelling (Woronowicz, [63]).Voor groepen volstaat minder concrete informatie over de representatiesook nog. Doplicher en Roberts bewezen immers dat je uit de abstractetensor C∗-categorie van een compacte groep (dus zonder de concrete rep-resentatie in de categorie van Hilbertruimtes te beschouwen), de groepvolledig kan bepalen. Voor kwantumgroepen is dit niet langer meer waar.Dit leidt tot de volgende definitie.

Definitie N.3.1. We noemen twee compacte kwantumgroepen G1 =(C(G1),∆) and G2 = (C(G2),∆2) monoidaal equivalent als er eenbijectie ϕ : Irred(G1) → Irred(G2) bestaat waarvoor ϕ(ε) = ε, en lin-eaire isomorfismen

ϕ : Mor(x1 ⊗ · · · ⊗ xr, y1 ⊗ · · · ⊗ yk)

→ Mor(ϕ(x1)⊗ · · · ⊗ ϕ(xr), ϕ(y1)⊗ · · · ⊗ ϕ(yk))

N.3 Ergodische acties met grote multipliciteit 131

die aan de volgende voorwaarden voldoen:

ϕ(1) = 1 ϕ(S ⊗ T ) = ϕ(S)⊗ ϕ(T )

ϕ(S∗) = ϕ(S)∗ ϕ(ST ) = ϕ(S)ϕ(T )(N.3)

wanneer de formules zin hebben. In de eerste formule beschouwen we1 ∈ Mor(x, x) = Mor(x ⊗ ε, x) = Mor(ε ⊗ x, x). Een dergelijke collectieafbeeldingen noemen we een monoidale equivalentie tussen G1 en G2.

Als twee kwantumgroepen monoidaal equivalent zijn, hebben ze dus zekerdezelfde fusieregels. Dit is echter niet genoeg. Ze moeten namelijk ookdezelfde 6-j symbolen hebben (zie [17]).

We zullen in de volgende delen aantonen dat er natuurlijke voorbeeldenzijn van monoidale equivalenties waar dim(ϕ(x)) 6= dim(x). Het is echterwel makkelijk in te zien dat de kwantumdimensies altijd bewaard worden,dus dimq(ϕ(x)) = dimq(x). Voor sommige klassen van compacte kwan-tumgroepen (de universele unitaire), is dit zelfs de enige voorwaarde voormonoidale equivalentie.

Notatie N.3.2. Als twee compacte kwantumgroepen G1 en G2 monoidaalequivalent zijn, noteren we dit door G1 ∼

monG2.

N.3.2 Ergodische acties van volle kwantummultipliciteit

In deze sectie vatten we de belangrijkste resultaten van hoofdstuk 3samen.

Stelling N.3.3. Beschouw twee compacte kwantumgroepen G1 en G2.Zij ϕ : G1 → G2 een monoidale equivalentie.

• Er bestaat een unieke unitale *-algebra B met daarop een trouwetoestand ω en unitaire elementen Xx ∈ L(Hx,Hϕ(x))⊗ B voor allex ∈ Irred(G1) zodat:

1. Xy13X

z23(S⊗1) = (ϕ(S)⊗1)Xx voor alle S ∈ Mor(y⊗z, x) ,

2. de matrixcoefficienten van de Xx-en vormen een lineaire basisvoor B,

3. (id⊗ ω)(Xx) = 0 als x 6= ε.


• Er bestaat een uniek paar van commuterende acties

δ1 : B → B ⊗alg C(G) en δ2 : B → C(G2)⊗ B

zodat

(id⊗ δ1)(Xx) = Xx12U

x13 en (id⊗ δ2)(Xx) = U

ϕ(x)13 Xx

23

voor alle x ∈ Irred(G1).

• De toestand ω is invariant voor δ1 en δ2. We noteren door Br denormsluiting van B in de GNS-representatie ten opzichte van ω endoor Bu the universeel omhullende C∗-algebra van B. De acties δ1en δ2 hebben unieke uitbreidingen tot acties van G1 en G2 op Br,resp. Bu.

Deze acties zijn gereduceerd, resp. universeel en bovendien ergodischen van volle kwantummultipliciteit.

• Omgekeerd is elk paar commuterende gereduceerde, respectievelijkuniversele ergodische acties van volle kwantummultipliciteit afkom-stig van een monoidale equivalentie van compacte kwantumgroepen.

Het bewijs verloopt grotendeels op dezelfde manier als het bewijs van deTannaka-Krein stelling van Woronowicz ([63]).

N.3.3 Monoidale equivalentie voor Ao(F )

Stelling N.3.3 stelt ons nu in staat om ergodische acties van compactekwantumgroepen te construeren waarbij de multipliciteit de dimensie vaneen irreducibele representatie overschrijdt. Dit toont aan dat de multiplic-ity bound stelling niet meer waar blijft voor compacte kwantumgroepen.

De volgende stelling beschrijft volledig de monoidale equivalentie voor deuniversele orthogonale compacte kwantumgroepen.

Stelling N.3.4. Zij F1 ∈ Mn1(C) en F2 ∈ Mn2(C) zodat F1F 1 = ±1 enF2F 2 = ±1.

• De compacte kwantumgroepen Ao(F1) and Ao(F2) zijn monoidaalequivalent als en slechts als F1F 1 en F2F 2 hetzelfde teken hebbenen Tr(F ∗

1F1) = Tr(F ∗2F2).

N.3 Ergodische acties met grote multipliciteit 133

• Stel dat Ao(F1) and Ao(F2) monoidaal equivalent zijn. ZijCu(Ao(F1, F2)) de universele unitale C∗-algebra voortgebracht doorde coefficienten van

Y ∈Mn2,n1(C)⊗Ao(F1, F2) met relaties

Y unitair en Y = (F2 ⊗ 1)Y (F−11 ⊗ 1) .

Dan is Ao(F1, F2) 6= 0 en bestaat er een uniek paar van com-muterende universele ergodische acties, δ1 van Ao(F1) en δ2 vanAo(F2), zodat

(id⊗ δ1)(Y ) = Y12(U1)13 en (id⊗ δ2)(Y ) = (U2)12Y13 .

Hier noteren we door Ui de fundamentele representatie van Ao(Fi).

• (Ao(F1, F2), δ1, δ2) is isomorf met de C∗-algebra Bu en de actiesdaarop gegeven door stelling N.3.3 .

We bekomen nu ook dat Ao(F ) ∼mon

SUq(2) als en slechts als Tr(F ∗F ) =

|q + 1/q| en FF = −sgn (q). Dit levert ons voorbeelden van acties dieniet voldoen aan de klassieke bovengrens voor de multipliciteit.

Gevolg N.3.5. Zij 0 < q ≤ 1. Voor elk even natuurlijk getal n waar-voor 2 ≤ n ≤ q + 1

q bestaat er een ergodische actie van SUq(2) van vollekwantummultipliciteit zodat de multipliciteit van de fundamentele repre-sentatie n is. Voor n oneven geldt hetzelfde voor elk natuurlijk getal nmet 2 ≤ n ≤ |q + 1

q |.

N.3.4 Monoidale equivalentie voor Aaut(B, ω)

Net zoals voor de universele orthogonale kwantumgroepen, kunnen we ookvoor de kwantumautomorfismegroepen een volledige beschrijving gevenvan de monoidale equivalenties en bijhorende link algebra’s. Kwantum-automorfismegroepen zijn een veralgemening van de klassieke automor-fismegroepen van C∗-algebra’s. Om de definitie zinvol te maken, hebbenwe als extra gegeven een invariante toestand nodig, die in ons geval eenδ-vorm is.

Definitie N.3.6. Zij (B,ω) een eindigdimensionale C∗-algebra met daaropeen δ-vorm. We definieren de compacte kwantumgroep G = Aaut(B,ω)als volgt.


• Cu(G) wordt gedefinieerd als de universele C∗-algebra voortgebrachtdoor de coefficienten van U ∈ L(B) ⊗ Cu(G) met relaties zodat Uunitair wordt, η ∈ Mor(U, ε) en µ ∈ Mor(U,U T© U). Hier is µ devermenigvuldiging en η de eenheidsafbeelding op B.

• (id⊗∆)(U) = U12U13

Deze relaties op U zorgen er precies voor dat U een actie levert op B metinvariante toestand ω.

Stelling N.3.7. Zij B1 en B2 eindig dimensionale C∗-algebra’s en ω1 enω2 respectievelijk een δ1-vorm en een δ2-vorm op B1, respectievelijk B2.

• De compacte kwantumgroepen Aaut(B1, ω1) en Aaut(B2, ω2) zijnmonoidaal equivalent als en slechts als δ1 = δ2.

• Stel dat Aaut(B1, ω1) en Aaut(B2, ω2) monoidaal equivalent zijn.Noteer door Cu(Aaut((B1, ω1), (B2, ω2))) de universele C∗-algebravoortgebracht door de coefficienten van een unitaal *-homomorfisme

γ : B1 → B2 ⊗ Cu(Aaut((B1, ω1), (B2, ω2)))

met relaties (ω2 ⊗ id)γ(x) = ω1(x)1 voor alle x ∈ B1 .

Dan is Cu(Aaut((B1, ω1), (B2, ω2))) 6= 0 en bestaat er een uniekpaar van commuterende ergodische acties van volle kwantummulti-pliciteit: δ1 van Aaut(B1, ω1) en δ2 van Aaut(B2, ω2), zodat

(id⊗ δ1)γ = (γ ⊗ id)β1 en (id⊗ δ2)γ = (β2 ⊗ id)γ ,

waarbij β1 : B1 → B1 ⊗ Cu(Aaut(B1, ω1)) en β2 : B2 → B2 ⊗Cu(Aaut(B2, ω2)) de natuurlijke acties van de kwantum automorfis-megroepen zijn.

• Cu(Aaut((B1, ω1), (B2, ω2))) is isomorf met de C∗-algebra Bu en deacties daarop gegeven door stelling N.3.3 en de monoidale equiva-lentie Aaut(B1, ω1) ∼

monAaut(B2, ω2).

Dit impliceert dat elke kwantumautomorfismegroepAaut(B,ω), met ω eenδ-vorm, monoidaal equivalent is met Aaut(M2(C),Tr(·F ) met Tr(F−1) =δ2.

N.4 Acties van monoidaal equivalente kwantumgroepen 135

N.4 Acties van monoidaal equivalente compacte

kwantumgroepen

De resultaten in dit en het volgende deel kwamen tot stand in samenwer-king met Nikolas Vander Vennet.

We merkten reeds op dat voor gewone groepen, monoidale equivalentieimpliceert dat de groepen isomorf zijn. Dit is de inhoud van de Doplicher-Roberts dualiteitstheorie uit [19]. Voor kwantumgroepen is dit niet langermeer waar. We kunnen ons dan afvragen welke informatie over een com-pacte kwantumgroep er gehaald kan worden uit zijn monoidale categorievan representaties. Het blijkt dat, ruwweg gesproken, monoidaal equiva-lente kwantumgroepen ‘dezelfde acties’ hebben.

Beschouw twee monoidaal equivalente compacte kwantumgroepen G1 enG2 en een C∗-algebra D1. Stel dat we een actie α1 : D1 → D1 ⊗ C(G1)gegeven hebben. We kunnen net zo goed met de onderliggende Hopf*-algebra actie α1 : D1 → D1 ⊗ C(G1) werken. Beschouw een monoidaleequivalentie ϕ : G2 → G1. Volgens stelling N.3.3 bekomen we een linkalgebra B, unitairen Xx ∈ L(Hx,Hϕ(x)) ⊗alg B en twee commuterendeergodische acties

δ1 : B → C(G1)⊗alg B en δ2 : B → B ⊗alg C(G2) ,

gegeven door

(id⊗ δ1)(Xx) = Uϕ(x)12 Xx

13 en (id⊗ δ2)(Xx) = Xx12U

x13 .

In de volgende stelling construeren we een actie van G2 met dezelfdespectrale structuur als α1.

Stelling N.4.1. De beperking van id⊗ δ2 tot de algebra

D2 := a ∈ D1 ⊗alg B | (α1 ⊗ id)(a) = (id⊗ δ1)(a)

geeft een Hopf *-algebra actie α2 van G2 op D2 waarvoor geldt:

• a 7→ a ⊗ 1 levert een *-isomorfisme tussen de vastepuntsalgebra’svan α1 en α2.


• De afbeelding Tx : Kϕ(x) → Kx : v 7→ v12Xx13 is een bimodulair iso-

morfisme tussen de spectrale deelruimten van α1 en α2. Bovendienis T een unitair element van L(Kϕ(x),Kx) voor de inproducten 〈·, ·〉len 〈·, ·〉r.

• De verzameling (Tx)x∈Irred(G2) respecteert de monoidale structuur inde zin dat voor x, y, z ∈ Irred(G2) en V ∈ Mor(x⊗ y, z),

Tx(X)13Ty(Y )23(V ⊗ 1) = Tz(X13Y23(ϕ(V )⊗ 1))

voor alle X ∈ Kϕ(x), Y ∈ Kϕ(y).

• Als α1 een ergodische actie is, is α2 dat ook. Bovendien geldt vooralle x ∈ Irred(G2) dat multq(x) = multq(ϕ(x).

We merken hierbij op dat deze stelling niet meteen kan geformuleerdworden op C∗-algebra niveau. Op von Neumann algebra niveau echter,is er geen probleem. Dit zullen we dan ook uitvoerig gebruiken in devolgende sectie.

De constructie van de bovenstaande stelling is inverteerbaar.

Propositie N.4.2. De constructie van stelling N.4.1, toegepast op deinverse monoidale equivalentie ϕ−1 : G1 → G2 en de actie α2 : D2 →D2⊗alg C(G2) levert, op isomorfisme na, terug de actie α1 : D1 → D1⊗alg

C(G1).

Bovenstaande resultaten geven ons nu een natuurlijke bijectieve corres-pondentie tussen de acties van twee monoidaal equivalente kwantum-groepen G1 en G2.

We kunnen dit alles ook toepassen op het speciale geval van een homogeneruimte. Dit zal zijn nut bewijzen in de sectie over Poissonranden.

Zij G1 ∼mon

G2 zoals hierboven. Beschouw een kwantumdeelgroep H vanG1 en de bijhorende actie

∆H\G1: C(H\G1) → C(H\G1)⊗alg C(G1)

op de homogene ruimte. Nu kunnen we aan deze homogene ruimte opnatuurlijke wijze een homogene ruimte van de link algebra associeren.

N.5 Poissonranden via monoidale equivalentie 137

Definitie N.4.3. We definieren de homogene ruimte BH door

BH := a ∈ B | (rH ⊗ id)δ1(a) = 1⊗ a .

Dan is δ2 een actie op BH omdat δ1 en δ2 commuteren.

Stelling N.4.4. De constructie van stelling N.4.1, toegepast op D1 =C(H\G1) en α1 = ∆H\G1

geeft ons een C∗-algebra die isomorf is met BH.Bovendien intertwinet dit *-isomorfisme de actie α2 en de beperking vanδ2 tot BH.

N.5 Poissonranden via monoidale equivalentie

Een belangrijke toepassing van de bijectieve correspondentie tussen actieszoals voorgesteld in het vorige deel, vinden we in de studie van Poisson-randen voor discrete kwantumgroepen. Deze werden in het kwantumgevaleerst bestudeerd door Biane ([9]) voor dualen van compacte groepen enlater door Izumi ([30]) voor willekeurige discrete kwantumgroepen. Inzijn artikel identificeerde hij de Poissonrand van SUq(2) met de Podlessfeer. Izumi, Neshveyev en Tuset berekenden de Poissonrand van de dualevan SUq(n) in [31].

Een volgende stap werd gezet door Tomatsu, die er recentelijk in slaagdeom de Poissonranden van alle amenable discrete kwantumgroepen G waar-bij G commutatieve fusieregels heeft te identificeren. De Poissonrandblijkt in dit geval samen te vallen met de homogene ruimte van G tenopzichte van de maximale gesloten kwantumdeelgroep van Kac type.

Alle bovenstaande berekeningen van Poissonranden, beschouwen enkelamenable kwantumgroepen. De eerste identificatie van de Poissonrandvan een niet-amenable discrete kwantumgroep werd gegeven door Van-der Vennet en Vaes in [52]. Zij bewezen dat de Poissonrand van Ao(F )geıdentificeerd kan worden met een hoger dimensionale Podles sfeer. Hunmethode maakt doorgedreven gebruik van de monoidale equivalentie vanAo(F ) met SUq(2) voor de geschikte q. Op die manier brachten ze hetprobleem terug tot een reeds gekend, namelijk de Poissonrand van SUq(2).

De resultaten in deze laatste sectie werden weer gevonden in samenwer-king met Nikolas Vander Vennet. We geven een systematische methodeom de Poissonranden van de dualen van monoidaal equivalente kwantum-groepen met mekaar te verbinden. De relatie gaat als volgt. Er is een


natuurlijke actie van G op de Poissonrand van een discrete kwantumgroepG. Als nu G1 en G2 monoidaal equivalent zijn, zijn hun Poissonrandenmet mekaar verbonden door de bijectieve correspondentie tussen de actiesvan G1 en G2 zoals beschreven in de vorige sectie.

Stelling N.5.1. Beschouw twee monoidaal equivalente compacte kwan-tumgroepen G1 en G2. Zij µ een voortbrengende kansmaat op Irred(G1).De Poissonrand H∞(G2, µ) is isomorf met

D2 := a ∈ H∞(G1, µ)⊗B | (αG1 ⊗ id)(a) = (id⊗ δ1)(a) .

Bovendien intertwinet dit isomorfisme de actie α2 met de toegevoegdeactie αG2.

Het isomorfisme wordt gegeven door een zogenaamde veralgemeende Izumioperator. Het bewijs loopt in verschillende stappen. Eerst tonen we aandat de beeldruimte van deze operator wel degelijk harmonische elementenoplevert. Vervolgens tonen we aan dat hij bijectief is, wat door de com-plete positiviteit meteen oplevert dat het een *-isomorfisme is.

Tomatsu bewees in een recent artikel het volgende resultaat:

Stelling N.5.2 (Tomatsu, [45]). Zij G een coamenable compacte kwan-tumgroep met commutatieve fusieregels en H zijn kanonieke Kac-deelgroep.Zij µ een voortbrengende maat op Irred(G). De Izumi operator

L∞(H\G) → H∞(G, µ) : a 7→ (id⊗ h)(V∗(1⊗ a)V)

is een *-isomorfisme dat de toegevoegde actie αG en de actie ∆H\G inter-twinet.

We kunnen dit combineren met onze bovenstaande stelling en stellingN.4.3. We bekomen dan onderstaand resultaat.

Stelling N.5.3. Beschouw een coamenable compacte kwantumgroep G1

met commutatieve fusieregels. Zij G2 een compacte kwantumgroep diemonoidaal equivalent is met G1. Noteer door B de link algebra horendebij de monoidale equivalentie. Zij H1 de canonieke Kac-deelgroep van G1.Beschouw een voorbrengende maat µ op Irred(G1). Dan is(H∞(G2, µ), αG2) isomorf met (BH1 , δ2). Het isomorfisme wordt gegevendoor de volgende veralgemeende Izumi operator

Θ : BH1 → H∞(G2, µ) : a 7→ (id⊗ ω)(X∗(1⊗ a)X .

Dit isomorfisme intertwinet de toegevoegde actie αG2 en de actie δ2.

N.5 Poissonranden via monoidale equivalentie 139

Dit maakt het ons mogelijk de Poissonrand te identificeren van de dualevan een aantal niet coamenable compacte kwantumgroepen die monoidaalequivalent zijn met coamenable compacte kwantumgroepen. Een eersteklasse voorbeelden wordt gegeven door de universele orthogonale kwan-tumgroepen Ao(F ). Als de dimensie van F groter is dan 3, is Ao(F ) nietcoamenable. Maar zoals hierboven opgemerkt, zijn de kwantumgroepenAo(F ) en SUq(2) wel monoıdaal equivalent voor een geschikte q. Boven-dien was de Poissonrand van SUq(2) reeds berekend door Izumi en op eenandere manier door Tomatsu (SUq(2) is coamenable). Met behulp vanonze constructie kunnen we nu de Poissonrand van Ao(F ) identificeren.Dit resultaat was ook al behaald in [52] met een andere methode.

Een tweede klasse voorbeelden vinden we bij de kwantumautomorfisme-groepen Aaut(B,ϕ), waarbij B een eindigdimensionale C∗-algebra is metdimensie groter dan 4 en ϕ een zogenaamde δ-toestand op B. Deze zijnco-amenable als en slechts als de dimensie van de C∗-algebra B gelijk isaan 4. Bovendien komen we tot volgend resultaat.

Stelling N.5.4. Stel F 6= id. De canonieke Kac deelgroep vanAaut(M2(C),Tr(·F )) is isomorf met de torus T.

Bijgevolg is de Poissonrand van G = Aaut(M2(C),Tr(·F )) dus isomorfmet L∞(T\G). Omdat ook de link algebra tussen Aaut(M2(C),Tr(·F ))en Aaut(B,ϕ) volledig gekend is in termen van voortbrengers en relaties,kunnen we op deze manier de Poissonrand van alle dergelijke kwantum-automorfismegroepen identificeren.

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