Tannaka-Krein duality for compact quantum group coactions (survey)

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Tannaka-Krein duality for compact quantum group coactions (survey)

Leonid Vainerman

To cite this version:

Leonid Vainerman. Tannaka-Krein duality for compact quantum group coactions (survey). METH-

ODS OF FUNCTIONAL ANALYSIS AND TOPOLOGY, 2015. �hal-02146820�

Tannaka-Krein duality for compact quantum group coactions (survey)

Leonid Vainerman February 1, 2014

Abstract

The last decade saw the appearance of a series of papers containing a very interesting development of the Tannaka-Krein duality for com- pact quantum group coactions on C-algebras. The present survey is intended to present the main ideas and constructions underlying this development.

Dedicated to Professor Yurij M. Berezansky on the occasion of his 90-th anniversary.

1 Introduction

According to the Pontryagin’s duality for abelian locally compact groups, such a group can be reconstructed if its dual (i.e., the group of its continuous characters) is known. Extending this result to non-commutative groups, T.

Tannaka [30] showed that a compact group G can be reconstructed if the set Rep p G q of its continuous finite dimensional representations is known. In 1949, M.G. Krein [13] gave an abstract description of Rep p G q .

Later on, mainly due to works by A. Grothendieck, P. Deligne, and N.

Saavedra Rivano, these results referred to as ”Tannaka-Krein duality for compact groups” where formulated in the language of symmetric monoidal (or tensor) categories and extended to affine algebraic groups.

After the discovery of quantum groups by V.G. Drinfeld and M. Jimbo,

there was a number of papers on the Tannaka-Krein duality for this new class

of objects in a purely algebraic context - see [12] and the references therein.

Here the representation categories were not symmetric, in general.

Motivated by superselection principles in quantum field theory, S. Do- plicher and J.E. Roberts introduced the notion of a C

-tensor category with conjugates [8] and proved that any such a category with permutation symme- try is equivalent to the representation category of a unique compact group. In the setting of compact quantum groups (CQG) [40], [42], the S.L. Woronow- icz’s Tannaka-Krein duality [43] claims that any C

-tensor category with conjugates and a unitary tensor functor to the category H

of finite dimen- sional Hilbert spaces (fiber functor), is equivalent to the category Rep p G q of unitary finite dimensional representations of a unique CQG G with the canonical fiber functor sending any representation to the Hilbert space where it acts. This can be viewed as a reconstruction of a regular coaction of G via its coproduct on the C

-algebra C p G q ”of all continuous functions on G”.

As for a general continuous coaction α of a CQG G on a unital C

-algebra B , the problem is to find a structure on the category Rep p G q which would replace the above mentioned canonical fiber functor in order to be able to reconstruct B and α from this structure.

The results by A. Wassermann [37], [38] and M.B. Landstad [15] concern- ing usual group actions on C

-algebras mean that there is a bijection between ergodic actions of full multiplicity of a compact group G and arbitrary fiber functors Rep p G q Ñ H

. In particular, A. Wassermann [39] proved that any erogdic action of SU p 2 q is obtained by induction from a projective represen- tation of its closed subgroup. Thus, there are countably many isomorphism classes of such actions. The Hopf algebraic version of the above mentioned general result is due to K.-H. Ulbrich [34] and P. Schauenburg [28].

The general theory of ergodic coactions of CQG’s on C

-algebras has been initiated by F. Boca [5] and Landstad [16]. P. Podle´s [25] showed that there is a continuum of non-isomorphic ergodic coactions of the famous compact quantum group SU

p 2 qp| q |   1 q [41] on the coideal C

-subalgebras of C p G q called ”quantum 2-spheres” which do not necessarily correspond to quantum subgroups of SU

p 2 q . More generally, R. Tomatsu [32] classified all the coideal C

-subalgebras of C p SU

p 2 qq .

The bijection between unitary fiber functors of Rep p G q and ergodic coac- tions of full multiplicity of a CQG G has been established by J. Bichon, A.

De Rijdt, and S. Vaes [4]; using this, they classified all such coactions for

universal orthogonal and unitary quantum groups and constructed, for | q |

small enough, a family of such coactions of SU

p 2 q which are not coideals.

It is clear that in order to extend the categorical duality to wider classes of coactions, one has to replace fiber functors by some more general structure.

In particular, C. Pinzari and J.E. Roberts [24] introduced spectral functors F : Rep p G q Ñ H

which are not tensor, but only quasi-tensor which means that, for any U, V P Rep p G q , F p U q b F p V q is isometric to a subspace of F p U b V q and not to the whole space. They showed that spectral functors exactly correspond to all ergodic coactions of G.

Later on, A. De Rijdt and N. Vander Vennet [9] obtained similar result for non-ergodic coactions, but in other terms. Finally, S. Neshveyev [18] intro- duced weak tensor functors functors and established the categorical duality for arbitrary CQG coactions. Note that this new notion in ergodic case is equivalent to the one of a quasi-tensor functor and that in both [24], [18] the reconstruction procedure follows the lines of [43].

There is an alternative formulation of the Tannaka-Krein duality for CQG coactions motivated by reconstruction theorems for finite semisimple tensor (or fusion) categories [11]. In particular, the notion of a module category [23]

has been adopted by K. De Commer and M. Yamashita to C

-tensor category context [6]. They showed that there is a one-to-one correspondence between ergodic coactions of a CQG G and semisimple irreducible module categories over Rep p G q with a simple generator, which enabled them to classify all ergodic coactions of SU

p 2 q in terms of weighted graphs.

This approach was extended by S. Neshveyev [18] to general coactions, he also showed that the two versions of the duality are equivalent. Finally, S. Neshveyev and M. Yamashita [21] proved that the module category in question can be defined by a unitary tensor functor if and only if G coacts on a Yetter-Drinfeld C

-algebra.

In this short survey, we briefly discuss the main ideas and constructions contained in the cited papers, especially, in the recent ones because they give the most clear and general picture of the categorical duality for CQG coactions. The survey is organized as follows. Section 2 contains necessary definitions, results and notations concerning CQG’s, their representation cat- egories, and also abstract C

-tensor categories and unitary tensor functors, our main reference is [19]. Here we freely use without special explanations the standard language of the tensor category theory [17]. Next we remind the reader of the relations between continuous and algebraic CQG coactions on C

-algebras - see [5], [32] and [6] for ergodic and [18] for nonergodic case.

Section 3 is devoted to the presentation of the categorical duality. First,

to the category of pairs p M, M q , where M is a Rep p G q -module category and M its generator. The discussion is based on the ideas from [6] and uses essentially G-equivariant Hilbert modules introduced earlier in K-theory [1].

Conversely, such a pair p M, M q gives rise to an algebraic G-coaction on a unital -algebra B - see [6], [18], and then to a continuous G-coaction.

These two functors define an equivalence of the categories in question, i.e., the Tannaka-Krein duality for G-coactions. Then we present the alternative construction in terms of weak tensor functors introduced in [18], which are, in the ergodic case, equivalent to spectral functors studied earlier in [24].

If a C

-algebra carries coactions of both, CQG and its dual, compatible in a special way (Yetter-Drinfeld algebra), the Tannaka-Krein duality can be formulated in terms of a unitary tensor functor. The discussion of this situation is based on [21].

Section 4 presents two applications of the Tannaka-Krein duality. The first of them is a characterisation of subalgebras of C p G q invariant with re- spect to coactions of a CQG G and its dual as coideals of quotient type, which was obtained in [21]. Next we briefly explain, in which way the categorical duality was used in [7] in order to reduce the classification of ergodic SU

p 2 q - coactions to the classification of weighted oriented graphs. The discussion of the last problem is beyond the scope of this survey, so we only mention some of the nice results obtained in this way.

We call the triple p B, G, α q , where α is a continuous coaction of a CQG G on a unital C

-algebra B, a G C

-algebra, this terminology is different from the one used in some other papers, for example, in [6], [7], and [32]. We consider only coactions on C

-algebras and do not discuss their von Neumann algebraic counterparts.

All Hilbert spaces and C

-algebras are supposed to be separable.

Notations. For an object X of a category C , we write X P C . The set of

morphisms between X, Y P C is denoted by C p X, Y q except for C Rep p G q ,

where we write M or p X, Y q . If E is a right Hilbert module over a C

-algebra

A, L p E q is the space of all bounded adjointable A-linear operators acting on

E and K p E q is its closed ideal of ”compact” operators [14]. In the context of

C

-algebras, b means the minimal C

-tensor product. The algebraic tensor

product is denoted by b

. M p A q is the multiplier algebra of a C

-algebra

A and r B s is the norm closure of a vector subspace B € A. K p H q is the

C

-algebra of all compact operators on a Hilbert space H, B p H, K q is the

space of all bounded operators between Hilbert spaces H and K . ω

is the

linear functional on B p H q : B p H, H q defined by ω

p A q   Aη, ξ ¡ .

Acknowledgements. I am grateful to Kenny de Commer for helpful dis- cussions on the Tannaka-Krein duality.

2 Preliminaries

2.1 Representation category of a CQG

1. A compact quantum group G [40] - [43] is defined by a unital C

-algebra C p G q equipped with a coproduct, i.e., a unital C

-homomorphism ∆ : C p G q Ñ C p G q b C p G q satisfying the coassociativity p ∆ b id q ∆ p id b ∆ q ∆ and the cancellation properties: rp 1 b C p G qq ∆ p C p G qqs C p G q b C p G q rp C p G q b 1 q ∆ p C p G qqs . There is a unique state h on C p G q satisfying p h b id q ∆ p a q h p a q 1 (and/or p id b h q ∆ p a q h p a q 1), for all a P C p G q , called the Haar state.

If h is faithful, G is called reduced.

A unitary representation of G on a Hilbert space H

is a unitary U P M p K p H

q b C p G qq such that p id b ∆ qp U q U

U

(we use the standard leg notation). In particular, 1 is the trivial representation of G on C , and the unitary W defined by W

p a b b q : ∆ p b qp a b 1 q p a, b P C p G qq is the regular representation of G acting on the GNS Hilbert space L

p G, h q . We have

W

W

W

W

W

and ∆ p a q W

p 1 b a q W,

for all a P C p G q . The tensor product of unitary representations U and V acting on Hilbert spaces H

and H

, respectively, is defined by U j V : U

V

. Let M or p U, V q be the set of intertwiners between U and V :

M or p U, V q : t T P B p H

, H

q| V p T b 1 q p T b 1 q U u .

We say that U and V are equivalent (resp., unitarily equivalent) if M or p U, V q contains an invertible element (resp., a unitary). In general, U j V and V j U are not equivalent. By definition, dim p U q : dim p H

q . U is called irreducible if M or p U, U q C id.

The regular -subalgebra Cr G s € C p G q is defined as the linear span of matrix coefficients u

: p ω

b id qp U q p ξ, η P H

q , where U runs through the finite dimensional unitary representations of G. Denote by ˆ G the (count- able) set of equivalence classes of irreducible unitary representations of G (all of them are finite dimensional), and choose a representative U

of x P G ˆ acting on a Hilbert space H

. As a linear space,

Cr s ` b ` p q

It is known that Cr G s is a Hopf -algebra with coproduct ∆, antipode S and counit ε, it is norm dense in C p G q , and the state h is faithful on Cr G s . Every finite dimensional representation U defines a -representation π

of the algebraic dual of Cr G s acting on H

: µ ÞÑ p id b µ qp U q .

We also denote by Cr G s

the linear span of matrix coefficients of U (in particular, Cr G s

: Cr G s

), and by 1 the class of the trivial representation.

The C

-algebra C

p G q generated by C p G q in the GNS-representation defined by h, is called the reduced form of G. The universal C

-algebraic envelope C

p G q of Cr G s is called the universal form of G.

The C

-algebra of the dual (discrete) quantum group denoted by c

p G ˆ q is isomorphic to c

`

B p H

q and can be viewed as part of the algebraic dual of Cr G s with respect to the duality

  u

, e

¡ δ

δ

δ

, @ x, y P G, ˆ

where u

are matrix coefficients of U

and e

are matrix units in B p H

q with respect to some orthogonal basis in H

.

We say that a CQG G is of Kac type, if either the antipode S of Cr G s satisfies S

id or h is a tracial state. In this case, it is a G.I. Kac algebra in the sense of [10].

2. From an abstract point of view, the category Rep p G q of unitary finite dimensional representations of G with the above mentioned tensor product and morphisms is a semisimple rigid tensor C

-category in the sense of the following definitions (for the details see [19]).

Definition 2.1 A C

-category D is a C -linear category whose morphism spaces D p X, Y q are Banach spaces such that || S T || ¤ || S |||| T || ( is a composition of morphisms), admitting antilinear contravariant conjuga- tion : D p X, Y q Ñ D p Y, X q : T ÞÑ T

, and satisfying the C

-condition

|| T

T || || T ||

for any morphism T . A linear functor between two C

- categories preserving the -operation is called a unitary C

-functor. An object X P D is called simple if End p X q : D p X, X q C . D is called semisimple if it admits finite direct sums and if any its object is isomorphic to a finite direct sum of simples.

Definition 2.2 A strict C

-tensor category C p D, b , 1 q is a C

-category

D with a bilinear C

-functor b : D D Ñ D and an object 1 P D, which

is also a strict tensor category in the sense of [17]. We say that an object

X P D admits a conjugate or dual if there exists a triple p X, R

, R

q , where

X P C and R

: 1 Ñ X b X, R

: 1 Ñ X b X are morphisms satisfying the conjugate equations:

p R

b id

qp id

b R

q id

, p R

b id

qp id

b R

q id

. The category C is called rigid if any its object admits a conjugate.

Remark 2.3 (see [19], 2.2). A rigid C

-tensor category C is automatically semisimple and End p X q is finite dimensional, for all X P C. If X is a simple object of such a category, then the number dim

p X q || R

|||| R

||

is independent on the solution of the conjugate equations and is called the quantum (or intrinsic) dimension of X. In general, if X `

X

with X

simples, put dim

p X q : Σ

dim

p X

q .

Example 2.4 1. The category H of Hilbert spaces with H p H, K q : B p H, K q , 1 C and usual tensor product, is a C

-tensor category. Its maximal rigid subcategory H

is formed by finite dimensional Hilbert spaces. If H P H

, its complex conjugate space H gives a dual,

R

: H b H ÞÑ C : ζ b η ÞÑ  η, ζ ¡ , R

: H b H ÞÑ C : ζ b η ÞÑ  ζ, η ¡ . Here   ζ, η ¡ is the scalar product in H. Then dim

p H q dimH, @ H P H

.

2. Rep p G q with intertwiners as morphisms, and j as a tensor product, is a semisimple rigid C

-tensor category. If U P Rep p G q , its conjugate is

U : p j p π

p ρ q

q b 1 qp j b id qp U

qp j p π

p ρ q

q b 1 q P B p H

q b C p G q ,

R

p 1 q : Σ

p ζ

b π

p ρ

qp ζ

qq , R

p 1 q : Σ

p π

p ρ

qp ζ

q b ζ

q , (2) where ρ f

is the Woronowicz character [40], j : B p H

q Ñ B p H

q is the canonical -anti-isomorphism defined by j p T q ζ T

ζ and t ζ

u

is an orthonormal basis in H

(see [19]). Note that dim

p U q ¥ dim p H

q .

3. A correspondence over a C

-algebra A is a right Hilbert A-module E

(see [14]) with a -homomorphism ϕ : A Ñ L p E q such that ϕ p A qp E q is dense

in E . A-correspondences form a C

-tensor category Corr p A q with tensor

product b

and adjointable A-bilinear maps as morphisms [18].

Definition 2.5 A unitary tensor functor F : C

Ñ C

between C

-tensor categories is a -functor C

Ñ C

which is also a linear tensor functor whose natural transformations F p X q b F p Y q Ñ F p X b Y q and 1

Ñ F p 1

q are unitary. It is called a tensor C

-equivalence if F is an equivalence.

CQG’s G and G

are said to be monoidally equivalent if Rep p G q Rep p G

q . Example 2.6 1. Let G be a CQG and F : Rep p G q Ñ H

the forgetful functor f : U ÞÑ H

(H

is the underlying Hilbert space of U ) acting as the identity on morphisms. The above transformations are identities and F is a faithful unitary tensor functor.

The Woronowicz’s Tannaka-Krein duality claims that any rigid C

-tensor category admitting a unitary tensor functor to H

(called a fiber functor), is equivalent to the category Rep p G q of some CQG G. The corresponding Hopf -algebra Cr G s is unique up to isomorphism - see [19], Theorem 2.3.2.

2. By a closed quantum subgroup of a CQG G we mean a CQG H together with a surjective homomorphism P

: Cr G s Ñ Cr H s of Hopf -algebras (this definition is slightly less restrictive than the one in [26]). Then the restriction functor Rep p G q Ñ Rep p H q is a unitary tensor functor.

Any CQG G has a unique maximal Kac quantum subgroup, i.e., a CQG of Kac type K and a surjective homomorphism P

as above, and if p H, P

q is any other closed quantum subgroup of Kac type of G, then P

factors through P

[29]. Indeed, the ideal I € Cr G s generated by the elements a S

p a q , for all a P Cr G s , is clearly a Hopf -ideal. Hence the quotient Cr K s : Cr G s{ I is a unital Hopf -algebra with involutive antipode, so it defines a closed quantum subgroup K of Kac type of G. Clearly, K is maximal in the above sense, and Cr K s is called the canonical Kac quotient of Cr G s .

Definition 2.7 [19] The fusion ring of a rigid tensor C

-category C is the universal ring R p C q generated by the equivalence classes of its objects with operations of the direct sum and of the tensor product.

Let I be the set of the equivalence classes of simple objects of C . Choose a representative U

in every class, then the fusion rule is the decomposition

U

b U

`

N

U

@ ζ, η P I,

where the multiplicities N

P Z , only finitely many nonzero. The conjuga- tion α ÞÑ α is a bijection of I which extends to a Z -linear anti-multiplicative involution of R p C q . We also have the Frobenius reciprocity:

N

N

ζ,α

N

@ ζ, η, α P I,

which follows from the Frobenius reciprocity for morphisms in C [19], Theo- rem 2.2. The map dim

: I Ñ r 1, 8r is such that d

p ζ q d

p ζ q and extends to a Z -linear multiplicative map R p C q Ñ R .

Remark 2.8 A fusion ring R p C q is an example of a hypercomplex system with discrete basis in the sense of Yu.M. Berezanskij and A.A. Kaljuzhnyi [2] (also called a hypergroup) with the structural constants

C

N

d p α q d p ζ q d p η q .

2.2 G C

-algebras

Definition 2.9 A unital left G C

-algebra is a triple p B, G, α q , where B is a unital C

-algebra and α : B Ñ C p G q b B is a left continuous coaction of a CQG G on B, i.e., an injective unital -homomorphism such that p id b α q α p ∆ b id q α and rp C p G q b 1 q α p B qs C p G q b B. The C

-subalgebra B

: t b P B | α p b q 1 b b u is called the fixed point subalgebra of B .

An algebraic coaction of a CQG G on a unital -algebra B is a Hopf - algebra coaction α : B Ñ Cr G s b

B such that p ε b

id q α id, the fixed point subalgebra B

: t b P B | α p b q 1 b b u is a unital C

-algebra, and the map E p h b id q α : B Ñ B

is completely positive. So, B is a right pre-Hilbert B

-module with inner product   a, b ¡ E p a

b q . In this case, we call p B, G, α q a unital left G -algebra.

Right G C

-algebras and G -algebras are defined similarly.

For any U P Rep p G q , H

is a left Cr G s -comodule via δ

: ξ

ÞÑ Σ

u

b ξ

, where t ξ

u is an orthonormal basis in H

. Given a unital G C

-algebra B, the spectral subspace B

of B is the linear span of the images of all comodule maps S : H

Ñ B. Put B

B

, so B

B

and α p B

q € Cr G s

b

B

with Cr G s

spanned by matrix coefficients of U

. It was proved essentially in [5], [26] that B

B

€ span t B

u , @ x, y P G, where ˆ z are the classes of direct summands U

of U

j U

and that B : `

B

is a unital -algebra B norm dense in B. It is also characterized either as B : t b P B | α p b q P Cr G s b

B u or, equivalently, as B : tp h b id qrp a b 1 q α p b qs| @ a P Cr G s , b P B u . The conditional expectation E as above is faithful on B and clearly E p B q E p B q . Thus, B is a unital G -algebra called the regular subalgebra of B .

Conversely, see [6], Proposition 4.4:

Proposition 2.10 Given an algebraic coaction α of a CQG G on a unital -algebra B, there is a unique C

-completion B of B to which α extends as a coaction of the reduced form of C p G q with the same fixed point subalgebra.

Indeed, for any b P B, there are U P Rep p G q , an intertwiner S : H

Ñ B, and ξ P H

such that b Sξ, so there are b

P B and z

P C such that b Σ

z

b

and α p b

q Σ

u

b b

. Then one shows that Σ

b

b

P B

which implies that all b

and b can be faithfully represented by left multiplication operators on the right pre-Hilbert B

-module B, which defines a C

-completion B r B s . Further, one shows that the map V defined on Cr G s b B by V p a b b q : α p b qp a b 1 q extends to a unitary on a Hilbert B

-module L

p G, h q b B, and that α p b q V p 1 b b q V

, so α extends to a continuous coaction of G on B . Finally, if ˜ B is any C

-completion of B for which the assertion of the proposition holds, then the map E as above is faithful because h is faithful on C p G q . Then ˜ B is presented faithfully on the Hilbert B

-module B , from where the unicity follows.

A coaction α is said to be universal (resp., reduced) if B is the universal enveloping C

-algebra of B (resp., E is faithful on B). In particular, if B C p G q and α ∆, we have B Cr G sq which leads to the usual notions of universal and reduced forms of G.

2.3 The ergodic case

A G C

-algebra p B, G, α q is called ergodic if the coaction α is ergodic, i.e., if B

C 1

. In this case, the state ω on B defined by ω p b q 1

E p b q , for all b P B, is the unique α-invariant state (i.e., such that p id b ω q α p b q ω p b q 1, @ b P B), the spectral subspaces are finite dimensional and orthogonal for both scalar products ω p b

c q and ω p cb

q p b, c P B q [5]. An ergodic coaction is reduced if and only if the above ω is faithful on B.

Example 2.11 Let G be a CQG and H its closed quantum subgroup (see Example 2.6, 2). Then the norm closure C p G { H q of the -algebra Cr G { H s : t x P Cr G s|p id b P q ∆ p x q x b 1 u is called an algebra of continuous functions on the quantum homogeneous space G { H. The triple p C p G { H q , G, ∆ |

q is an ergodic G C

-algebra. Indeed, C p G { H q

is the same for both, continuous and algebraic coactions, so it suffices to consider b P Cr G { H s

for which we have p id b P q ∆ p b q b b 1 and ∆ p b q 1 b b, so b P C 1.

However, there is an example of a commutative ergodic G C

-algebra,

where α is not a quotient action as above [36].

Let p B, G, α q be an ergodic G C

-algebra and x P G. We call ˆ dim p B

q the multiplicity of x in α. It is known [5] that dim p B

q ¤ dim

p x q .

3 Categorical duality

3.1 Tannaka-Krein reconstruction in terms of Rep p G q - module categories

Definition 3.1 [6] Let C be a C

-tensor category with unit object 1. A C

-category M is called a right C -module C

-category if there is a bilinear - functor b : M C Ñ M with natural unitary transformations M bp X b Y q Ñ p M b X qb Y and M b 1 Ñ M p X, Y P C, M P M q making M a right module category over C viewed as tensor category, i.e., making some pentagonal and triangular diagrams commutative - see [23]. We say that M is strict (resp., indecomposable) if these natural transformations are identities (resp., if, for all non-zero M, N P M, there is X P C such that M p M b X, N q 0).

We say that an object M P M generates M if any object of M is isomor- phic to a subobject of M b X for some X P C . M is said to be semisimple if the underlying C

-category is semisimple.

We will always consider C

-categories closed with respect to subobjects, i.e., such that for any object M and any projection p P End p M q , there are an object N and isometry v P M p N, M q satisfying p vv

(if necessary, one can complete given C

-category with respect to subobjects).

One naturally defines a morphism F : M

Ñ M

between two C-module C

-categories as a morphism of the underlying C

-categories equipped with a unitary natural equivalence F p M b X q Ñ F p M q b X, @ X P C , M P M satisfying some coherence conditions (see [6], 2.17).

Example 3.2 Let F : C Ñ D be a unitary tensor functor. Then M D has a structure of a C-module C

-category with M b X : M b F p X q , for all X P C, M P D. For particular examples of this kind see Example 2.6.

Definition 3.3 [1]. Let p G, α, B q be a unital G C

-algebra . A G-equivariant Hilbert B -module is a right Hilbert B -module E with a coaction α

: E Ñ C p G q b E (C p G q b E is considered as a right Hilbert C p G q b B -module), such that rp C p G q b 1 q α

p E qs C p G q b E , and

p q p q p q @ P P

2.   α

p ζ q , α

p η q ¡

α p  ζ, η ¡

q , @ ζ, η P E . E is called irreducible if dim p L

p E qq 1, where

L

p E q : t T P L p E q| α

p T ζ q p 1 b T q α

p ζ q , @ ζ P E u .

Example 3.4 Let M

be the category of finitely generated G-equivariant right Hilbert B-modules, its morphisms are G-equivariant maps of Hilbert B -modules (they are automatically adjointable since Hilbert B-modules are finitely generated). If E P M

and U P Rep p G q , construct a new object E b U P M

by taking the right Hilbert B-module H

b E with the coaction

˜

α : H

b E Ñ C p G q b p H

b E q : ˜ α p ζ b η q U

p ζ b α

p η qq

. One can check that p E b U qb V E bp U j V q and that this operation is natural in both U and E , so it defines a strict right Rep p G q -module C

-category structure on M

. If α is ergodic, M

is semisimple and indecomposable [6], Proposition 3.11. The equivariant version of Kasparov’s stabilisation theorem (see [35], 3.2 or [20], Lemma 3.2) shows that B is its generator.

The next theorem has been proved in ergodic case in [6], and then in general case in [18]:

Theorem 3.5 Let G be a reduced CQG. Then the following two categories are equivalent:

(i) The category G Alg of unital G C

-algebras with unital G-equivariant -homomorphisms as morphisms.

(ii) The category Rep p G q M od of pairs p M, M q , where M is a right Rep p G q -module C

-category and M is its generator, with equivalence classes of unitary Rep p G q -module functors respecting the generators as morphisms.

The coactions in (i) are ergodic if and only if the Rep p G q -module C

- categories in (ii) are semisimple, indecomposable, and with simple generators.

The idea of the proof is as follows (for the details see [6], [18]).

1. The construction of the functor T : G Alg Ñ Rep p G q M od on the

level of objects was explained in Example 3.4: given B P G Alg, construct

the category M

with generator B. Now, if f : B

Ñ B

is a morphism in

G Alg, then the morphism T p f q : M

Ñ M

is given by E ÞÑ E b

B

.

2. Next, in order to define the functor S : Rep p G q M od Ñ G Alg p G q ,

we have to construct a unital G C

-algebra B from a given pair p M, M q .

According to Proposition 2.10, it suffices to construct a unital G -algebra B.

Looking at the construction (1) of Cr G s , considering H

as a Rep p G q -module category with generator C via the forgetful fiber functor U ÞÑ H

, @ U P Rep p G q (see Example 3.2), and identifying H

with H

pC , H

q , it is natural to define B as a vector space by

B `

p H

b M p M, M b U

qq .

The next reasoning follows [18], [21]. In order to simplify notations, it is convenient to replace M by an equivalent Rep p G q -module C

-category N as follows. Identify M with 1 P Rep p G q , M b U with 1 b U U , and define N p U, V q : M p M b U, M b V q , @ U, V P Rep p G q . We have also to complete N with respect to subobjects, so that N can be considered as the completion of Rep p G q with larger morphism spaces than in Rep p G q . In particular, the generator 1 is not necessarily simple in N , and the definition of B gives:

B `

p H

b N p 1, U

qq . (3) Now, it is convenient to introduce a much larger auxiliary vector space:

B ˜ `

p H

b N p 1, U qq , (4) which is an associative unital algebra with respect to the product defined by p ζ b T qp η b S q p ζ b η qbp T b id q S, @ ζ P H

, η P H

, T P N p 1, U q , S P N p 1, V q . Note that p T b id q S P N p 1, U j V q . Define now an antilinear map : ˜ B Ñ B ˜ :

p ζ b T q

: p id b ζ q R

p 1 q b p T

b id q R

,

where R

and R

were defined in (2). Note that p T

b id q R

P N p 1, U q . Then, for any U P Rep p G q , choose isometries w

: H

Ñ H

defining the decomposition of U into irreducibles and define the projection p : ˜ B Ñ B by

p p ζ b T q Σ

p w

ζ b w

T q , @ ζ P H

, T P N p 1, U q , (5) which is independent of the choice of w

. One checks that B is a unital - algebra with the prduct p p b q p p c q : p p bc q and the involution p p b q

: p p b

q , for all b, c P B. It is also a ˜ Cr G s -comodule via the map α : B Ñ Cr G s b B defined as follows. If t ξ

u is an orthonormal basis in H

(U P Rep p G q ), put

p p b qq p b p b qq p b q

One shows that α is an algebraic coaction with B

End

p 1 q End

p M q that gives rise to a continuous G-coaction. Clearly, this construction allows to get a morphism in G Alg from a given morphism in Rep p G q M od.

3. Equivalence. In order to explain the equivalence of the Rep p G q - module categories N and M

, replace the last one by an equivalent Rep p G q - module category N

, which is a completion of Rep p G q with respect to subob- jects, with generator 1. Now it suffices to explain why N

p U, V q N p U, V q , for all U, V P Rep p G q . One can check, as in [18], that the first of them con- sists of elements S P B p H

, H

qb B, such that V

p id b α qp S q U

S

, and if T P N p U, V q , the needed isomorphism can be given explicitly by:

S Σ

θ

π

p ρ

q b p pp ζ

b η

q b p T b id q R

q ,

where t ζ

u and t η

u are orthonormal bases in H

and H

, respectively, and θ

p η q :   η, η

¡ ζ

is a rank 1 operator from H

to H

(see [21]).

Example 3.6 1. The regular coaction ∆ of G on C p G q corresponds to the Rep p G q -module category M H

with the generator C via the forgetful fiber functor U ÞÑ H

. Namely, identifying M pC , H

q with H

, we see that the algebras B ˜ and B constructed from the pair p H

, Cq are isomorphic to Cr ‚ G s `

p H

b H

q and to Cr G s , respectively. Then the map p : ˜ B Ñ B turns into the map p

: Cr ‚ G s Ñ Cr G s sending ζ b η (where ζ, η P H

) to the matrix coefficient p ω

b id q U of U.

2. If G is a CQG and H its closed quantum subgroup H (see Example 2.6, 2), the G C

-algebra C p G { H q corresponds to the category Rep p H q with generator 1 viewed as a Rep p G q -module category via the restriction tensor functor Rep p G q Ñ Rep p H q . Namely, identifying M or

pC , H

q with a sub- space of H

, we can view the corresponding algebra B ˜ as a subalgebra of Cr ‚ G s . Then the above map p

induces a G-equivariant isomorphism B Cr G { H s .

3.2 Spectral functors

Let p B, G, α q be a unital G C

-algebra and U P Rep p G q . Consider the tensor product comodule H

b B and its subcomodule of G-invariant vectors

p H

b B q

: t X Σ

p ξ

b b

q| α p b

q Σ

u

b b

u ,

where t ξ

u

is an orthonormal basis in H

and u

are the matrix coefficients

of U with respect to this basis. The spectral subspaces can be recovered from

p H

b B q

using the canonical surjective maps

H

b p H

b B q

Ñ B

: ξ b X ÞÑ p ξ b id qp X q ,

which are isomorphisms for irreducible U . The space p H

b B q

is a B

- bimodule and a right Hilbert B

-module with inner product   X, Y ¡ Σ

b

c

for X Σ

p ξ

b b

q and Y Σ

p ξ

b c

q (note that the element   X, Y ¡ is in B

and is independent on the choice of t ξ

u

- we will write it as X

Y ). So, p H

b B q

P Corr p B

q (see Example 2.4, 3).

Remark also that for all U, V P Rep p G q , the map X b Y ÞÑ X

Y

is an isometry from p H

b B q

b

p H

b B q

to p H

b B q

.

Definition 3.7 [18] Given a G C

-algebra p B, G, α q , the associated spectral functor is the unitary functor F : Rep p G q Ñ Corr p B

q defined by

F p U q p H

b B q

, for all U P Rep p G q ,

F p T q T b id, for all morphisms, together with the B

-bilinear isometries F

p U, V q : F p U q b

F p V q Ñ F p U j V q : X b Y ÞÑ X

Y

.

The spectral functor is not, in general, a tensor functor because the maps F

are not surjective. It was abstractly characterized in [18], Definition 2.1 as a weak tensor functor F : C Ñ Corr p A q (where C is a strict C

-tensor category and A is a C

-algebra). The only difference between such F and a unitary tensor functor is that F

are not isomorphisms but just A-bilinear isometries satisfying some coherence condition.

Let us explain the relation between Rep p G q -module C

-categories and spectral functors. Let G be a reduced QCG G and M be a strict right Rep p G q -module C

-category with generator M . Put A End p M q and con- sider the functor F : Rep p G q Ñ Corr p A q defined by F : U Ñ M p M, M b U q with the right and left A-module structures on F p U q given by, respectively, composition of morphisms and by aX p a b id q X, and the inner product   X, Y ¡ X

Y . The action of F on morphisms is defined by F p T q X p id b T q X, and F

: F p U q b

F p V q Ñ F p U b V q is given by X b Y ÞÑ p X b id q Y . Then it was shown in [18], Theorem 2.2 that p B

, G, α

q is a unital G C

-algebra, where

B ` H b F p U

q , α p ξ b X q Σ p u b ξ b X q .

Here t ξ

u

is an orthonormal basis in H

and u

are the corresponding matrix coefficients of U

. Moreover, A B

, M M

with M B, and the functor F is a spectral functor (there are canonical isomorphisms p H

b B q

M

p B, B b U q sending Σ

p ξ

b b

q to the morphism b ÞÑ Σ

p b

b b ξ

q ).

The module B b U is just H

b B discussed above.

Remark 3.8 1. The main part of the proof of [18], Theorem 2.2 is the con- struction of a G C

-algebra p B

, G, α

q from a given weak tensor functor F which can be viewed as another version of the Tannaka-Krein reconstruction.

It follows the lines going back to [43].

2. In the ergodic case, the definition of a weak tensor functor is equivalent to the one of a quasitensor functor introduced earlier in [24]. An example of such a functor is the functor sending any x P G ˆ to the corresponding spectral subspace - see [24], Definition 7.2 and Theorem 7.3. These results imply that isomorphism classes of ergodic coactions of monoidally equivalent CQG’s are in canonical correspondence with each other.

In particular, when F

p U, V q are unitary, F is a unitary tensor functor and the corresponding ergodic coaction is said to be of full quantum multi- plicity. This situation was studied systematically in [4] in other terms.

3.3 Yetter-Drinfeld G C

-algebras [21]

Yetter-Drinfeld (YD) C

-algebras constitute a special class of G C

-algebras for which one can formulate the categorical duality in terms of a unitary tensor functor. Let p B, G, α q be a unital G C

-algebra equipped with a continuous right coaction β : B Ñ M p B b c

p G ˆ qq of the dual discrete quantum group ˆ G, where c

p G ˆ q is viewed as a subalgebra of the algebraic dual of Cr G s . Similarly to the definition of a YD algebra for a general locally compact quantum group G given in [22], Definition 3.1, we will say that p B, G, α, β q is a unital YD C

-algebra, if

p α b id q β p b q W

p id b β q α p b q W

,

where W is the fundamental unitary of G and b P B. However, it is more convenient to formulate this condition, like in [21], in terms of the left action

™ : Cr G s b

B Ñ B defined by u ™ b p id b u q β p b q , for all u P Cr G s , b P B.

Then B is a left Cr G s -module unital -algebra, i.e., for all u P Cr G s , b, c P B:

u ™ p bc q p u

™ b qp u

™ c q , u ™ b

p S p u q

™ b q

, u ™ 1 ε p u q 1. (6)

Definition 3.9 We say that p B, G, α, ™q is a YD C

-algebra if, for all u P Cr G s and b from the regular subalgebra B:

α p u ™ b q u

b

S p u

q b p u

™ b

q . (7) We used here the Sweedler’s leg notations: ∆ p u q u

b u

, α p b q b

b b

. A YD C

-algebra is said to be braided-commutative if

ab b

p S

p b

q ™ a q , for all a, b P B. (8) The coaction β can be reconstructed from the action ™ as follows:

Proposition 3.10 ([21], Proposition 1.3.) Let p B, G, α q be a G C

-algebra, where G is a reduced CQG, such that B is a left Cr G s -module unital - algebra and (7) holds. Then there is a unique continuous right coaction β : B Ñ M p B b c

p G ˆ qq such that u ™ b : p id b u q β p b q , for all u P Cr G s , b P B.

Indeed, if U Σ

p e

b u

q P Rep p G q and b P B, the map β

p b q : Σ

p u

™ b qb e

defines a unital -homomorphism B Ñ B b B p H

q satisfying

U

p α b id q β

p b q U

p id b β

q α p b q .

Then the map y ÞÑ U

p α b id qp y q U

extends to a continuous left coaction of G on r β

p B qs P B b B p H

q . This implies that α viewed as an algebraic coaction of G on B, extends to a continuous coaction of G on the completion of B with respect to the C

-norm max t|| β

pq|| , || ||u , but due to the unicity statement in Proposition 2.10, this completion equals to B .

Since c

p G ˆ q c

`

B p H

q , the family of homomorphisms β

p x P G ˆ q define a unital -homomorphism β : B Ñ M p B b c

p G ˆ qq l

`

p B b B p H

qq such that p id b u q β p b q u ™ b for all u P Cr G s , b P B. Finally, one checks that it is a continuous coaction, and the unicity is clear.

Remark 3.11 1. One can regard a YD G C

-algebra as a D p G q C

- algebra for the Drinfeld double D p G q of G - see [22], Proposition 3.2.

2. If b P B

, then (8) gives ab ba, so B

€ Z p B q .

Example 3.12 Let G be a CQG and H its closed quantum subgroup (see Example 2.6, 2). Then B C p G { H q is a braided commutative YD G C

-

| ™ p q Cr s

Let us formulate the Tannaka-Krein duality for Yetter-Drinfeld G C

- algebras.

Theorem 3.13 ([21], Theorem 2.1). Let G be a reduced compact quantum group. Then the following two categories are equivalent:

(i) The category Y D

p G q of unital braided-commutative YD G C

- algebras with unital G- and G-equivariant ˆ -homomorphisms as morphisms.

(ii) The category T ens p Rep p G qq of pairs p C , F q , where C is a tensor C

- category and F : Rep p G q Ñ C is a unitary tensor functor such that C is generated by the image of F . Its morphisms p C, F q Ñ p C

, F

q are equivalence classes of pairs p F , η q , where F : C Ñ C

is a unitary tensor functor and η : F p F q Ñ F

is a natural unitary monoidal isomorphism of functors.

Moreover, given a morphism p F , η q : p C , F q Ñ p C

, F

q , the corresponding homomorphism of YD G C

-algebras is injective (resp., surjective) if and only if F is faithful (resp., full).

Here is the idea of the proof (for the corresponding calculations see [21]).

1. Functor T ˜ : Y D

p G q Ñ T ens p Rep p G qq . Apply the functor T from the proof of Theorem 3.5 to B P Y D

p G q and turn the corresponding category M

into a tensor C

-category by showing that any M P M

is an object of the tensor C

-category Corr p B q - see Example 2.4, 3. As B is a generator of M

, it suffices to consider M H

b B, @ U P Rep p G q , and to construct a non-degenerate -homomorphism ϕ

: B Ñ End

p H

b B q B p H

q b B. As B P Y D

p G q , one can check that the map

b ÞÑ ϕ

p b q : Σ

e

b p u

™ b q ,

where b P B, U Σ

e

b u

and e

are the matrix units in B p H

q re- lated to an orthonormal basis t e

u in H

, gives the needed -homomorphism.

Here ϕ

p b q P L p M q acts on any regular vector ζ P M (i.e., such that α

p ζ q P Cr G s b

M q as follows: ϕ

p b q ζ ζ

p S

p ζ

q ™ b q . We have also α

p ϕ

p b q ζ q p id b ϕ

p b qq α p b q α

p ζ q - see [21], Lemma 2.2.

Thus, M is a G-equivariant B-correpondence. Moreover, if M, N P M

, then M b

N P M

, and C p M

, b

, B q is a full tensor C

-subcategory of Corr p B q (see [21], Theorem 2.4). Note that C is generated by the image of the unitary tensor functor F : Rep p G q Ñ M

: U ÞÑ H

b B.

If f : B

Ñ B

is a morphism in Y D

p G q , the morphism T p f q : M

Ñ

M

maps any M to M b

B

(see the proof of Theorem 3.5), and the

tensor structure of T is described by the isomorphisms

p M b

B

q b

p N b

B

q p M b

N q b

B

, for all M, N P M

, which are defined, for all a, b P B

, and regular ξ P M, ζ P N , as follows:

p ξ b a q b p ζ b b q ÞÑ ξ b ζ

b p S

p ζ

q ™ a q b.

Together with the obvious isomorphisms η

: p H

b B

q b

B

Ñ H

b B

, we define a morphism p M

, F

q Ñ p M

, F

q .

2. Functor S ˜ : T ens p Rep p G qq Ñ Y D

p G q . Conversely, given a pair p C , F q as above, C has a structure of a singly generated Rep p G q -module C

- category coming from F . As in the proof of Theorem 3.5, identify C with a completion N of Rep p G q , and its generator with 1. The functor F is automatically faithful because the category Rep p G q is rigid (indeed, F p 1 q is the direct summand of F p U q b F p U q , for any U P Rep p G q ). So we can assume that it is just the embedding functor.

Now, define a Cr G s -module algebra structure on B (see (3)) using the algebra homomorphisms p : B r Ñ B from (5) and p

: Cr ‚ G s Ñ Cr G s from Example 3.6. First, define a linear map ˜ ™ : Cr ‚ G s b r B Ñ r B by

p ξ b ζ q ™p ˜ η b T q p ξ b η b π

p ρ

qp ζ qq b p id b T b id q R

,

where ξ, ζ P H

, η P H

, U, V P Rep p G q , T P N p 1, V q , R

was introduced in (2). Note that p id b T b id q R

P N p 1, U j V j U q . Then define an action ™ : Cr G s b B Ñ B by p

p u q ™ p p b q : p p u ™ ˜ b q , for all u P ‚ Cr G s and b P B. Finally, one shows by direct but tedious calculations that the norm ˜ completion B of B is in Y D

p G q .

Considering the category of pairs p C , F q as above, we may assume that the restriction of any morphism F : C Ñ C

on Rep p G q is the identical tensor functor, and that η id. Then the construction of the algebras B and B

shows that the maps H

b C p 1, U q Ñ H

b C

p 1, U q , defined by ξ b T ÞÑ ξ b F p T q , give a unital -homomorphism B Ñ B

that respects the Cr G s -comodule and Cr G s -module structures. It extends to a homomorphism f of C

-algebras by [6], Prop. 4.5. The above definition of morphisms in the category of pairs p C, F q shows that f depends only on the equivalence class of p F , η q , and this gives the needed functor.

Furthermore, it is clear from the construction that the morphism f : B Ñ B

defined by a morphism p F , η q : p C , F q Ñ p C

, E

q is injective (resp., surjec-

p q Ñ

p q ÞÑ p q

(resp., surjective), for all x P G. It follows from the Frobenius reciprocity for ˆ morphisms and the fact that the categories C and C

are generated by Rep p G q , that f is injective (resp., surjective) if and only if F is faithful (resp., full).

Finally, it can be proved by direct calculations that the functors ˜ T and S ˜ define an equivalence of the categories in question.

Example 3.14 If H is a closed quantum subgroup of a CQG G, then B C p G { H q P Y D

p G q with α ∆ |

and u ™ b u

bS p u

q - see Exam- ple 3.12. It corresponds to the tensor C

-category Rep p H q with generator 1 viewed as a Rep p G q -module category via the restriction unitary tensor functor Rep p G q Ñ Rep p H q - see Example 3.6. Indeed, identifying M or

pC , H

q with a subspace of H

, we can view the corresponding algebra B ˜ as a sub- algebra of Cr ‚ G s `

p H

b H

q . Then the map p

: Cr ‚ G s Ñ Cr G s induces a G-equivariant isomorphism B Cr G { H s .

Conversely, the action of Cr G s on Cr G { H s defined by the tensor functor Rep p G q Ñ Rep p H q is exactly the adjoint one. Indeed, it suffices to consider the trivial H corresponding to the functor Rep p G q Ñ H

, while the general case corresponds to the intermediate functor Rep p G q Ñ Rep p H q . Let U, V P Rep p G q , t ξ

u P H

and t η

u P H

be orthonormal bases, and u

(resp., v

) their matrix coefficients. Then the definitions of the maps ™ ˜ and ™ imply

u

™ v

Σ

u

v

S p u

q (see [21], 3.1), which is exactly our claim.

For more examples of braided commutative YD G C

-algebras see [22].

4 Some applications

4.1 Invariant subalgebras of compact quantum groups

Theorem 4.1 [21], [27], [29] Let G be a CQG. Then any unital left G- and right G-invariant ˆ C

-subalgebra B of C p G q is of the form C p G { H q for some closed quantum subgroup H of G.

To prove this, note that B is a subobject of C p G q in Y D

p G q , and applying the functor ˜ T to this inclusion, we get a morphism in T ens p Rep p G qq :

p C

, F

q Ñ p C

, F

q p H

, F q ,