lecture notes on tensor categories, aqft, subfactors ...€¦ · 2 tensor categories the subject of...

Lecture Notes on Tensor Categories, AQFT,

Subfactors, Quantum Groups

Claudia Pinzari

1 Introduction

These are rough lecture notes on Tensor Categories, Algebraic Quantum FieldTheory, Subfactors and Quantum Groups, still under construction. They are notintended for publication, at least in the present form. Writing a complete reporton all this would be a huge task. Moreover, excellent books already exist. Myaim is rather that of introducing the young student to this area. Therefore I havelimited myself to indicate some streamlines, mainly on subjects I feel more familiarwith. Many important developments are not considered here. My aim is to be oflittle help in organizing further reading. (However, my reference list is far frombeing complete.) Comments are welcome.

2 Tensor categories

The subject of tensor categories and quantum groups has become extremely vast,because of the many applications in various areas of mathematics and physics.

In this survey I will select some of their aspects. The guiding principle will bethat of describing aspects that may be of interest, even in a very idealized way,for low dim QFT in the algebraic approach, originally settled, for 4d spacetimedimensions, by Haag and Kastler in the 60s [40].

Thus, because of its fundamental role in quantum theories, the notion of aHilbert space plays a role in the categories and quantum groups I will consider.

However, there is an enormous literature, where deep results have been ob-tained, where tensor categories are thought of as deprived with the analytic struc-tures. This more general framework turns out useful to derive far–reaching con-nections to geometry and topology.

Of course, all this work is of interest also for quantum theories, as it is knownhow to construct in some cases from an algebraic framework, the Hilbert spaceframework. I am thinking, for example, of the work of Wenzl and Rosso.

(Strict) tensor categories

The first examples of tensor categories are those associated to representationsof noncommutative finite groups, where, obviously, the term tensor refers to thetensor product between representations. These categories admit a natural repre-sentation into categories of vector spaces, which is often called a fiber functor or an

1

2 TENSOR CATEGORIES 2

embedding. The classical Tannaka’s theorem shows how the reconstruct the groupfrom the associated category of representations and the embedding functors. Forthis reason, tensor categories endowed from the start with an embedding functorsare called Tannakian or, sometimes, concrete.

Perhaps lack of examples of non-Tannakian categories delayed an abstract no-tion till the 60’s, when it was formalized under the name of monoidal category (see[73]). With time, the more familiar term of tensor category has been introducedand used with the same meaning, this is the term that we shall use.

We shall work only with C–linear categories, meaning that the arrow spacesare complex vector spaces and composition of arrows is bilinear.

A strict tensor category is defined requiring that the set of objects forms anassociative semigroup with unit ι, and that for each pair of arrows T ∈ (u, v),T ′ ∈ (u′, v′), there is a tensor product arrow

T ⊗ T ′ ∈ (u⊗ u′, v ⊗ v′)

satisfying1ι ⊗ T = T ⊗ 1ι = T.

The tensor product is associative, bilinear and compatible with composition,

(S ⊗ T ) (S′ ⊗ T ′) = (S S′)⊗ (T T ′).

The arrow space (ι, ι) is a commutative algebra. We shall always assume

(ι, ι) = C.

As is well known, strict tensoriality refers to the associativity of tensor prod-ucts. In a more general situation, one assumes that associativity holds up to anatural transformation. A well known theorem of MacLane asserts that any tensorcategory is equivalent to a strict one [74]. We shall always work with strict tensorcategories, and drop the adjective strict. This is very convenient for notationalpurposes.

Tensor C∗–categories The notion of C∗–category T has been formalized, to myknowledge, in [38] However, se also references in Muger Lectures [78]. Each ar-row space (u, v) is a complex Banach space and there is an antilinear involutivecontravariant functor,

∗ : T → T

such that for any pair of arrows,

‖S T‖ ≤ ‖S‖‖T‖,

‖T ∗ T‖ = ‖T‖2.

In a tensor C∗–category compatibility between ∗–involution and tensor productsis required:

(S ⊗ T )∗ = S∗ ⊗ T ∗.


If we drop the requirement of the existence of a C∗–norm, we are then leftwith the weaker notion of a ∗–category. In general, a ∗–category (even a tensor∗–category) may have no C∗–structure even assuming finite dimensionality of itsarrow spaces. An example is the Hecke category with parameter q at roots ofunity with its standard ∗–structure. This is an important example, that we shallconsider in more detail later. For the time being, I want to mention that the studyof Hecke algebras or Hecke categories in the above sense flourished in the late 80’sand it is strongly related to the theory of subfactors. Hecke algebras are finitedimensional, but at roots of unity they are not semisimple, and for this reasonthey can not be made into C∗–algebras. However, the work of Wenzl shows thatthey admit a semisimple quotient admitting a C∗–structure.

The objects of the Hecke category are the elements of N0 and the arrow space(n, n) is the Hecke algebra Hn(q).

A necessary and sufficient condition for a ∗–category with finite dimensionalarrow spaces be a C∗–category is that the ∗–operation be faithful, i.e. the equationT ∗T = 0 is satisfied only by the zero arrow, or, equivalently that there is a positivefaithful state on each arrow space (ρ, ρ). This fact is used without explicit mentionby Wenzl in his papers where he constructs tensor C∗–categories from quantumgroups.

Duality in tensor categories If u and ∗u are objects in a strict tensor category,∗u is called a left dual of u if the functor of tensoring on the left by u has thefunctor of tensoring on the left by ∗u as a right adjoint. This amounts to requirethe existence of two arrows, d ∈ (ι, ∗u⊗ u), e ∈ (u⊗ ∗u, ι) satisfying

1∗u ⊗ e d⊗ 1∗u = 1∗u,

e⊗ 1u 1u 1u ⊗ d = 1u.

A left dual is unique up to isomorphism.A right dual u∗ of u is defined in a similar way, requiring that the functor of

tensoring on the right by u∗ is a right adjoint of the functor of tensoring on the rightby u. It may be expressed be existence of arrows e′ ∈ (u∗ ⊗ u, ι), d′ ∈ (ι, u ⊗ u∗)satisfying similar equations.

If every object has a left (right) dual, the category is called left (right) rigid. Achoice of left duals u→ (∗u, dv, ev) is called a left duality. In this case, the choiceextends to a contravariant functor, If T ∈ (u, u′) the associated arrow is

T c := 1∗v ⊗ ev 1∗v ⊗ T ⊗ 1∗v′ 1dv ⊗ 1∗v′ .

Clearly, u is a left dual of u′ if and only if u′ is a right dual of u, i.e. wemay write ∗(u∗) = v = (∗u)∗. However, a right dual and a left dual of v arenot necessarily isomorphic. When they are, we may choose an object u which isat the same time a right and left dual of u. In this situation, every right bidualu∗∗ is isomorphic to u. Conversely, if u and some u∗∗ are isomorphic in a tensorcategory with right duals then u∗, being a left dual of u∗∗, is also a left dual of u.In conclusion, a right dual of u is also a left dual of u if and only if u and u∗∗ areisomorphic.


And in tensor C∗–categories Although less general than their algebraic counter-part, abstract tensor C∗–categories arose already in the late 60’s in the frameworkof algebraic quantum field theory on a 4–dimensional Minkowski spacetime [41].

Note that, thanks to the ∗–involution, a right dual is always a left dual, knownas a conjugate object is specified by the assignment of arrows

R ∈ (ι, u⊗ u), R ∈ (ι, u⊗ u)

satisfying the conjugate equations

R∗ ⊗ 1u 1u ⊗R = 1u,

R∗ ⊗ 1u 1u ⊗R = 1u.

The main reference paper is [68], motivated by Jones index theory. Manyresults and methods of the symmetric case could be generalized without the as-sumption of any symmetry. These categories are semisimple if subobjects anddirect sums exist, and one may always assume this. Standard solutions of theconjugate equations often play the role of left or right duality. These are solutionsso normalized on the irreducible objects:

‖R‖ = ‖R‖

or direct sums of them on reducible objects. They are unique up to unitaryequivalence. Moreover, left and right associated traces

trv : T ∈ (ρ, ρ)→ R∗ 1ρ ⊗ T R, T ∈ (ρ, ρ)→ R∗ T ⊗ 1ρ R

coincide for standard solutions. Every object has an intrinsic, or categorical, di-mension. On irreducible objects, it is defined by any solution, not necessarilynormalized,

d(u) = ‖R‖‖R‖,

in that it does not depend on the choice of the conjugate. On reducible objectswe may use standard solutions and use the same formula. The fundamental resultof Jones implies that

d(u) ∈ 2 cosπ/n, n ≥ 3 ∪ [2,∞.

Dimension in non-involutive tensor categories Muger has extended dimension tocertain non-involutive tensor categories. These are the spherical categories of [6].This is a unifying framework for representation categories of Hopf algebras withinvolutive coinverse and ribbon Hopf algebras of Reshetikhin and Turaev. In thesecategories, duals are two sided and the associated left and right traces coincide.The dimension on irreducibles is defined as

d(u) = (eu d′u)

and one has d(u) = d(u).


Braided symmetry An additional structural element is the braiding. The notion ofa braided tensor category is due to Joyal and Street [51]. It is a generalization ofthe previous notion of symmetric tensor category, where the braid group replacesthe permutation group.

A braided symmetry makes the tensor product commutative up to a naturalisomorphism. More precisely, to each pair u, v of objects of T there is an invertibleε(uv, vu) natural in u, v, in the sense that for T ∈ (v, u), T ′ ∈ (v′, u′),

ε(u, u′) T ⊗ T ′ = T ′ ⊗ T ε(v, v′),

andε(ι, u) = ε(u, ι) = 1u,

ε(u⊗ u′, v) = ε(u, v)⊗ 1u′ 1u ⊗ ε(u′, v),

ε(u, v ⊗ v′) = 1v ⊗ ε(u, v′) ε(u, v)⊗ 1v′ .

Together with ε, we also have the inverse braided symmetry ε−1(u, v) := ε(v, u)−1.If T is a braided tensor C∗–category, we also have the adjoint braided symmetryε∗(u, v) := ε(v, u)∗ and the dual εd := ε−1∗.

The relation with the braid group is well known. Fix an object ρ and setσ1 = ε(ρ, ρ)⊗ 1ρn−2 , σ2 = 1ρ⊗ ε(ρ, ρ)⊗ 1ρn−3 , . . . . Then, σ1, . . . , σn−1 satisfy thepresentation relations of the Braid group Bn

σiσj = σjσi, |i− j| ≥ 2,

σiσi+1σi = σi+1σiσi+1.

The braided symmetry ε is called a permutation symmetry if ε = ε−1. In thiscase, the relation with the braid group reduces to a relation with the permutationgroup.

Ribbon categories Ribbon categories are braided tensor categories with duals anda compatible twist. This notion for abstract tensor categories was introduced in[52, 53, 55], with different names (tortile, balanced,...). For Hopf algebras thenotion of a twist appeared in [103].

A twist in a braided tensor category is a collection of invertible arrows θv ∈(v, v), natural in v,

θι = 1ι,

θv⊗w = ε(w, v) ε(v, w) θv ⊗ θw,

A ribbon category is a braided tensor category with a left duality and a choiceof compatible twists

1∗v ⊗ θv dv = θ∗v ⊗ 1v dv.

This property amounts to require that contravariant functor associated to the leftduality takes the value θ∗v on θv.

In a ribbon category a left dual is also a right dual and there is an associatedright duality (v∗ = ∗v, e′v, d

′v) given by

ev = e′v 1v∗ ⊗ θv ε(v, v∗),


dv := ε(v, v∗) θv ⊗ 1v∗ d′v.

As is well known, we may then define a scalar valued right trace

trv : T ∈ (v, v)→ ev 1v∗ ⊗ T d′v,

coinciding with the left trace T ∈ (v, v) → e′v T ⊗ 1v∗ dv by naturality of thebraiding and the above relations.

The twist can be computed from the braided symmetry and trace,

(θv)−1 = trv ⊗ 1v(σ−1(v, v)).

The case of braided tensor C∗–categoriesBraided tensor C∗–categories arose from low dimensional AQFT in the late

80’s [34], and most of the notions of the symmetric case could be adapted. Thesenotions have well known analogues in the algebraic approach where they are knownwith different names, including, for example the left inverse (known as the cate-gorical trace in the algebraic setting), statistics dimension or intrinsic dimension(quantum dimension), statistics parameter or invariant κ (the twist).

There is an analogue of the twist, that becomes here a notion derived frombraided symmetry and conjugation.

Specifically, it is shown in [68] that, if ε is an invertible braided symmetry and(R,R) is a standard solution, there is a unique class function v → κ(v) ∈ (v, v)such that

ε(v, v) R = 1v ⊗ κ(v) Rand it does not depend on the choice of the standard solution. If the braidedsymmetry is unitary, κ is unitary. This may be shown starting from irreducibleobjects. κ may also be computed as

(κ(v))−1 = trv ⊗ 1v(ε(v, v)).

This equation is related to the relation between statistics parameter, statisticsphase and statistics dimension in AQFT. Moreover, κ satisfies the same compati-bility properties of a twist with respect to the tensor structure.

An approach to the twist as a derived notion is also possible in the algebraiccase, in the framework of the spherical categories of [6]. Braided spherical tensorcategories are ribbon categories.

The following easy proposition compares the invariant κ of a braided tensorC∗–category M with the twist of a braided spherical category containing M as afull subcategory.

Proposition ([98]) Let N be a spherical braided tensor category and M a tensorC∗–category with conjugates embedded in N as a full tensor subcategory. Thenfor any irreducible object v of M,

θv = ±κr(v)−1

with respect to any spherical structure on N. The plus sign corresponds to spher-ical structures with positive traces.

3 ALGEBRAIC QUANTUM FIELD THEORY 7

3 Algebraic Quantum Field Theory

Historically, tc appeared in local quantum physics in the 70’s. I shall be extremelybrief here, the main reference is Haag’s book [41]. See also the lectures of Roberts[102].

In QM observables are described by operators in Hilbert space, and corre-spondingly fields φ ought to correspond to operator valued distributions on testfunctions. For a function f with support in a spacetime region O, φ(f) is a possi-bly unbounded operator on a Hilbert space. The mathematical axiomatization ofa quantum field in this sense is known as the Wightman axioms.

Distributions and unbounded operators cause mathematical difficulties. Forexample for an Hermitian field, φ(f) ⊂ φ(f)∗ and the algebra generated by thefields is not a ∗–algebra. The domain of an unbounded operator on a Hilbert spacedepends on the operator, and this causes difficulties already for simple algebraicoperations.

The starting point of the algebraic approach, started by Haag and Kastler, isto replace unbounded operators by bounded ones. The mathematical propertiesthat allows this are the spectral theorem, which tell us that we may reconstruct aselfadjoint unbounded operator from its spectral projections, a theorem which saysthat the double commutant of a selfadjoint operator is the von Neumann algebraof bounded Borel functions on that operator, and von Neumann bicommutanttheorem M = M ′′ for a von Neumann algebra M . These bounded operators mayinformally be regarded as bounded functions on the unbounded operators φ(f).

They generate a von Neumann algebra F(O), the field algebra, which is definedas a sort of ‘double commutant’ of the set of φ(f) where f varies in the set of testfunction with support in O. We thus have a net,

O→ F(O)

known as the net of field local algebras.The first step towards AQFT is that is not the fields that are essential, but the

fact that we have a net O→ F(O). (Roberts lectures, Martina Franca, 2000)Not all quantum fields are observables, hence the net of local observables is a

subnet of the net of field algebras:

A(O) ⊂ F(O).

A(O) may often be characterized as the set of quantum fields which are invariantunder a compact gauge group G.

Observables are operators on a Hilbert space. Mathematically it is convenientto consider the von Neumann algebra generated by the observables. There aremany inequivalent representations of the net, and therefore one is led to considerthis as a net of abstract operator algebras, with no reference to a specified Hilbertspace.

It is useful to form the ∪OA(O), which is a unital ∗–algebra, and complete it inits unique C∗–norm. We get what is called the C∗–algebra A of local observables,which has the same states of the net. A may be understood, ideally, as an algebraof observables, even though its elements are not necessarily localized.


The net of local observables characterizes the physical theory and it satisfies alist of physically motivated properties which I shall briefly recall here, known asthe Haag and Kastler axioms, [41].

The leading physical principle in the algebraic approach is that of locality, firstformulated by Haag (1959), meaning that measurements are localized in spacetime.Let A(O) be understood as the algebra generated by observables that can bemeasured in a bounded region of Minkowski spacetime O. Formally, we have anassociation

O→ A(O).

It is sufficient to consider double cones

(v+ − V+) ∩ (v− + V+),

where V+ = x · x > 0, x0 > 0 is the forward light cone at the origin, v+ apoint that can be influenced by v− (v+ positive timelike w.r.t. v−).

Minkowski space has inner product

x · y = x0y0 −3∑1

xiyi,

and the corresponding invariance group is the Poincare group

P↑+ = L = (Λ, a) : x→ Λx+ a, Λx · Λy = x · y a ∈ R4.

P↑+ has the translation group T = (I, a) and Lorentz group L

↑+ = (Λ, 0)

as subgroups. The meaning of the symbols ↑ and + is that one restricts to theconnected component of the origin of the Lorentz group imposing the conditions

det(Λ) > 0, e0 · Λe0 > 0.

The covering group P↑+ of the restricted Poincare group may be realized as

P↑+ = (A, a), A ∈ SL(2,C), a ∈ T.

The net A is assumed to satisfy the following axioms, first formulated in [40], seealso [22, 23]. 1) Localization of measurements (Isotony)

It is formalized byA(O1) ⊂ A(O2), if O1 ⊂ O2.

2) Principle of Locality

In QM not all observables can be measured simultaneously, and this correspondsmathematically to the fact that not all operators commute. On the other handby Einstein’s theory of relativity no signal can propagate faster than the speedof light. As a result, observables localized in regions that are spacelike separatedshould commute:

A1A2 = A2A1, A1 ∈ A(O1), A2 ∈ A(O2), O1 ⊂ O2′,


whereO′ = x : (x− y) · (x− y) < 0, y ∈ O

is the spacelike complement of O.

Poincare covariance

There should be a unitary of represented by a group of automorphisms of thenet,

αL(A(O)) = A(LO).

Vacuum representation The abstract net A should possess an irreducible repre-sentation π0, the vacuum representation, such that in this representation αL isimplementable by a unitary representation U of the universal cover of P

↑+:

U(L)π0(A)U(L)∗ = π0(αL(A)),

where LO is the image of O under the image of L via the covering map P↑+ → P

↑+.

Furthermore, there is a unique, up to a phase, normalized vector Ω such thatU(L)Ω = Ω.

A basic problem posed in [23] was the following: if we are only given thevacuum representation, can we construct all other sectors (physically relevant rep-resentations) and define a field net F and a compact group G in such a way thatA = FG?

Spectrum condition The representation U satisfies the following condition: Thegenerator of the translation group T has spectrum in V+.

Duality In the vacuum representation every selfadjoint element of B(Hπ0) can beconsidered in an idealized way an observable (π0 is irreducible). Duality assertsthat if such an element commutes with all the observables that can be measuredin regions spacelike to a double cone O then it can be measured in O,

π0(A(O)) = π0(A(O′))′.

As Haag and Kastler noted already in their 1964 paper, physically relevant statesare not necessarily normal in the vacuum state representation, hence we needto consider inequivalent Hilbert space representations of the net. The relevantnotion is that of ‘physical equivalence’ which is weaker than unitary equivalenceand mathematically corresponds to Fell’s notion of weak equivalence, meaning thatthe two representations have the same kernel.

Unitary equivalence classes of irreducible representations of the net correspond-ing to the states of relevance are called superselection sectors.

We may thus recognize that already in this paper, C∗–categories emerge fromAQFT. However, for their tensor structure revealing their non-Tannakian charac-ter, we need perhaps wait until the DHR papers of 1971–1974.

In elementary particle physics, irreducible representations of the net are inter-preted as charges. As such, it is expected that they can be added and composed,that we may express a product of sectors ξ, ξ′ as

ξξ′ = ξ1 + · · ·+ ξn,


that to every sector ξ there is a conjugate sector ξ characterized by

ξξ = ι+ . . .

and that every sector should have a an associated statistics. These fusion rules wererecognized analogous to the fusion rules betwteen representations of a compactgroup and an important question was whether that was the case. A positiveanswer, given in 1989 by Doplicher and Roberts, led to the reconstruction ofthe field net from the observable net. However, fusion rules are not enough todetermine the group uniquely. Already for finite groups, we may have same fusionrules for different groups. On the other hand, the superselection structure givesmore. Not only we have the fusion rules, but also the intertwiners. This wasreassuring, as in the theory of compact groups one has the Tannaka–Krein dualitytheorem, asserting that we may reconstruct uniquely the group from the categoryof its f.d. unitary representations. However, that theorem was not applicable tocategories from AQFT as in the classical TK reconstruction, categories come witha given embedding

T → Hilb

to the category of Hilbert spaces (fiber functor), while the categories from AQFThad not one. The main task in fact was that of constructing such an embedding,leading Doplicher and Roberts to a new duality theory for compact groups. Toexplain in some more detail the structure of these categories, let us go back brieflyto the selection criterion. A will stick to the simplest, called the DHR selectioncriterion.

DHR Sectors For each double cone O,

π A(O′)' π0 A(O′) .

In other words, for each O there is a unitary U such that for each O1 ⊂ O′,

Uπ(A) = π0(A)U, A ∈ A(O1).

This unitary defines a representation

ρ(A) := Uπ(A)U−1,

equivalent to π, now acting on the same Hilbert space as π0 and trivially onA(O′). Because of Haag duality, ρ is actually an endomorphism of A, referred toas a localized endomorphism. Unlike representations, endomorphisms of an algebracan be composed, and irreducible components of compositions of localized endo-morphisms are still localized endomorphisms. Furthermore, localized morphismscommute if their supports are spacelike.

The associated tensor C∗–category Since localized endomorphisms are, afterall,representations, we may compute intertwiners:

(ρ, ρ′) := T ∈ B(Hπ0) : Tρ(A) = ρ′(A)T,A ∈ A.


Localization of the endomorphisms reflects in localization of the intertwiners:

(ρ, ρ′) ⊂ A(O), ρ, ρ′ localized in O.

Therefore the category with objects localized endomorphisms and arrows inter-twiners is a full subcategory of the category of endomorphisms of A. On the otherhand the C∗–structure of A passes to the arrows, making this category into aC∗–category.

Composition of endomorphisms gives rise to a product between intertwiners

T ⊗ T ′ := Tρ(T1) ∈ (ρρ1, ρ′ρ′1), T ∈ (ρ, ρ′), T1 ∈ (ρ1, ρ

′1),

behaving just like the tensor product of linear maps between Hilbert spaces.

DHR analysis Given ρi, i = 1, 2, pick unitaries Ui ∈ (ρi, ρ′i), ρ′i with spacelike

supports (so ρ′1 and ρ′2 commute). Set

ε(ρ1, ρ2) := U∗2 ⊗ U∗1 U1 ⊗ U2 ∈ (ρ1ρ2, ρ2ρ1).

This does not depend on the choice of ρ′i or Ui and defines a unitary permutationsymmetry in the category. A localized endomorphism is not an automorphismin general. However, it admits a left inverse, meant as a positive linear mapΦρ : A→ A such that

Φ(ρ(A)B) = AΦ(B).

If ρ is irreducible,λρ := Φρ(ε(ρ, ρ))

is a scalar, called the statistics parameter, it is a class function:

λρ = λρ′ , if ρ ' ρ′.

A major result is that for each irreducible ρ, the quasiequivalence class of theassociated representation

Pn → (ρn, ρn)

is determined by the statistics parameter, which can only take a restricted set ofvalues,

λρ ∈ 0,±1d, d = 1, 2, . . . .

It should be noted that the proof of this restriction result makes crucial use of pos-itivity, i.e. of the C∗–structure. The value λ = 1

d is realized by the representationsof Pn that permute the factors of H⊗n, where H is a Hilbert space of dimensiond,

ψ1 ⊗ · · · ⊗ ψnp−→ ψp−1(1) ⊗ · · · ⊗ ψp−1(n),

while λ = −1/d is realized by the tensor product of this representation withthe sign representation. The quasiequivalence classes of these representations areequivalently characterized by identifying the kernels, they correspond respectivelyto the ideal generated by the antisymmetric and symmetric projections E±d ∈CPd ⊂ CPn. The two invariants associated with a localized endomorphism,

4 THE DR RECONSTRUCTION THEOREM 12

d(ρ) := |λ−1ρ | =∈ N ∪ ∞, κ(ρ) = ±1

are called the statistics dimension and the statistics phase of ρ. Most importantly,it may be shown that finiteness of the statistical dimension is equivalent to theexistence of a conjugate endomorphism ρ, namely of a localized endomorphismcharacterized by

ρρ = ι⊕ . . . .Now, it is a classical and well known fact that the permutation representation ofPn on H⊗n and the nth tensor power of the basic representation of SU(d) enjoy aremarkable duality, in that they are one the commutant of the other. In this model,the representation theory of SU(d) appears! This was a clear indication that thecase with phase statistics κ(ρ) = 1, the categories from AQFT were related torepresentation theory of compact groups.

BF analysis Starting from a different selection criterion, Buchholz and Freden-hagen derived a more general selection criterion, now for non compactly localizedendomorphisms [16].

4 The DR reconstruction theorem

The problem was solved in 1989 by Doplicher and Roberts, actually in the com-pletely general case. DR duality theorem asserts that every tensor C∗–categorywith conjugates, permutation symmetry and irreducible tensor unit is the repre-sentation category of a unique compact group.

To this aim, the original proof of Doplicher and Roberts emphasizes an alge-braic approach, in the sense that the DR algebra Oρ associated to an object ρ ofthe category and a corresponding endomorphism

ρ : Oρ → Oρ

play a role. This approach, natural from the QFT origin, allows to reduce theproblem to categories of endomorphisms of an algebra. Although purely categoricalproofs are also available, the original proof has its own interest for the great insightto subsequent research. I will briefly sketch the main ideas.

Oρ is a Z–graded C∗–algebra. The homogeneous part together with ρ may bedescribed as follows.

C⊗1ρ // (ρ, ρ)

⊗1ρ // (ρ2, ρ2)⊗1ρ // ... // O(0)

ρ

C⊗1ρ //

1ρ⊗

OO

(ρ, ρ)

1ρ⊗

OO

⊗1ρ // ... // O(0)ρ

ρ

OO

If H is an object of the category of f.d. Hilbert spaces, OH is just the Cuntz algebraOd, d = dim(H), and H the canonical shift endomorphism. Most importantly,

ρ ∈ T → ρ ∈ End(Oρ)

4 THE DR RECONSTRUCTION THEOREM 13

is a full tensor ∗–functor in cases of interest, hence we may replace the category T

with the algebraic datum (Oρ, ρ).Now, arguments used to show integrality of the statistics dimension in AQFT

are purely categorical, in that every object in a symmetric tensor C∗–category hasa dimension d which still turns out to be an integer. Similarly, the (statistics)phase is still ±1. We should thus start looking at the case where all the statisticsphases are 1.

Every object has a determinant, and the category has sufficient objects ofdeterminant 1, namely objects such that the antisymmetric projection of orderthe dimension

Ed ∈ (ρd, ρd)

is nonzero and of smallest order and it should correspond to the trivial object.Sufficiency means that every full tensor subcategory generated by finitely many

objects is in fact a subcategory generated by an object of determinant 1.The advantage of starting with objects of determinant 1 is that the tensor cat-

egory it generates has conjugates, as the conjugate ρ of ρ is realized as a subobjectof a tensor power of ρ.

Every special object ρ of dimension d and statistics phase 1 gives rise to acanonical embedding of symmetric tensor C∗–categories

Rep(SU(d))→ Tρ.

The important step is the identification of a closed subgroup G ⊂ SU(d),d = d(ρ) together with the construction of a canonical equivalence of tensor C∗–categories

Tρ → Rep(G)

such thatRep(SU(d))→ Tρ ' Rep(G)

u −→ u G .

With an integer d we may consider the Cuntz algebra, regarded as an algebraicversion of all the linear maps between tensor powers of a d–dimensional Hilbertspace. Every d–dimensional unitary representation of a compact group G becomesan automorphic action on Od with the fixed points OGd playing the role of theintertwiners between tensor powers of the representation.

The next step is the construction of a cross product C∗–algebra

Oρ ⊗RepSU(d) Od, d = d(ρ),

which, like any crossed product, is an algebra just big enough to make ρ inner,here in the sense ρ is implemented by a Hilbert space of isometries.

Most important for future developments is the fact that this construction en-codes also aspects of tensor product of bimodules, over the arrows of the represen-tation theory of SU(d), as the notation suggests.

5 SUBFACTORS 14

To get the perception of this, we just recall that since ρ has determinant 1, thedeterminant S ∈ (ι, ρd) corresponds to the determinant of SU(d), and hence inthe crossed product construction relations of the type

AS ⊗ T = A⊗ ST

just express associativity of the product in the crossed product algebra. SU(d)acts on Od and hence on the cross product algebra as well,

α = trivial⊗ canonical.

Its center Z(Oρ ⊗RepSU(d) Od) turns out to be SU(d)–ergodic under the action,hence its spectrum Ω may be identified with a classical compact homogeneousspace,

Ω = G\SU(d)

with G an isotropy subgroup.The important step is the identification of a closed subgroup G ⊂ SU(d) such

thatOρ ' OGd , d = d(ρ)

leading to the desired equivalence

T → Rep(G).

5 Subfactors

Textbooks: A good starting point is the book by Jones and Sunder [50]. A classicaland excellent reference is [39]. Subsequent excellent monograph is [33].

The Jones index Let N be a II1 factor with separable predual (equivalently, with afaithful representation on a separable Hilbert space). By the theory of dimensionof Murray and von Neumann, N has a unique normalized tracial state. Therepresentation theory of N is described by the theory of dimension of Murray andvon Neumann. Consider the Hilbert space L2(N, tr)⊗`2 regarded as a left moduleover N . This carries a right module structure over the II∞ factor N ⊗B(`2). Leftand right actions are one the commutant of the other,

πl(N)′ = πr(N ⊗B(`2)).

We may form more left N–modules with selfadjoint projections of N ⊗ B(`2).Indeed, for each such projection p,

H := (L2(N, tr)⊗ `2)p

is still a left N–module. The commutant of the left action is now the reducedfactor of the original II∞ factor by p,

πl(N)′ = πr(p(N ⊗B(`2))p).

5 SUBFACTORS 15

Hence πl(N)′ is a II1 factor if and only if Tr(p) < ∞, where Tr is the uniquesemifinite positive trace of N ⊗B(`2) normalized so that the minimal projectionsof B(`2) have trace 1. Now, any left N–module H on a separable Hilbert space isisomorphic to one of the form (L2(N, tr)⊗ `2)p, with p ∈ N ⊗B(`2) a projectionuniquely determined up to Murray and von Neumann equivalence. Hence such amodule is determined, up to unitary equivalence, by the trace of the associatedprojection, and this trace is the coupling constant of Murray and von Neumann,

dimN (H) := Tr(p).

Thus dimN (H) <∞ iff N ′ is a II1 factor. We thus have a bijective correspondencebewteen equivalence classes of normal representations of N on separable Hilbertspaces and [0,∞].

Jones replaced the representation theory of a single factor N by the repre-sentation theory of a pair of factors N , M . This amounts to replace modulesby bimodules. Indeed, given an inclusion of II1 factors with the same identity,N ⊂ M , L2(M) is both a left module over N and a right module over M inthe natural way. Jones used the coupling constant to measure the size of a thesubfactor N in M ,

[M : N ] := dimN (L2(M)).

This is the Jones index of N ⊂M . As before, [M : N ] <∞ iff the commutant ofN in the representation L2(M) is a II1 factor.

Jones basic construction Represent N and M on L2(M, tr) and regard the uniquetrace invariant conditional expectation E : M → N as an orthogonal projectione1 ∈ B(L2(M)). The basic construction is

M1 :=< M, e1 > .

If [M : N ] <∞, M1 is a II1–factor and [M1 : M ] = [M : N ]. We may thus iterateand setting Mi+1 :=< Mi, ei+1 >, and obtain the Jones tower

N ⊂M ⊂M1 ⊂M2 ⊂ . . .

satisfyingdimN ′ ∩Mr−1 ≤ [M : N ]r.

The projections (ei) satisfy the relations

eiej = ejei, |i− j| ≥ 2,

eiej±1ei = τ−1ei,

τ = [M : N ], and the normalized trace satisfies the Markov property

tr(wei) = τ−1tr(w), w ∈< 1, e1, . . . ei−1 > .

Jones fundamental result asserts the above relations can be realized in a vonNeumann algebra with a finite trace if and only if

τ ∈ 4 cos2 π/n, n = 3, 4, . . . ∪ [4,+∞).

5 SUBFACTORS 16

Therefore [M : N ] is bound to take one of the above values. Wenzl showed thatthe same holds without assuming the trace [121]. The index values in the discretepart correspond to irreducible inclusions: N ′ ∩M = C. Conversely, a system ofselfadjoint projections in a von Neumann algebra with a trace satisfying the aboverelations for a correct value of τ and the Markov property gives rise to a subfactorof the hyperfinite II1 factor R

Rτ :=< 1, e2, · · · >′′⊂< 1, e1, e2, · · · >′′= R

with [R : Rτ ] = τ .

Example Given an inclusion of the fixed point algebra N = MG ⊂ M under anouter action of a finite group G, [M : N ] = |G| and M1 = M oG.

Jones subfactor A subfactor of this form with index 4 cos2 π/n is constructed start-ing from an inclusion of finite dimensional C∗–algebras with the correct norm ofits inclusion matrix, apply the construction of the tower in the finite dimensionalcase and construct the subfactor Pτ ⊂ P with the resulting Jones projections.This is the Jones subfactor.

Actually, Jones construction was much deeper than how I sketched it above. Itinvolves ideas that revealed important in the development of the theory and clas-sification of subfactors, such as the principal graph of a subfactor and its relationwith Kronecker’s classification of Coxeter graphs of norm ≤ 2. This is materialthat can be found in the book [39].

Commuting squares An inclusion of algebras with conditional expectations

BEB←− // D

A //

OO

C

EC

OO

such that EB(C) ⊂ A (or EC(B) ⊂ A). Main examples: if A ⊂M1 ⊂M2,

M1// M2

A′ ∩M1//

OO

A′ ∩M2

OO

Appearance of the braid group In 1983 Pimsner and Popa found a representationof the algebra

An =< 1, e1, . . . , en−1 >

presented by the above relations for τ > 4 on the Hilbert spaceH⊗n, whereH = C2

[97], and in the mid 80’s Evans pointed out that the same representation of thealgebraic relations of the ei’s had been discovered in the 70’s by Temperley andLieb, in statistical mechanics. de la Harpe pointed out the similarity between theTemperley–Lieb relations and the presentation relations of the braid group. The

5 SUBFACTORS 17

relation with the braid group is given interpreting the ei’s as spectral projectionsof the σi’. More precisely, [?], writing

τ = q + 2 + q−1,

σi := qei − (1− ei)

thene2i = ei iff σ2

i = (q − 1)σi + q,

τ > 4↔ q > 0,

τ = 4 cos2 π/n↔ q = e±2πi/n.

The quotient of CBn by the above relations satisfied by the generators σ1, . . . , σn−1

of the braid group is the Hecke algebra Hn(q) of type A. We thus have a surjectivehomomorphism

CBn → Hn(q).

In turn, the quotient of Hn(q) by the ideal generated by the element

E3 := 1 + σ1 + σ2 + σ1σ2 + σ2σ1 + σ2σ1σ2

is precisely the Temperley–Lieb algebra An [39]. We thus have another surjectivehomomorphism

Hn(q)→ An.

Now for q = 1, this quotient describes the quasiequivalence class of the representa-tion of Pn on H⊗n, H = C2 and recall that this is a dual way of looking at SU(2).In this sense, the Temperley–Lieb algebras are closely related to SU(2), and theJones subfactor with index < 4 to the roots of unity.

Wenzl’s work on Hecke algebras The algebras Hn(q) are semisimple for genericvalues of q while they are not if q is a root of unity. Since the Jones idempotents(ei) are selfadjoint, the associated representations of the braid group are unitaryprecisely if q is a root of unity and selfadjoint if q is real. In particular, they havesemisimple image.

Wenzl constructed an irreducible subfactor of finite index from Hecke algebrasin a similar relationship with SU(d) for each root of unity q. Irreducible repre-sentations of the Hecke algebras in the generic case were known since the 70’s,they had be constructed by Hoefsmit in his unpublished thesis and rediscoveredby Ocneanu in the framework of subfactors as reported in [122]. Most impor-tantly, Wenzl constructed semisimple representations in the case of root of unity.He showed that there are nontrivial C∗–representations of Hn(q) for n arbitrarilylarge if and only if q takes one of the above values and he classified the irreducibleones. For the real values of q, Hoefsmit representations are easily seen to be C∗–representations with σi selfadjoint. Most importantly, Wenzl classified irreducibleC∗–representations of Hn(q) for q a root of unity. His method is based on thegeneralization of the notion of Markov trace to Hecke algebras, a trace on H∞(q)such that

tr(xσn) = ηtr(x), x ∈ Hn−1(q),

5 SUBFACTORS 18

and he showed that tr is nonnegative for the natural ∗–involution if and only if ηis restricted to the values

[0, q − 1] ∪ qd(q − 1)qd − 1

, d ∈ Z, for q ≥ 1,

qd(q − 1)qd − 1

, |d| ≤ n− 1, for q = e2πin .

Next I am going to recall the notion of the principal graph. I will not followthe original notion, but rather an equivalent notion in the framework of tensorcategories. To do this, I need to pause for a moment and talk about two papersof Kosaki and Longo.

The papers of Kosaki and Longo Kosaki extended Jones index theory to type IIIfactors [66]. This work was important, as the the algebras that arise from AQFTare typically type III1 factors. Longo [67] related Jones index theory to AQFT,showing that if we have an endomorphism ρ of an infinite factorM , and we considerthe inclusion ρ(M) ⊂ M then [M : ρ(M)] < ∞ if and only if ρ has a conjugateendomorphism ρ. Longo showed the equality

d(ρ)2 = [M,ρ(M)].

The composite γ := ρρ may be described only in terms of modular operatorsderived from the Tomita–Takesaki theory for M and the subfactor. It is calledLongo’s canonical endomorphism. As mentined above, Longo’s work was impor-tant also because it established a clear connection between index theory an tensorcategories.

Principal graph, Ocneanu’s paragroup and Popa’s work [96] We describe the prin-cipal graph and the standard invariant of a subfactor following Ocneanu’s descrip-tion in terms bimodules, the correspondences of of Connes [19], and called it theparagroup. The great advantage of bimodules as compared to modules is that theyadmit tensor products. To see this, let us start with the bimodule

ρ := NL2(M)M .

This is a Hilbert space with a left action of N and a right action of M . Thanksto the ∗–operation, together with ρ we also have a contragradient bimodule,

ρ := ML2(M)N .

It acts on the conjugate Hilbert space and the bimodule structure is given by

mξ n := n∗ ξ m∗.

There is a notion of tensor products of bimodules of this form, every time rightand left algebras match,

ρ⊗M ρ⊗N ρ⊗M . . . ,

5 SUBFACTORS 19

ρ⊗N ρ⊗M ρ⊗N . . .

These are the Connes correspondences and the tensor product is also referred toas the fusion tensor product. Formally, this has the structure of a tensor 2–C∗–category.

Pimsner and Popa observed that finiteness of the Jones index implies thatL2(M) is finite projective as an N–module (and also as an M–module, but this isa trivial fact). It turns out that as bimodules,

NL2(Mr)N ' (ρ⊗ ρ)r+1,

NL2(Mr)M ' (ρ⊗ ρ)r ⊗ ρ,

leading to an interpretation of the tower of relative commutants as inclusions ofarrows of the category. Watatani remarked that, thanks to the finiteness property,this category is equivalent to the one obtained dropping the scalar valued innerproduct while keeping N and M–valued inner products given by the trace invariantconditional expectations. Namely, we just need the notion of Hilbert bimodulesover noncommutative C∗–algebras N , M . The theory of Hilbert bimodules thusshows that intertwiners can be computed in a completely algebraic way, they arejust the bimodule intertwiners.

Now, finiteness of the Jones index is equivalent to the fact that ρ is a conjugateof ρ in the tensor 2–C∗–category of Hilbert bimodules over N and M , and this isanother way of explaining finite dimensionality of the higher relative commutants.

The principal graph is the graph with vertices the irreducible submodules oftensor products ρ ⊗ ρ ⊗ ρ ⊗ . . . . Hence, if the irreducible submodule arises froma tensor product of even order, it is an N–N bimodule, otherwise it is an N–M bimodule. An even vertex α connected to odd vertex β by r lines if r =dim(α, β ⊗ ρ).

The dual principal graph is similarly defined replacing the roles of ρ and ρ.A subfactor is called finite depth if both these graphs are finite. Subfactors

with index < 4 are automatically finite depth.

The paragroup of a subfactor is a richer invarian and it is also known withother names, Popa’s standard invariant, Jones’ planar algebra. The following isthe categorical description.

C⊗1ρ // (ρ, ρ)

⊗1ρ // (ρρ, ρρ)⊗1ρ //

C⊗1ρ //

1ρ⊗

OO

(ρ, ρ)

1ρ⊗

OO

⊗1ρ //

From this viewpoint, the principal graph corresponds to the fusion multiplicities,while the standard invariant keeps track of the intertwiners and traces.

Popa introduced a notion of strong amenability for a subfactor and showedthat strongly amenable subfactors are completely classified by their paragroup. Inparticular, subfactors of N ⊂ R with [R : N ] ≤ 4 satisfy the requirements. Popa’s

5 SUBFACTORS 20

result establishes a relation between classification of strongly amenable subfactorsand classification of tensor C∗–categories with conjugates.

Example Consider the example N = MG ⊂M , with G finite acting outerly on M .This inclusion is known to have depth 2. The full tensor subcategory generatedby ρρ corresponds to the category generated by the regular representation λ of G.The depth 2 condition can be read off from the fundamental property

λ⊗ λ = λ⊕ · · · ⊕ λ, (|G| times).

The full subcategory generated by ρρ gives the representation category of its dualG. Hence the standard invariant encodes information of both G and G. Moregenerally, a theorem of Ocneanu asserts that depth 2 inclusions are precisely fixedpoint inclusions MG ⊂ M under the action of a Hopf algebra, with the tensorcategory generated by the double object ρρ similarly described in terms of therepresentation theory of G. A proof has been given by Szymanski [109] For ageneral inclusions N ⊂ M , NMN is always a selfconjugate real object. A naturalquestion is whether it may still be interpreted as generating the representationcategory of a group-like object.

Example Subfactor theory covers representations theory of compact groups as fol-lows. A unitary representation π of a compact group G of dimension d definescanonically an automorphic action on R = ⊗∞1 Md. The fixed point algebra andthe right shift endomorphism σ define an inclusion σ(RG) ⊂ RG with index d2.The associated category of bimodules identifies naturally with a full subcategoryof the category of representations of G under the identifications ρ → π, ρ → π.In particular, it is irreducible if π is irreducible. Such subfactors are stronglyamenable [93]. For G = SU(2) they have been studied by [57], the index is 4,the graph A∞, the remaining subfactors of index 4 appear in [39] they correspondto closed subgroups of SU(2), the corresponding graph is the McKay correspon-dence. For general groups these have been considered by Wassermann. Note thatthis model is related to DR model on the Cuntz algebra. The correspondencebetween type II1, realized as ⊗∞1 Md and type III1/d, given by O′′d irreducibilityin the type III case is more frequent thanks to richness of isometries. For exam-ple, the Jones subfactor of index > 4 is always reducible, with graph A∞,∞ whilethe corresponding extension to O′′d is irreducible with graph A∞. Note that thesesubfactors arise from the compact quantum group SµU(2) of Woronowicz.

Classification of subfactors with small indicesThe principal graphs of possible subfactors with index ≤ 4 has been obtained

in [39]. They are Coxeter graphs of type ADE. This is related to Kroneckerclassification of Coxeter graphs of norm ≤ 2. Realizations: An is unique [57] E6

[8], E(1)6 , E(1)

7 , E(1)8 , D(1)

n [39].Complete classification of paragroups of the hyperfinite II1 factor with index

< 4 was announced by Ocneanu in 1987. A proof is due to [60]. In particular,D2n+1 and E7 are not realized, unique with graph D2n, 2 paragroups with E6 andE8. Kawahigashi proved that E(1)

6,7,8 have unique paragroup, Popa that A(1)2n−1,

A(1)∞ , have respectively n, 1 and D

(1)n has n− 2 paragroups [?, ?].

6 DHR CATEGORY IN LOW DIMENSIONS 21

While the subfactors with index 4 correspond to the closed subgroups of SU(2),those with index < 4 are understood in the framework of quantum groups. Theycorrespond in a similar way quantum subgroups subgroups of SU(2) at roots ofunity.

Beyond index 4, for every index value there is an irreducible non-hyperfinitesubfactor with principal graph A∞ [95]. It is not known whether the same holdsin the hyperfinite case. Haagerup listed the the principal graphs of possible fi-nite depth subfactors with index < 3 +

√3 [42]. Many were excluded in [10].

Two of them were uniquely realized in [?]. Notably, they do not correspond therepresentation theory of quantum groups.

Etingof, Nikshych and Ostrik proved that the Jones index of a subfactor withfinite depth is a cyclotomic integer [31]. As a consequence Asaeda and Yasuda [4]excluded all the other graphs in Haagerup’s except one. This last one has beenrealized by by a unique planar algebra by Bigelow, Morrison, Peters and Snyder[15].

6 DHR category in low dimensions

It seems that in Physics, one dimensional representations of the braid group hadappeared since the 70’s, see, e.g. [41]. However, the study of DHR sectors for lowdimensional theories, was initiated by Fredenhagen, Rehren and Schroer.

FRS analysis The basic difference in d = 2 spacetime dimensions is that the causalcomplement of a double cone is disconnected. This reflects into the fact that thepermutation group is replaced by the braid group. We get a tensor C∗–categorywith conjugates and unitary braided symmetry. Now,

λ−1ρ = κ(ρ)d(ρ)

the statistics phase κ(ρ) may be a complex number and the statistics dimension,is not an integer in general. An analysis similar to the classification of statisticsin 4 dimensions can be carried over for particular representations of the braidgroup. This has been first worked out by Wenzl in the framework of the theory ofsubfactors [122] and by FRS for low dim QFT [34].

In the case of BF sectors, braid group symmetry arises already in dimension 3[16].

Nets of subfactorsLongo and Rehren started the study of nets of subfactors A(O) ⊂ B(O), re-

garded as a generalization of the case where B is the field net. No gauge group isassumed, and the theory is developed mainly to treat low dimensional cases. Theytheory of subfactors is applied and the main assumption is a finite index condition.The notion of restriction and induction of localized endomorphisms developing aformer idea of Roberts is considered. However, here it makes use of the braiding.While the restriction of a localized endomorphism is localized, the same does nothold for induction. Most importantly, extensions of A of finite index are classified


by the so called Q–systems in A, triples (ρ, S, T ) of a localized endomorphism ρof A and isometries satisfying suitable conditions.

Induction in this sense has been further studied later by Xu and Bockenhauer,Evans and Kawahigashi. This is a procedure which makes use of the braiding ofthe smaller net and allow to construct all objects of the tensor category associatedto B from objects of that associated to A.

However, despite some analogies, α–induction does not seem to be related tothe induction process between representation of a group and a subgroup. In factnot even a quantum group is invoked in the construction. It makes use of thebraiding.

Starting from a 2–dimensional Minkowski space, one is naturally led to studychiral theories on a light ray. This has been initiated by [?]. Correspondingly, onehas subnets in 1–dimensional theories. A set of axioms similar to HK axioms isassumed, where Poincere’ group is replaced either by the Mobius group or by thediffeomorphism group of the circle. The light ray is then compactified, and onepasses to a net defined on the circle. This structure is called local conformal net.It is also shown that the Jones index is constant over the net algebras.

Xu worked out examples with finitely many irreducibles [126, 127]. Wasser-mann constructed examples of local conformal field theories from loop group rep-resentations [?, ?]. Xu proved it is of finite index (and hence completely rationalby later result of KLM [129]. Xu constructed examples associated with inclusionsof the form SU(N)n+m ⊂ SU(N)m × SU(N)m [128].

Kawahigashi, Longo and Muger studied conformal inclusions associated to dou-ble intervals

A(I1) ∨A(I3) ⊂ (A(I2) ∨A(I4))′

where I1–I4 the circle counterclockwise, and showed that the associated tensorcategory has only finitely many irreducible objects and is modular [61]. The mainassumption is finiteness of the Jones index. The strong additivity assumption wasshown to be redundant by Longo and Xu [69]. Non-degeneracy of the braiding hadbeen introduced by Rehren in this framework [?]. This result shows that for animportant class of DHR categories, the associated braiding has properties oppositeto a permutation symmetry.

Kawahigashi and Longo classified nets on the circle covariant under the diffeo-morphism group and with central charge < 1 [62]. For a given such net B, thediffeomorphism group has a role similar to the of the permutation symmetry forsymmetric tensor categories, in the it gives rise to a minimal subnet, the Virasorosubnet A ⊂ B. The problem is then that of classifying extensions of this minimalmodel. It relies on the classification of the modular invariants of the Virasoro netby Cappelli-Itzykson-Zuber [18] and Longo’s Q–systems.

These techniques have been applied to classification of 2–dimensional conformalfield theories [63].

More recent constructions are due to Dong and Xu [21] and Xu [130]. Thelatter makes use of Longo’s Q–systems.

DR reconstruction for non-symmetric tensor categories


We are interested in cases where the category T has no permutation symmetry.It may or may not have a braided symmetry. However, in this more general casevarious unpleasant problems arise, and a result as strong as DR reconstruction isnot possible.

a) There are important situations arising from subfactors where no symmetry,not even braided, is present. If permutation symmetry is dropped, and we donot assume any other kind of symmetry, uniqueness of the predual is lost. Twodifferent finite groups may have isomorphic representation categories even as tensorC∗–categories [47, 30]. As a consequence, uniqueness may be recovered only atthe price of fixing more data.

b) A braided symmetry does not fix the uniqueness problem. Quantum groupsprovide interesting examples of braided tensor categories and hence are suitablepreduals. However, there are non-isomorphic quantum groups with isomorphicrepresentation categories even as braided tensor C∗–categories. These are thequantum groups Ao(F ) of Wang and Van Daele for very many matrices F ’s ascompared to the SµU(2).

c) However, the impact of the examples in b) may be not too dramatic, atleast if we are interested in the categories arising from low dim QFT, as in thatcase braided symmetries are unitary. In generic situations, these categories do notadmit an embedding functor. Unitary braided symmetries and compact quantumgroups belong to two different worlds. However, if we look at their common littleintersection, the integer dimensions may still be uniquely recovered.

d) Although quantum groups provide interesting examples of braided tensorcategories, one needs to widen the category of the possible preduals. More generalstructures like quantum groupoids , provide rigid tensor categories. In the operatoralgebraic approach, a corresponding direction is that of ergodic actions of compactquantum groups.

e) Things do not look better if one tries more generally to classify braided tensorcategories, as this problem contains the problem of classifying finite groups. How-ever, interesting results have been obtained in the algebraic approach by Etingov,Ostrik, Nychshik, Drinfeld and many others in this direction. In a very recent pa-per, the authors introduce a new invariant, the core of the category, which allowsto separate the group-like part of the category.

The insight gained from DR reconstruction it that an inclusion of tensor C∗–categories

S→ T

ought to be understood as a restriction functor associated to an inclusion of cor-responding group-like objects (in the reverse order), GT ⊂ GS. Hence T may beregarded as an imprimitive category in the sense of Mackey, in that it may berecovered from the smaller category S via induction. From this viewpoint, classifi-cation of extensions S→ T ought to correspond to classification of ‘subgroups’ ofGS.

7 QUANTUM GROUPS 24

7 Quantum groups

The theory of quantum groups was initiated in the early 80’s by the Leningradschool to provide solutions to the Yang-Baxter equations.

In the mid 80’s Drinfeld and Jimbo understood quantum groups as a gener-alization of the theory of simple Lie algebras, while Woronowicz appealed to theGelfand-Naimark theorem to describe a group with a noncommutative structure,thus pursuing an analyic approach based on C∗–algebras.

Quantum groups almost immediately revealed to be important in various ar-eas such as e.g. low dimensional QFT, tensor categories, topology of knots, 3–manifolds, representation theory. Moreover, they often play a unifying or clarifyingrole.

The various applications of quantum groups led to an enormous developmentof the theory. In the framework of the Lie theory, we mention quantum affineLie algebras and quantum KZ equations and the relation with CFT, see e.g. [29],while the operator algebraic setting has developed the compact matrix quantumgroups of Woronowicz towards locally compact quantum groups, first pursued byBaaj and Skandalis. This setting also led to an elegant and simple formulation ofcompact quantum groups (Woronowicz).

We shall focus here on the two approaches that we have already mentioned,namely, the quantized universal enveloping algebras of Drinfeld and Jimbo and thecompact quantum groups of Woronowicz. For a comprehensive treatment see thestandard textbooks, [70, 17, 58, 65] for the algebraic approach or [111] for theoperator algebraic approach.

The theory of quantum groups at roots of unity so far is described in the Liealgebra approach, while the operator algebraic approach seems more suitable totreat quantum groups which are not deformation of classical groups.

Although they have only finitely many irreducible inequivalent representations,quantum groups at roots of unity have attracted much attention for various rea-sons, their representation theory is very different from the classical representationtheory, they give rise to the Reshetikin-Turaev invariants of links and 3-manifolds,unitary representations of the braid group, important in physics, subfactors withfinite index, they arise in conformal field theory.

On the other hand, compact quantum groups have a representation theorythat resembles very much the representation theory of compact groups. They pro-vide examples of (non-unitarily) braided tensor C∗–categories with infinitely manyirreducible objects. They have attracted attention in the theory of subfactors, non-commutative geometry and in the theory of noncommutative homogeneous spaces.

Quantum groups rely on the notion of a Hopf algebra. We shall always workover the complex numbers (although this was not Drinfeld’s original approach).In the algebraic setting, a Hopf algebra A is a unital associative algebra and acounital coassociative coalgebra,

∆ : A→ A⊗A, ε : A→ C.

If A is in addition a ∗–algebra with a ∗–preserving ∆, then A is called a Hopf

7 QUANTUM GROUPS 25

∗–algebra. There should also be a coinverse S : A→ A a linear map satisfying

m(ι⊗ S)∆(a) = m(S ⊗ ι)∆(a) = ε.

ε is automatically a ∗–homomorphism, S is antimultiplicative, and, if A is a Hopf∗–algebra, (S ∗)2 = 1

A basic example is the algebra of functions on a finite group G, with comulti-plication

∆ : F (G)→ F (G)⊗ F (G), ∆(f)(g, h) = f(gh)

and counitε : F (G)→ C, ε(f)(g) = f(e), S(f) = f(g−1)

describing the group structure.With any algebra A we may associate the category Rep(A) of finite dimensional

representations. If A is a Hopf algebra, Rep(A) becomes a tensor category withduals as follows. If ρV and and ρV ′ are representations on spaces V and V ′, thetensor product is the representation on V ⊗ V ′,

ρV⊗V ′(a) := (ρV ⊗ ρV ′)(∆(a)),

ι = ε is the tensor unit and a left and right dual of ρV act on the dual vector spaceV ∗ by

ρV ∗(a) := (ρV (S(a)))∗, ρV ∗(a) := (ρV (S−1(a)))∗.

and they are not equivalent in general. A sufficient condition is that S2 is inner,i.e.

S2(a) = uau−1

and this is the case in presence of an R–matrix [27] or a ∗–structure.Another basic example is the group algebra CG, where now

∆(g) = g ⊗ g, ε(g) = δg,e, S(g) = g−1.

These two examples are related by a natural duality isomorphism,

F (G)′ ' CG

that determines the cooperations of one from the operations of the other andviceversa. Hence, commutativity of F (G) reflects into cocommutativity of CG.

More generally, the category of f.d. Hopf algebras is self-dual, in that given afinite dimensional Hopf algebra A, the dual space A′ becomes a Hopf algebra withdual operations, and a Hopf ∗–algebra if so was A.

This self-duality is regarded as a recover of the symmetry described by Pontr-jagyn duality in the category of locally compact abelian groups. Symmetry thatwas broken in the classical theory of noncommutative groups.

A consequence of self-duality is that a representation π of A on a f.d. vectorspace, has a dual notion, that of a corepresentation of A′. This is an invertiblematrix with entries in A′ satisfying

π = (πij) ∈Mn ⊗A′, ∆(πij) =∑k

πik ⊗ πkj .

7 QUANTUM GROUPS 26

Representations of A and corepresentations of A′ are in bijective correspondence.If A is a Hopf ∗–algebra, this correspondence takes ∗–representations into unitarycorepresentations.

This is already an important enough reason for wishing to determine classes ofself-dual infinite dimensional quantum groups. This class has been determined byVan Daele, they are called the algebraic quantum groups.

However, there is also another just as important reason to extend duality be-yond finite dimensional Hopf algebras. Drinfeld discovered that a good dualityallows to construct universal R–matrices, and, as a consequence, braided tensorcategories. I would like to explain the idea in the simplest finite dimensional case.

Universal R–matrices and braided tensor categories A universal R–matrix for af.d. Hopf algebra A is an invertible element R ∈ A⊗A satisfying

∆op(a) = R∆(a)R−1, a ∈ A,

ι⊗∆(R) = R13R12, ∆⊗ ι(R) = R13R23.

The category Rep(A) becomes a braided tensor category setting

ε(u, v) = FR : u⊗ v → v ⊗ u.

Moreover, as it will be clear from the discussion on Tannaka reconstruction, everybraided symmetry for Rep(A) is of this form.

Quantum double constructionLet A be a f.d. Hopf algebras and let (A′)op be the dual vector space with dual

multiplication but opposite dual comultiplication,

∆′(φ)(a1, a2) = f(a2a1), (∆′(φ) = F∆(φ)).

Theorem([?, 28], see, also [17, ?, 59])Consider the vector space

D(A) := A⊗ (A′)op

as a coalgebra with the tensor product coalgebra structure. Regard A and A′ assubspaces of D(A) with the obvious embeddings,

a ∈ A→ a⊗ I ∈ D(A), φ ∈ A′ → I ⊗ φ ∈ D(A).

If ai ∈ A and φj ∈ A′ are dual bases, consider the element

R :=∑i

(ai ⊗ I)⊗ (I ⊗ φi) ∈ D(A)⊗D(A)

Then D(A) has a unique algebra structure such that

a) the embeddings a ∈ A → a⊗ I ∈ D(A), φ ∈ A′ → I ⊗ φ ∈ D(A) are unitalalgebra homomorphisms,

7 QUANTUM GROUPS 27

b) A becomes a Hopf algebra with universal R–matrix R.

The algebra structure of D(A) is given by

(a⊗ I)(I ⊗ φ) = a⊗ φ,

(I ⊗ φ)(a⊗ I) =∑

< φ(1), S−1a(1) >< φ(3), a(3) > a(2) ⊗ φ(2).

The key idea is that if B is a Hopf algebra and if R ∈ B ⊗ B is an invertiblesolution of the Yang-Baxter relation

R12R13R23 = R23R13R12

satisfying∆⊗ ι(R) = R13R23, ι⊗∆(R) = R13R12,

then twisting the original coproduct with R,

∆R(b) := R∆(b)R−1

gives another coassociative coproduct for B for the same algebra structure. Onthe other hand, if Aop is the Hopf algebra with opposite multiplication, we mayform the tensor product Hopf algebra A′ ⊗Aop, and if (ai) and φi are dual bases,and

R′ := (I ⊗ φi)⊗ (ai ⊗ I)

then R′−1 satisfies the required relations. Thus we obtain a new coproduct onA′ ⊗Aop twisting the old coproduct with R, see also [101], [59]. Dualizing gives anew product on A⊗A′op, and this is the algebra structure described in the abovetheorem.

Dual pairs of Hopf algebras and R–matrices The quantum double method is effec-tive to construct R–matrices and braided tensor categories from infinite dimen-sional Hopf algebras. However, some care is needed. In fact, in general, A′ is analgebra, but not a Hopf algebra, as the ‘comultiplication’ of A′ takes values in(A ⊗ A)′ which contains A′ ⊗ A′ properly. One thus needs to restrict to a subal-gebra. The maximal subspace Ao ⊂ A′ on which we get a comultiplication on Ao

is called the restricted dual. This is now a Hopf algebra, or even a Hopf ∗–algebraif A was.

This restricted dual may be rather small in some cases [14]. Moreover, it isoften difficult to determine and it does not yield a desired duality, i.e.

Aoo 6= A.

However, Ao is quite rich in many important cases, in that finite dimensionalrepresentations of A provide Hopf subalgebras of Ao which separate the points ofA. We shall later illustrate this with an important class of examples studied byRosso who was motivated by the desire of relating the compact quantum groupSµU(d) with the quantized universal enveloping algebra Uµ(sl(d)). As a result, heconstructed compact quantum groups of other Lie types.

7 QUANTUM GROUPS 28

For this reason, it is natural to introduce the notion of a dual pairing betweenHopf ∗–algebras

A×B → C,

satisfying obvious relations between operations and cooperations. When this du-ality is non-degenerate, then we have faithful inclusions of Hopf ∗–algebra

A→ Bo, B → Ao.

Another subtlety in the construction of R–matrices for infinite dimensionalHopf algebras is that R will lie in some completion of A⊗Bop. However, a gener-alized quantum double construction holds for nondegenerate dual pairings, givingrise to braided symmetries of the associated tensor categories of finite dimensionalrepresentations, see, e.g., [59].

Drinfeld-Jimbo quantum groups associated to a Lie algebra for a generic parameterLet g be a simple f.d. complex Lie algebra of rank r, h a Cartan subalgebra.

Let R be the root system, Φ = α1, . . . , αr the set of simple roots, < ·, · > aninvariant symmetric bilinear form normalized by < α,α >= 2 for short roots. Weshall identify h and h∗ via this form. Then di :=< αi, αi > /2 ∈ N. A = (aij)the Cartan matrix, aij := 2<αi,αj>

<αi,αi>. For a nonzero complex q, which is not a root

of unity, the quantum universal enveloping algebra Uq(g) is the unital complexalgebra with generators E±i , Ki such that Ki are invertible, and

KiKj = KjKi, KiE±j = q±<αi,αj>E±Ki,

[E+i , E

−j ] = δi,j

Ki −Ki

qi − q−1i

,

where qi := q<αi,αi>/2, and the quantum Serre relations,

1−aij∑0

(−1)k(

1− aijk

)qi

(E±i )kE±j (E±i )1−aij−k = 0, i 6= j,

where(1−aijk

)qi

are suitable deformed binomial coefficients. The Hopf algebrastructure is given by

∆(Ki) = Ki ⊗Ki, ∆(E±i ) = E±i ⊗K1+ω±i +K

ω±i ⊗ E±i ,

S(Ki) = K−1i , S(E±i ) = −K−ω±i E±i K

−(1+ω±)i ,

ε(Ki) = 1, ε(E±i ) = 0,

where ω+ = 0, ω− = −1.Let P+ be the cone of dominant integral weights: P+ := λ ∈ h∗ : λ(α∨i ) ∈

Z+, where α∨ = 2α<α,α> .

A highest weight module with highest weight λ ∈ h∗ is a representation ona f.d. vector space with a cyclic vector vλ (in fact unique up to a scalar) such

7 QUANTUM GROUPS 29

that E+i vλ = 0 and Kivλ = q<αi,λ>vλ. As in the classical case, there is a unique

irreducible highest weight module with highest weight λ, denoted L(λ).The representation theory of these quantum groups generalizes the classical

representation theory of complex simple Lie algebras.

Theorem ([72])Finite dimensional representations are completely reducible. L(λ)is f.d. if and only if λ ∈ P+. Any irreducible f.d. representation is of the formL(λ)⊗ γε, where

γε : E±i → 0, Ki → εi ∈ 1,−1.

This theorem allows to restrict attention to the full tensor subcategory gener-ated by the L(λ) (type 1 representations).Example Irreducible representations of Uq(sl(2)) of type 1 are classified by positiveintegers n ∈ N, with n+ 1 the dimension of the representation,

πn(E+)vr = [n− r + 1]qvr−1, πn(E−)vr = [r + 1]qvr+1, πn(K)vr = qn−2rvr,

on a linear basis v0, . . . , vn where [k]q := qk−q−kq−q−1 .

Universal R–matrix for Uq(sl(d)) Assume that either q is a formal variable oris not a root of 1. Consider now the complex Hopf algebra U+ with generatorsE+

1 , . . . E+d−1, K±1

1 , . . . ,K±1d−1 satisfying all the presentation relations of Uq(sl(d))

involving these generators, except the quantum Serre relations, and let U− be sim-ilarly presented with generators E−1 , . . . E

−d−1, K ′1

±1, . . . ,K ′d−1

±1. The coalgebrastructure for U+ is defined using the same formula as for Uq(sl(d)), while that ofU− is the opposite. Then U+ and U− form a dual pair of Hopf algebras with

< Ei,K′j >=< Ki, E

−j >= 0,

< E+i , E

−j >= − δij

q − q−1, < Ki,K

′j >= q−<αi,αj>/2.

It may be shown (see [59], see also [113]) that this duality is degenerate, thatthe corresponding ideals I+ and I− in U+ and U− are generated by the quantumSerre relations, and that they are Hopf ideals. Therefore the generalized quantumdouble construction may be applied to get a Hopf algebra D(U+/I+). It turns outthat there is a canonical surjective homomorphism of Hopf algebras

u+ ⊗ u− ∈ D(U+/I+)→ u+u− ∈ Uq(sl(d))

with kernel the ideal generated by Ki − K ′i. The image of the the universal R–matrix for the quantum double is a universal R–matrix for Uq(sl(d)). This matrixcan be written down explicitly. It involves infinite sums, and it lies in a suitabletopological completion of the tensor product algebra [106]. However, the associatedbraided symmetry on the category of representations takes a simple form. If u isthe d–dimensional representation of Uq(sl(d)) corresponding to the fundamentalweight ω1, i.e.

E+i → Ei,i+1, E−i → Ei+1,i, Ki → qEi,i + q−1Ei+1,i+1 +

∑j 6=i,i+1

Ej,j ,

7 QUANTUM GROUPS 30

then on a basis of (Cd)⊗2 the representative ε(u, u) := Fu ⊗ u(R) of the braidedsymmetry is a scalar multiple of the Jimbo–Woronowicz representation of theHecke algebras,

ε(u, u) = q1d−1 ηJW (g1).

Note that this scalar can not be suppressed, it is related to the naturality of thebraided symmetry. It may be shown that, if q is a parameter, different roots of qyield inequivalent braided symmetries.

The non-generic case There are various approaches. I shall say a few words onlyof Lusztig’s approach, known as the restricted specialization, [71]. Assume forsimplicity that the Lie algebra is simply laced.

The algebraic relations among the generators show that if q` = 1, all the theelements E`i , F

`i are central. In the restricted specialization they become zero.

To do this, replace q by a formal variable x and define Ux(g) as an algebra overC(x) (in fact, a suitable extension C(x1/N ) is needed to construct R–matrices).Consider the subalgebra of UA

x (g) ⊂ Ux(g) over A := Z[x±1/N ] generated by theKi’s and the divided powers

E±i(n)

:=(E±i )n

[n]!xi.

This is a complex Hopf algebra that may be presented with generators and rela-tions. Set

Uq(g) := UAx (g)⊗A C.

Now (E±i )k` = 0 but (E±i )(`) do not vanish.Irreducible highest weight modules L(λ) of Ux(g) may be specialized to Uq(g),

hence we still have the specialized Weyl modules L(λ). However, these specializedmodules neither stay irreducible nor are completely reducible. For example, ifg = sl(2), L(`− 1) is irreducible but

d(L(`− 1)) = [`] = 0.

Even worse, if ` ≤ m ≤ 2`− 1,

E−v`−1 = [`]v` = 0.

In this case, the elements vr, m−` < r ≤ `−1 span the unique invariant subspace.Hence L(m) is reducible but not completely reducible.

However, L(0), . . . , L(`− 2) are irreducible with nonzero quantum dimension.In particular,

d(L(1)) = [2] =q2 − q−2

q − q−1= 2 cosπ/`.

Reshetikhin and Turaev [104] have shown that L(0), . . . , L(`− 2) give rise to asemisimple tensor category with ‘truncated tensor products’. The fusion rules are

L(m)⊗ L(n) =min2(`−2)−(m+n), m+n⊕

k=|m−n|

L(k),

7 QUANTUM GROUPS 31

where the sums goes in steps of 2. A universal R–matrix can be defined (Lusztig,Rosso, see references in Kassel’s book) and the associated category is modular.

Note that truncated tensor products had also appeared in the physics literature[?], see also [37].

The general situation has been considered by Andersen [1] and studied in detailin a series of papers by him and his collaborators. One can determine a finite subsetC of P+, called the Weyl alcove, such that if λ ∈ C, L(λ) is irreducible and ithas nonzero quantum dimension. This finite set of irreducibles gives rise to asemisimple ribbon tensor category (with truncated tensor products) with finitelymany simple objects, the L(λ)’s, λ ∈ C.

The truncated tensor products have a categorical interpretation in terms of theMacLane construction of quotient category, see [64]. This material is discussed inalmost every book on quantum groups. For a short paper where all the necessarydetails are treated leading to ribbon Hopf algebras, see [108]. Note that this finitecategory is even a modular category, see, e.g.[5].

Compact quantum groups In the original definition Woronowicz assumed the exis-tence of a generating finite dimensional representation [123]. The new perspectivegained from the work of Baaj and Skandalis [7] opened the way to treat the generalcase [124]. See also [?] for a beautiful comprehensive survey.

A compact quantum group is a unital C∗–algebra with a coassociative ∗–homomorphism A→ A⊗A into the minimal tensor product satisfying

∆(A)A⊗ C, ∆(A)C⊗A dense in A⊗A.

If G is a compact group, both axioms clearly hold for the commutative Hopfalgebra C(G). Conversely, if A is a commutative unital C∗–algebra, A = C(X)for a unique compact space X by the commutative Gelfand-Naimark theorem.Coassociativity of ∆ means that X is a compact semigroup, and that ∆(f)(g, g′) =f(gg′). The density axiom implies that X has left and right cancellation. Indeed,if gh = g′h then all the functions of ∆(A)A ⊗ C take the same values on (g, h)and (g′, h), and since they are dense, g = g′. On the other hand, the cancellationproperty characterizes compact groups among compact semigroups.

The great advantage of compact quantum groups of Woronowicz is that ex-istence and uniqueness of a Haar measure may be established. This is a state h(positive normalized functional) of A which is left and right invariant:

(ι⊗ h)(∆(a)) = h(a), a ∈ A,

(h⊗ ι)(∆(a)) = h(a), a ∈ A.

Theorem A compact quantum group admits a unique left invariant state, whichis also right invariant.

Infinite dimensional ‘strongly continuous’ unitary representations, such as theregular representation, may be introduced. They correspond to unitary opera-tors of the multiplier C∗–algebra u ∈ M(K(H) ⊗ A), with K(H) the algebra of

7 QUANTUM GROUPS 32

compact operators on the Hilbert space of u. They are completely reducible intofinite dimensional representations. Moreover, every invertible f.d. representationis equivalent to a unitary representation. In particular, if u is a unitary and f.d.,u∗ := (u∗i,j) is still an invertible representation, hence u := Fu∗F

−1 is unitaryfor some invertible matrix F . This is the conjugate of u, defined up to unitaryequivalence.

The following is a remarkable result.

Theorem Coefficients of unitary representations of the cqg G = (A,∆) span adense ∗–algebra A∞, which is a Hopf ∗–algebra, i.e. coinverse and counit exist andare unique.

The category Rep(G) of unitary finite dimensional representations of G is atensor C∗–category with conjugates, endowed with an obvious embedding functor

E : Rep(G)→ Hilb.

Moreover, the existence of an embedding functor characterizes representationcategories of cqg among tensor C∗–categories. (Tannaka–Krein duality theoremof Woronowicz).

Theorem If T is a tensor C∗–category with subobjects, direct sums and irreducibletensor unit and F : T → Hilb is a tensor ∗–functor, then there is a compactquantum group G such that (T, F ) is isomorphic to (Rep(G), E). G is uniquelydetermined up to the choice of a norm in A∞.

Tannaka-Krein duality is a powerful tool to construct explicit examples of cqg.This is how Woronowicz constructed SµU(d).

Example (Woronowicz) SµU(d) is the completion in the maximal C∗–norm of theWoronowicz dual of the tensor C∗–category generated by the deformed determi-nant

S :=∑p∈Pd

(−µ)i(p)ψp(1) ⊗ · · · ⊗ ψp(d).

It is presented as the universal unital C∗–algebra generated by the entriesui,j , i, j = 1, . . . , d of a unitary matrix satisfying∑

p∈Pd

(−µ)i(p)ui1,p(1)ui2,p(2) . . . uid,p(d) = 0, (i1, . . . , id) /∈ Pd,

∑p∈Pd

(−µ)i(p)uq(1),p(1)uq(2),p(2) . . . uq(d),p(d) = (−µ)i(q), otherwise.

Coproduct is defined requiring (ui,j) to be a corepresentation.This compact quantum group may be regarded as a compact real form of quan-

tum SL(d). The relationship between Drinfeld-Jimbo Uq(sl(d)) and WoronowiczSµU(d) has been first studied by Rosso, who also constructed compact quantumgroups of type BCD [107]. Some exceptional compact quantum groups have beenconstructed by Andruskiewitsch [2], See also [65].

7 QUANTUM GROUPS 33

Example (Wang, Van Daele–Wang [114]) Au(F ) Let F ∈Mn(C) be any invertiblematrix, and let Au(F ) be the universal unital C∗–algebra generated by the entriesof a matrix (ui,j) with relations making u and Fu∗F

−1 unitary. If the coproductis defined requiring (ui,j) to be a corepresentation, Au(F ) becomes a compactquantum group

Note thatAu(F ) = Au((F ∗F )1/2) = Au(λF )

for any scalar λ.Ao(F ) If FF is a nonzero scalar, then this scalar is real, and we may assumeFF = ±1. Ao(F ) is the quotient of Au(F ) by

u = Fu∗F−1.

The importance of these quantum groups relies in the fact that every compactmatrix quantum group G is a quantum subgroup of some Au(F ), and if the definingrepresentation is irreducible and selfconjugate, it is in fact a quantum subgroup ofsome Ao(F ).

Brauer-Schur-Weyl dualityJimbo and Woronowicz have found a remarkable representation of the Hecke

algebras Hn(q2) on Cd⊗n taking with the first generator g1 of Hn(q2) to

ηJW (g1)ψi ⊗ ψj = qψj ⊗ ψi, i < j,

= ψi ⊗ ψi, i = j,

= qψj ⊗ ψi + (1− q2)ψi ⊗ ψj , i > j.

This is a deformation of the classical representation of the permutation group.Most importantly, a generalized Schur–Weyl duality between the Hecke algebra

Hn(q2) and the quantum group Uq(sl(d).

Theorem a) (Jimbo, Woronowicz) The images of Hn(q2) and Uq(sl(d)) are onethe commutant of the other.

b) ([43] ) A similar duality holds for quantum groups of type BCD, if the Birman-Wenzl-Murakami algebra [9, 80] replaces the Hecke algebra.

Van Daele’s algebraic quantum groups Classical Pontrjagyn duality is a dualitywithin one category, the category of locally compact abelian groups. This theoryasserts that the dual G of a lca group G is a lca group and ˆ

G ' G canonically.On the other hand duality is important in Hopf algebras, as, as we have seen inthe important special situations of Drinfeld Jimbo quantum groups, it is useful torelate representations to corepresentations of the dual, and moreover it leads tothe construction of braided tensor categories via the quantum double.

Motivated by the search of a larger category where a similar duality holds, andcould explain these implications in an axiomatic framework of quantum groups,

7 QUANTUM GROUPS 34

Van Daele introduced the algebraic quantum groups. The original references are[115, 116].

These specific situations mentioned above indicate that it is necessary to workin an algebra larger that the algebraic tensor product A ⊗ A. Van Daele startswith the observation that for any pair of functions f, f ′ ∈ c00(G), the algebra offunctions on a discrete group G with finite support and comultiplication ∆ arisingfrom the group multiplication,

∆(f)f ′ ⊗ I, ∆(f)I ⊗ f ′ ∈ c00(G)⊗ c00(G).

Moreover, as in the compact case, the associated bilinear maps on the two variables

T1 : f1 ⊗ f2 → ∆(f1)f2 ⊗ I, T2 : f1 ⊗ f2 → ∆(f)I ⊗ f ′

keep all the information that G is a group. Specifically, this information is equiv-alent to the fact that 1 and T2 are are bijective on c00(G)⊗ c00(G). Indeed, theyare dual to the maps

(g, g′)→ (gg′, g′), (g, g′)→ (g, gg′).

This leads to the notion of multiplier Hopf algebra.If A is a (not necessarily unital) complex algebra, the multiplier algebra is ‘the

algebra of elements that multiply A into itself from the left and from the right’.Thinking of a left and a right multiplier by c as linear maps L : A → A, a → caR : a ∈ A→ ac ∈ A,

M(A) := L,R : A→ A, L(ab) = L(a)b, R(ab) = aR(b), R(a)b = aL(b).

It is a unital associative algebra containing A and coinciding with A if A wasunital.

A multiplier Hopf algebra is an algebra with a suitably coassociative comulti-plication

∆ : A→M(A⊗A)

such thatT1 : a⊗ b→ ∆(a)I ⊗ b, T2 : a⊗ b→ a⊗ I∆(b)

are bijective maps on A⊗A.Multiplier Hopf algebras are far too general objects to lead to a duality result.

A good subclass is given by those with invariant functionals.A nonzero left invariant functional is unique and faithful, and there is always

a unique and faithful right invariant functional.

If φ is a left invariant functional,

A := φ( · a), a ∈ A ⊂ A′

Theorem ([116]) The dual A of a regular mha with nonzero invariant functionalsbecomes a regular mha with nonzero invariant functionals. Moreover,

ˆA ' A,

8 TANNAKA–KREIN DUALITY, ASPECTS OF THE REGULAR REPRESENTATION35

canonically.If A is a multiplier Hopf ∗–algebra with positive invariant functionals, the same

is A.Van Daele extends Pontrjagin duality to the category of multiplier Hopf alge-

bras with left and right invariant functionals, which is thus a self-dual category.This is a purely algebraic approach. It includesa) Finite dimensional Hopf algebras,b) The algebraic (A∞,∆) of Woronowicz compact quantum groups,c) discrete quantum groups of Effros and Ruan [32]d) certain examples of root of unity algebras which skip to the operator algebraicdescription.e) However, it does not contain all locally compact groups.

It includes Drinfeld’s quantum double constructions. This is due to [26].the assumption that the coproduct should take values in A⊗A, which is already

too strong for abelian l.c. groups, and replaced it with

∆ : A→M(A⊗A),

where M(B) is the algebra of left and right multipliers of B. Regular multiplieralgebras (i.e. (A,F∆) is a mha as well) have counit and coinverse.

8 Tannaka–Krein duality, aspects of the regularrepresentation

Generally speaking, classical Tannaka–Krein theorem has two aims: it allows toreconstruct a given group from its representations,

Rep(G) −→ G,

and it also gives an abstract characterization of those concrete categories whichare of the form Rep(G), i.e. G is not given a priori. This understanding of TKduality owes also a lot to the work of Grothendieck.

This second process is explicit and in fact a constructive tool in the theory ofquantum groups, where it was initiated by Lyubashenko and Woronowicz [49, 125],and later widely used by many authors, see, e.g. [76]

References [54] and [79] are excellent review papers on Tannaka-Krein duality,and I will not comment more on this.

In quantum theory, TK duality is interpreted in a different way. It becomesa tool to construct construct quantum groups. For example, Woronowicz con-structed Sµ(d) in this way. Moreover, TK duality plays an important role in theconstruction of the free universal quantum groups Ao(F ) and Au(F ) due to Wangand Van Daele.

The problem we have focused in these lectures is that of understanding to whatextent general tensor C∗–categories with conjugates may be described as ‘repre-sentation categories’ of noncommutative group-like structures. We remark that

9 ABSTRACT TK DUALITY FOR NON-SYMMETRIC CATEGORIES 36

other aspects of this problem involve the structure of the regular representationthat has been studied in [24, 86, 7, 20, 25]; and inclusions of factors wih finiteindex and finite depth. More precisely, f.d. Kac algebras in their regular represen-tations naturally arise from the tensor categories derived from irreducible depth2 inclusions [109, 46, 48]. Tensor categories derived from reducible finite depthinclusions give rise to weak Hopf algebras (also named finite quantum groupoids)introduced in [11, 12, 13], see [81, 82].

In an algebraic framework, Hayashi and Ostrik proved that every semisimplerigid tensor category with finitely many irreducible objects is the representationcategory of a finite quantum groupoid [44, 83]. See also the paper [31] and refer-ences therein.

9 Abstract TK duality for non-symmetric cate-gories

The aim of this section is to give a brief introduction to the papers [90, 91]. Wefocus on the problem of describing the predual of abstract tensor C∗–category withconjugates, where the datum is the set of its irreducibles, rather then a regularrepresentation. This is a generalization of the datum of Doplicher-Roberts dualityto the case where the existence of permutation symmetry is removed and notreplaced by anything else, not even a braided symmetry.

Little seems to be known about the problem of classifying general non-embedabletensor C∗–categories, although there are plenty of examples arising in the theoryof subfactors and in low dimensional QFT.

For instance, the tensor category generated by NMN , with N ⊂ M an irre-ducible II1 inclusion with finite index, is a non-embedable tensor C∗–categorywhenever the index is not an integer and the inclusion is amenable in the senseof Popa, see [88]. There is a similar result for an amenable object in a tensorC∗–category with non-integral dimension [68].

As a trivial remark, if N ⊂ M is a finite index inclusion of II1 factors, M ,as a bimodule over N , has always a real structure given by the ∗–involution ofM , generalizing the reality property of the regular representation of a finite group.Hence not all tensor C∗–categories with a single generating object are of this form.

We have shown [91] that tensor C∗–categories with conjugates (with possiblyinfinitely many irreducibles) admitting a generating object, that may or may notbe selfconjugate, but are always of intrinsic dimension ≥ 2 (or Jones index ≥ 4,by [67], see also [68]), can be embedded faithfully into the tensor C∗–category ofHilbert bimodules over a noncommutative C∗–algebra, defined intrinsically by thecategory.

These bimodules are the noncommutative analogues of the bimodule of con-tinuous sections of G–equivariant Hermitian bundles over compact homogeneousspaces. They carry actions of the universal orthogonal and unitary compact quan-tum groups Ao(F ) or Au(F ) of Wang and van Daele and the embedding functoridentifies the given category as the category of bimodule intertwiners of the quan-tum group. The quantum group is not uniquely determined but the representation


category is always the same. The coefficient algebra of these bimodules, togetherwith the ergodic C∗–action of Ao(F ) or Au(F ) describes the noncommutativehomogeneous space.

Actually, ergodic C∗–systems have been constructed in previous papers froman inclusion of the representation category of a compact quantum group G intoan abstract tensor C∗–category M [90, 89].

We remark that, given an ergodic C∗–action of a compact quantum groupG, the natural bimodule structure suggested by the commutative case, does notgive rise to a bimodule G–representation structure in the noncommutative case.More precisely, if, e.g., the right module structure is given by right multiplication,the left module structure can not be given by left multiplication unless the Hopfalgebra of G has enough mutually commuting elements (see Prop. 4.3).

Our construction provides an intrinsic left module structure making the naturalright module representation of the quantum group into a bimodule representation.It reduces to the natural left module structure in the commutative case and it is notinner even for quantum quotient spaces. Furthermore, these bimodule representa-tions are full, in a sense that we will describe more precisely later. This propertyimplies that the ergodic C∗–system may be reconstructed from the category ofbimodule representations.

Roughly speaking, the assumption on the intrinsic dimension allows us to re-alize solutions of the conjugate equations in the category of Hilbert spaces, hencerepresentations of the universal compact quantum groups arise. The associatedmatrices F then describe all possible realizations. These Hilbert space representa-tions are then enlarged to Hilbert bimodule representations of the quantum groupover the coefficient C∗–algebra by a method that we may term abstract induction.

Our ergodic actions do not, in general, arise from quantum subgroups but fromwhat may be termed, following Mackey, a virtual quantum subgroup. Virtualsubgroups are needed to describe cases where the category is non-embedable.

We construct a family of noncommutative compact G–spaces from a givencategory. In the commutative case it is well known that G–spaces are specialcases of groupoids. Thus our result, although relying on different methods, shouldbe compared with that of [81, 82]. It would be interesting to find an explicitconnection when the category arises from a finite depth inclusion.

Our construction sheds light on the problem of recognizing which tensor C∗–categories with conjugates are embedable into the category of Hilbert spaces. Moreprecisely, combining the bimodule construction with the imprimitivity theoremof Takesaki for dynamical systems on operator algebras [110] and the work ofHøegh–Krohn, Landstad and E. Størmer [45] on ergodic actions, we show that ifthe category contains the representation category of a compact group and if thevon Neumann completion of the coefficient algebra in the GNS representation ofthe unique invariant trace is of type I, then the given category may be identifiedwith a full subcategory of the representation category of a closed subgroup.

The problem, posed in [45], of whether an ergodic action of a compact simplegroup is always of type I is still open.

In particular, Wassermann’s result about the non-existence of ergodic actionsof SU(2) on the hyperfinite II1 factor [120], allows one to conclude that a tensor


C∗–category with a pseudoreal object of dimension 2 is always embedable.

Relation with induced representations

Induced representations, introduced by Frobenius [36], have been studied bymany authors in the course of developing the theory of representations of groupsand algebras. In particular, Mackey, in the 50’s, gave a general definition ofinduction for locally compact groups, emphasized their role in Quantum Mechanicsand generalized and unified the work of his predecessors.

Notably, Mackey carried over concepts of group theory to measurable ergodictheory, and referred to this idea as the virtual group point of view in ergodic the-ory [75]. Introducing the concept of a measurable groupoid, as a generalization ofthe concept of a group, he unified measurable ergodic theory and group theory.Here Mackey was, in particular, motivated by wanting to give a conceptual expla-nation of certain previously known results in ergodic theory. Many authors thencontributed to proving analogues in ergodic theory of results in group theory. Inparticular we recall the work of Ramsay on induction for virtual groups [99, 100].

As is well known, ergodic theory played an important role in deep developmentsin operator algebras and noncommutative geometry. Many von Neumann algebrasand C∗–algebras were discovered to be associated with measurable or topologicalgroupoids, for a beautiful recent survey see [77].

Induced actions of locally compact groups on noncommutative von Neumannalgebras have been studied by Takesaki [110]. They played a role in his work onthe structure of such algebras. In particular, Takesaki obtained an imprimitivitytheorem reducing the study of ergodic actions of compact groups on finite vonNeumann algebras to actions of closed subgroups on factors.

Høegh-Krohn, Landstad and Størmer showed that a von Neumann algebrawith an ergodic action of a compact group has a finite trace and is hyperfinite.Furthermore the multiplicity of an irreducible representation is always bounded byits dimension [45]. Combining this with Takesaki’s results, shows that any ergodicaction of a compact group on a von Neumann algebra is always induced from anaction on a full matrix algebra or the hyperfinite II1 factor.

Wassermann developed a general method for studying ergodic actions of com-pact groups on operator algebras, characterized and classified ergodic actions withfull multiplicity and proved that SU(2) does not act ergodically on the hyperfiniteII1 factor [118, 119, 120].

Rieffel proposed a generalization of the theory of induced representations toC∗–algebras, where the inclusion of groups was replaced by an inclusion of C∗–algebras with a conditional expectation. He proved the theorem on induction instages and an imprimitivity theorem. However, the structure in the group caseis richer and parts of the theory of induction, such as Frobenius reciprocity andthe tensor product theorem, did not seem to generalize to the C∗–algebra setting[105].

Induced representations have been further studied in the framework of quan-tum groups. In particular, versions of the Frobenius reciprocity theorem for arepresentation of a subgroup were proved see, e.g. [84, 112, 87]. However, ourreference list is by no means complete.

10 THE PAPERS [?, ?] IN MORE DETAIL 39

We start with the following observation. Although tensor C∗–categories aregenerically non-embedable, they may contain an embedable subcategory A with‘few arrows’. More precisely, for every tensor C∗–category with conjugates M

generated by a single object, we may always find a universal tensor C∗–categoryA generated by solutions of the conjugate equations (a Temperley-Lieb type cat-egory), embedable if the intrinsic dimension of the generating object is ≥ 2. Uni-versality gives a tensor ∗–functor µ : A→M.

Hence we may start with a pair of functors Hilb τ← Aµ→M. The embedding τ

is not unique, but a classification is known in certain cases. Different τ in generalgive rise to different compact quantum groups Gτ .

The basic example comes from a closed subgroup K of a compact group G,with µ : Rep(G) → Rep(K) the restriction functor. The duality theorem of DRcharacterizes such inclusions: if A ⊂ M is an inclusion of tensor C∗–categories,with conjugates and permutation symmetry and closed under direct sums andsubobjects, then there is a unique inclusion K ⊂ G of compact groups such thatthe given inclusion corresponds to µ : Rep(G) → Rep(K) up to equivalence ofcategories.

When permutation symmetry is not assumed the framework becomes non-commutative. The special case of embedable inclusions µ : A ⊂ M describesquantum subgroups of compact quantum groups in the sense of Woronowicz [124]and Podles [92].

10 The papers [90, 91] in more detail

This last section is based on two talks given in Bedlewo and Vietri sul Mare in thesummer 2009.

We start with an alternative proof the DR duality theorem described in Sect.4, that will be generalized to the non-symmetric case.

Pimsner introduced the universal C∗–algebra OX of a Hilbert C∗–bimoduleX, generalizing both Od and A oα Z [?]. He motivated this construction with thetheory of subfactors, in that, as we have seen, bimodules over non-commutativeC∗–algebras arise naturally. The bimodule X plays in OX a role similar to thatof a Hilbert space in the Cuntz algebra.

With the same notation of Sect. 4, one can show that, in case of a single objectof determinant 1,

A⊗RepSU(d) Od = OX ,

where X = Cd ⊗ C(K\SU(d)) with SU(d)–action:

ρ(g) := fundamental(g)⊗ translation(g)

Recall that given K ⊂ G, for v ∈ Rep(K),

Ind(v) = right translation

on L2–completion of the space of continuous functions

Xv := f : G→ Hv, f(kg) = v(k)f(g)


Frobenius reciprocity gives a natural isomorphism of Banach spaces,

(u K , v) ' (u, Ind(v))

We may adopt a geometric viewpoint and regard Ind(v) as a representation onthe induced C(K\G)–bimodule Xv. Then,

Ind(v)⊗C(K\G) Ind(v′) = Ind(v ⊗ v′)

It follows thatInd : Rep(K)→ Bimod(G)

is a faithful tensor ∗–functor with full image.For the DR bimodule,

X ' Ind(fundamental K)

In fact, DR embedding is constructed as

DR : T → Bimod(SU(d)) Ind−1

→ Rep(K).

ρ→ X → fundamental K

We regard the first arrow as abstract inductionWe now drop permutation symmetry. The framework becomes noncommuta-

tive.If the category is not embedable, we can not find any quantum subgroup K

and hence the concrete induction functor Ind.This part of DR reconstruction for non-symmetric categories then amounts to

construct abstract inductionT → Bimod(G)

with G a cqg replacing SU(d), that should be produced intrinsically by the cate-gory. G will not be unique.

Analogous situation in measurable ergodic theory. Mackey: an ergodic actionof a groupG on a commutative von Neumann algebra L∞(X,ω) should be regardedas a virtual subgroup, and, as such, we may talk about induction (Ramsay).

Hence, we should be looking for G and also, following Mackey, for an ergodicG–action, now in a topological noncommutative setting.

This viewpoint may be regarded analogous to Connes–Takesaki flow of weightsMod(M) of a type III von Neumann algebra, where an R∗+–ergodic action on acommutative von Neumann algebra is intrinsically produced by M .

In our setting, both the space and the acting group will have noncommutativestructures.

Our strategy is the following.

a) Often, embedable categories appear as subcategories of non-embedable onesT (e.g. TL category in the category of Ocneanu’s bimodules in subfactortheory)


b) To facilitate applications, we start with

Aµ−→ T,

we regard A as a universal version of the embedable subcategory and T as abuilding block. Fix an embedding of A,

Aτ−→ Hilbert spaces

defining a compact quantum group G by Woronowicz duality.

We thus have a pair of functors

Rep(G) τ←− Aµ−→ T

and look for

c) C (G–ergodic algebra) replacing C(K\SU(d)) and depending on µ and τ ,

d) induced G–bimodules Ind(ρ) replacing XDR, for objects in the image of µ

Our main restriction isd(ρ) ≥ 2

For d(ρ) < 2, one would have to deal with quantum groups at roots of unity,by Jones fundamental result and Wenzl’s work.

Ergodic actions

Let G = (Q,∆) ba a compact quantum group, and

α : C→ C⊗ Q

an ergodic G–action,

Cα := c ∈ C : α(c) = c⊗ I = C.

on a unital C∗–algebra.The spectral space of u ∈ Rep(G) is defined as

Lu := (Hu ⊗ C)u⊗α =

(ci) : ci ∈ C, α(ci) =∑k

ck ⊗ u∗k,i.

Their entries (ci) span a dense ∗–subalgebra Cspectral (Podles).

Example K subgroup of G, C = C(K\G), α = G–translation then

Lu ' K − fixed vectors in Hu

viak → (< u(g)ψi, k >)


Ergodicity implies• Existence of a unique G–invariant state,• Lu are Hilbert spaces:

< c, d >:=∑

c∗i di

Spectrum and multiplicities

spec(α) := u ∈ Rep(G), irreducible : Lu 6= 0

mult(u) := dim(Lu)

This is not enough to reconstruct the ergodic C∗–action: an example of Todd(1950) gives two non-conjugate subgroups K1, K2 of a finite group G with K1\Gand K2\G isospectral. (Mackey, 1964)

If G is a compact group:• the invariant state is a trace and

mult(u) ≤ dim(u) (H–K L S)

• H–K, L, S problem: does any simple compact group act ergodically on R ?• Jones problem: does any classical compact Lie group act ergodically on R ?• SU(2) does not. It acts ergodically only on type I von Neumann algebras

(Wassermann). Wasserman’s invariant (multiplicity maps) is stronger than thespectrum• A complete answer for SU(3) is not knownIf G is a cqg:• Haar state is not tracial (Woronowicz), there exist examples of Au(n) on

R and of Au(F ) type III factors (Wang). Since first examples (Podles quantumspheres), one starts from the spectral spaces• mult(u) ≤ q−dim(u), generalizing inequality involving quantum and integral

dimension for a cqg (Boca)• Tomatsu has classified certain ergodic actions of SµU(2) embedable into the

translation action• Bichon–De Rijdt–Vaes have given examples of SµU(2) actions with

mult(u) > dim(u)

method: apply Woronowicz’ Krein reconstruction to the possible tensor embed-dings

Rep(SµU(2))→ Hilbert spaces

to construct ergodic actions of SµU(2) with

Lu⊗v ' Lu ⊗ Lv.

If G is a group,

Lu⊗v ' Lu ⊗ Lv ⇔ mult(u) = dim(u)∀u


Duality Theorem

For a general ergodic action,

(ci) ∈ Lu, (dj) ∈ Lv ⇒ (djci) ∈ Lu⊗v,

We only have isometriesSu,v : Lu ⊗ Lv → Lu⊗v.

Dual object of an ergodic action

The dual object of an ergodic G–action (C, α) is the pair (L, S), where L isregarded as a ∗–functor

L : Rep(G)→ Hilbert spaces

S as a natural transformation

Su,v : Lu ⊗ Lv → Lu⊗v

Although u→ Lu is not tensor in general, there are coherent rules that governthe behaviour of L under tensor products. Rules later recognized analogous toPopa’s commuting squares appearing in Jones index theory:

Proposition Given spectral spaces Lu, Lv, Lw of an ergodic action, the followingdiagram commutes

Lu ⊗ Lv ⊗ Lw1⊗S //

S⊗1

Lu ⊗ Lv⊗w

S

Lu⊗v ⊗ Lw S // Lu⊗v⊗w

and it is a commuting square in the sense of Popa:

ELu⊗v⊗wLu⊗v⊗Lw(Lu ⊗ Lv⊗w) = Lu ⊗ Lv ⊗ Lw

An abstract pair (L, S) with

L : S→ T, Su,v ∈ Lu ⊗ Lv → Lu⊗v

satisfying the commuting square condition is a quasitensor functor

More examples• T tensor C∗–category;

ρ ∈ T → (ι, ρ) ∈ Hilbert spaces

is a quasitensor functor (the minimal one)• In particular for a II1 inclusion N ⊂M , we get M⊗r → N ′ ∩Mr−1.


Quasitensor functors unify Jones and Boca inequalities:

dimN ′ ∩Mr−1 ≤ [M : N ]r, mult(u) ≤ q − dim(u)

Duality Theorem [90]a) The spectral functor (L, S) of an ergodic action allows to reconstruct(Cspectral, action, invariant state, maximal norm)

b) Any quasitensor functor (L, S) between

L : Rep(G)→ Hilbert spaces

is the spectral functor of an ergodic action of G on a unital C∗–algebra.

Remark: part b) generalizes BDV construction.

Corollary The spectral functor (L, S) is a complete invariant for ergodic C∗–actions of compact quantum groups over amenable algebras:

Cred = Cmax

Examples of amenability: classical compact transitive spaces, SµU(d) (Nagy),of non amenability: Ao(F ), n ≥ 3 (Skandalis) and Au(F ) (Banica)

The ergodic algebra of a pair of functors

Theorem Given a tensor C∗–category T, with


there is a unital G–ergodic C∗–algebra µCτ

Idea of Proof If τ is injective, we may compose:

Rep(G) τ−1

−→ Aµ−→ T

minimal−→ Hilbert spaces

and find a quasitensor functor.

However, injectivity of µ is not necessary. On arrows it is automatic: anytensor C∗–category with conjugation is simple [89]

Constructing the induced Hilbert bimodules and the embedding

G–Bimodules of cqg

The non commutative analogue of G–equivariant Hermitian bundles over com-pact spaces.

Let G = (Q,∆) be a compact quantum group acting on a unital C∗–algebra C,

α : C→ C⊗ Q

and X a Hilbert bimodule over C


A bimodule representation of G on X is a map

v : Xv → Xv ⊗ Q,

with Xv a Hilbert C–bimodule, such that

v(xc) = v(x)α(c), v(cx) = α(c)v(c)

< v(x), v(x′) >= α(< x, x′ >)

v ⊗ 1 v = 1⊗∆ v

v(Xv)1⊗ Q dense in Xv ⊗ Q

Example

Given a C∗–action (C, α) of G and v ∈ Rep(G), we may form X := Hv ⊗ C

with actionv ⊗ α replacing Ind(v K).

It is always a representation of G on the free right module Hv ⊗ C.

Due to noncommutativity, trivial left C–action

c(ψ ⊗ c′) := ψ ⊗ cc′

is not a good choice:

• If G is a group, u⊗ α is a bimodule representation. However,

(u⊗ α, u′ ⊗ α) ⊂ B(Hu, Hu′)⊗ Z(C).

• If G is a quantum group, u⊗ α is not even a bimodule representation.

Example (Quantum quotients) IfK is a quantum subgroup of a compact quantumgroup G then for v ∈ Rep(G),

Ind(v K) ' Hv ⊗ C(K\G),

There is a good left module structure, non-trivial precisely in the noncommu-tative cases:

< ψi ⊗ I, cψj ⊗ I >=∑h

v∗hicvhj .

It is not inner for K 6= trivial group.

ExampleFor some ergodic actions there isn’t any: C = M3, G = SU(2)

The induced bimodules


Theorem [91] Given


with µ surjective on objects, there are• induced bimodule G–representations Ind(µu) ' u⊗α over µCτ with a ‘good’

left action,

• a faithful Frobenius tensor ∗–functor with full image

Ind : T → Bimod(G)

The induced bimodule is constructed first algebraically, requiring validity ofFrobenius reciprocity for the abstract ‘restriction’ functor µ:

‘(µu, v) ' (u, Ind(v))’, u ∈ G

This determines the spectral decomposition of Ind(v). We start with

Ind(v) := ⊕u∈G(µu, v)⊗ τu.

‘Good’ left actions

Are those for which

cξ = ξc, ξ ∈ (Hu ⊗ C)u⊗α, c ∈ C.

We call such bimodule structures full, as they give rise to a full functor:

u⊗ α ∈ Bimod(G)→ u⊗ α ∈ Mod(G)

(v ⊗ α, v′ ⊗ α)Bimod = (v ⊗ α, v′ ⊗ α)Mod,

a property trivially satisfied in commutative case.

Corollary There is a one-to-one correspondence between isomorphism classes

[Rep(G)→ T]→ [G,C]

Hence different categories provide non-conjugate ergodic actions.

Problem: Which ergodic actions appear?

The ergodic actions from subfactors

If N ⊂M is a proper inclusion of II1 subfactors with finite Jones index,

X ⊗M X = NMN


is a real objectR =

∑i

ui ⊗ u∗i ,

with (ui) a module basis for MN .

‖R‖2 = [M : N ].

Hence there is

Rep(Ao(F ))µ−→ Ocneanu’s bimodules

µ :∑

ψi ⊗ Fψi → R

ifTrace(F ∗F ) = [M : N ], FF = I.

Hence there is an Ao(F )–ergodic C∗–algebra CN⊂M .

The spectral spaces are

Lu⊗r = N ′ ∩Mr−1.

Theorem [88] Generators and relations of CN⊂M are

T ⊗ ξ, T ∈ N ′ ∩Mr−1, ξ ∈ Hr,

r = 0, 1, 2, . . . , and relations,

a) (T ⊗ ξ)(T ′ ⊗ ξ′) = Tpr,sT ′ ⊗ ξξ′,

b) (T ⊗ ξ1 . . . ξr)∗ = T ∗ ⊗ jξr . . . jξ1,

for r ≥ s:

c) S ⊗ (1ur ⊗R∗u ⊗ 1usη) = λSp(2s)r−s,2 ⊗ η,

c’) S′ ⊗ (1ur ⊗Ru ⊗ 1usη′) = λEr+sEr+s+1(S′(p(2s)r−s,2)∗)⊗ η′,

Properties

• If N ⊂M is amenable in the sense of Popa the Ao(F )–ergodic action is notembedable into the translation action for [M : N ] /∈ N.• quantum multiplicities are integral• If [M : N ] = n ∈ N, we may choose Ao(n), of Kac type, and in this case we

find an ergodic algebra with a tracial invariant state.

Problems

• Jones: Is the construction of CN⊂M related to Guionnet–Jones–Shlyakhtenkoconstruction of a subfactor from a planar algebra?


• Determine the type of the associated von Neumann completions in the GNSrepresentation of the invariant state.• In particular, we get an action of S−µU(2) for µ+ µ−1 = [M : N ]. When is

the resulting CN⊂M amenable? They are perhaps distinguished by the standardinvariant of N ⊂M1.

Gaps

Not all ergodic actions arise from pairs of tensor functors


The simplest gaps are the ergodic actions of• G = SU(2) on Mn, n ≥ 2,

• finite groups on Mn with full spectrum but low multiplicities

We may include all ergodic actions starting with


with µ a quasitensor functor.

Remark Rather surprisingly, the C∗–algebra and bimodule construction may beconstructed (computations are more complicated) the only difference being thatInd(µu) is finite projective rather than free.

This generalization shows that there may be different induction theories onthe same noncommutative ergodic space, when we vary µ. As, if we have a tensorembedding

µ : Rep(G)→ T

we also have a quasitensor one

Rep(G)→ Tminimal→ Hilbert spaces

giving rise to an inner left module structure.However, constructing directly examples of quasitensor functors to non-embedable

categoriesRep(G)→ T

seems difficult. Perhaps insight gained from their structure might shed light. Thisgeneralization sounds like a predictin to me, it still needs to be understood.

Embedding a Temperley-Lieb subcategory

TL±d:=The universal tensor ∗–category with objects N0 and arrows generatedby a single R ∈ (0, 2),

R∗ ⊗ 1 1⊗R = ±1,

d = R∗R


is the Temperley-Lieb category, the categorical counterpart of the TL algebras

(often defined without reference to the ∗–operation).• TL±d is simple except for d = 2 cosπ/m, when it has a single non-zero proper

tensor ideal I.• TL±d for d ≥ 2 and TL±d/I for d = 2 cosπ/m, are tensor C∗–categories

(Goodman–Wenzl).

Generalization to the non self-conjugate case

Consider the universal tensor ∗–category Td generated by two objects x, x andtwo arrows R ∈ (ι, x⊗ x), R ∈ (ι, x⊗ x) s.t.

R∗ ⊗ 1x 1x ⊗R = 1x, R∗ ⊗ 1x 1x ⊗R = 1x,

R∗R = R∗R = d.

Goodman–Wenzl’s theorem extends to Td• For d = 2 cosπ/m, the quotient categories of TL±d and Td can not be

embedded into the Hilbert spaces, as these values are not taken by Woronowiczquantum dimension

• For d ≥ 2, TL±d and Td are embedable.

More precisely: For any F ∈Mn satisfying

Tr(FF ∗) = Tr((FF ∗)−1) = d,

(and FF = ±I, resp.) there is an isomorphism

Td → Rep(Au(F ))

(TL±d → Rep(Ao(F )), resp.)

Summary

Theorem Let T be a tensor C∗–category with a real or pseudoreal generatingobject ρ with d(ρ) ≥ 2. Then for any invertible matrix F s.t.

Trace(FF ∗) = d(ρ), FF = ±I,

there is a Frobenius tensor ∗–isomorphism

T → Bimod(Ao(F ))

Similar conclusions if the objects of T are generated by two conjugate objectsρ, ρ with d(ρ) ≥ 2. The cqg is now Au(F ).

The above result sheds some light on the problem of recognizing which non-permutation symmetric tensor categories are embedable into the Hilbert spaces:

REFERENCES 50

This problem is related to the H–K L S and Jones problems.

After Takesaki and H–K L S:

Theorem If we have Rep(G) → T with G a compact Lie group, and if C′′ is oftype I then there is a tensor embedding

T → Hilbert spaces

as a full subcategory of Rep(K), with K a closed subgroup of G.

Remark: It is proved by classifying the full bimodule representations of G.

After Wassermann:

Corollary A tensor C∗–category T with a distinguished pseudoreal generatingobject ρ with d(ρ) = 2 admits

T → Hilbert spaces

image is now a full subcategory of Rep(K) with K < SU(2).

Concluding remarks

• In a work in progress with Roberts, we are considering a theory of inductionfor a pair of quasitensor functors

Sτ←− A

µ−→ T.

This more general setting is perhaps helpful to relate to JGS construction from aplanar algebra.• I have tried to described a map

certain tensor C∗–categories →

nc G–spaces and their Hermitian G–bundles

The description of these spaces naturally emphasizes a spectral viewpoint. Itwould be interesting to try to pursue the geometric aspect.

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