# Partial compact quantum groupsAcademic research paper on "Mathematics"

CC BY-NC-ND
0
0
Share paper
Journal of Algebra
OECD Field of science
Keywords
{"Hopf face algebras" / "Tannaka reconstruction" / "Dynamical quantum groups"}

## Abstract of research paper on Mathematics, author of scientific article — Kenny De Commer, Thomas Timmermann

Abstract Compact quantum groups of face type, as introduced by Hayashi, form a class of quantum groupoids with a classical, finite set of objects. Using the notions of weak multiplier bialgebras and weak multiplier Hopf algebras (resp. due to Böhm–Gómez-Torrecillas–López-Centella and Van Daele–Wang), we generalize Hayashi's definition to allow for an infinite set of objects, and call the resulting objects partial compact quantum groups. We prove a Tannaka–Kreĭn–Woronowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion C⁎-categories. As examples, we consider the dynamical quantum SU ( 2 ) -groups from the point of view of partial compact quantum groups.

## Academic research paper on topic "Partial compact quantum groups"

﻿Contents lists available at ScienceDirect

Journal of Algebra

www.elsevier.com/locate/jalgebra

Partial compact quantum groups

Kenny De Commer a'*, Thomas Timmermann b

a Department of Mathematics, Vrije Universiteit Brussel, VUB, B-1050 Brussels, Belgium

b University of Munster, Einsteinstrasse 62, 4814-9 Munster, Germany

I CrossMark

a r t i c l e i n f o a b s t r a c t

Article history: Received 24 September 2014 Available online 29 May 2015 Communicated by Nicolás Andruskiewitsch

MSC: 81R50 20G42 16T15

Keywords: Hopf face algebras Tannaka reconstruction Dynamical quantum groups

Compact quantum groups of face type, as introduced by Hayashi, form a class of quantum groupoids with a classical, finite set of objects. Using the notions of weak multiplier bialgebras and weak multiplier Hopf algebras (resp. due to Bohm—Gomez-Torrecillas—Lopez-Centella and Van Daele— Wang), we generalize Hayashi's definition to allow for an infinite set of objects, and call the resulting objects partial compact quantum groups. We prove a Tannaka—Krem—Woro-nowicz reconstruction result for such partial compact quantum groups using the notion of partial fusion C*-categories. As examples, we consider the dynamical quantum su (2)-groups from the point of view of partial compact quantum groups.

Introduction

The concept of face algebra was introduced by T. Hayashi in [13], motivated by the theory of solvable lattice models in statistical mechanics. It was further studied in [14-20], where for example associated *-structures and a canonical Tannaka duality were developed. This Tannaka duality allows one to construct a canonical face algebra from

* Corresponding author.

E-mail addresses: kenny.de.commer@vub.ac.be (K. De Commer), timmermt@uni-muenster.de (T. Timmermann).

http://dx.doi.org/10.1016/j.jalgebra.2015.04.039

any (finite) fusion category. For example, a face algebra can be associated to the fusion category of a quantum group at root unity, for which no genuine quantum group implementation can be found.

In [32,36,37], it was shown that face algebras are particular kinds of x^-algebras [40] and of weak bialgebras [5,3,31]. More intuitively, they can be considered as quantum groupoids with classical, finite object set. In this article, we want to extend Hayashi's theory by allowing an infinite (but still discrete) object set. This requires passing from weak bialgebras to weak multiplier bialgebras [4]. At the same time, our structures admit a piecewise description by what we call a partial bialgebra, which is more in the spirit of Hayashi's original definition. In the presence of an antipode, an invariant integral and a compatible *-structure, we call our structures partial compact quantum groups.

The passage to the infinite object case requires extra arguments at certain points, as one has to impose the proper finiteness conditions on associated structures. However, once all conditions are in place, many of the proofs are similar in spirit to the finite object case.

Our main result is a Tannaka-Krein-Woronowicz duality result which states that partial compact quantum groups (with finite hyperobject set) are, up to the appropriate notion of equivalence, in one-to-one correspondence with concrete semisimple rigid tensor C*-categories. Here we do not assume the unit of the tensor C*-category to be irreducible - this situation can also be dealt with using the notion of C*-bicategory introduced in [21]. By a concrete tensor C*-category we mean a tensor C*-category realized inside a category of (locally finite-dimensional) bigraded Hilbert spaces. Of course, Tannaka reconstruction is by now a standard procedure. For closely related results most relevant to our work, we mention [48,35,16,33,12,39,34,9,29] as well as the surveys [22] and [30, Section 2.3].

As an application, we generalize Hayashi's Tannaka duality [16] (see also [33]) by showing that any module C* -category over a semisimple rigid tensor C* -category has an associated canonical partial compact quantum group. By the results of [9], such data can be produced from ergodic actions of compact quantum groups. In particular, we consider the case of ergodic actions of SUq (2) for q a non-zero real. This will allow us to show that the construction of [14] generalizes to produce partial compact quantum group versions of the dynamical quantum SU(2)-group [11,24], see also [38] and the references therein. This construction will provide the right setting for the operator algebraic versions of these dynamical quantum SU (2)-groups, which was the main motivation for writing this paper. These operator algebraic details will be studied elsewhere [7].

The precise layout of the paper is as follows.

The first section introduces the basic theory of the structures which we will be concerned with. We introduce the notions of a partial bialgebra, partial Hopf algebra and partial compact quantum group, and show how they are related to the notion of a weak multiplier bialgebra [4], weak multiplier Hopf algebra [46,45] and compact quantum group of face type [14]. We also briefly recall the notions of tensor category and tensor C*-category.

In the next two sections, our main result is proven, namely the Tannaka-Krein-Woro-nowicz duality. In the second section we develop the representation theory of partial compact quantum groups, and we show how it allows one to construct a concrete semisimple rigid tensor C* -category. In the third section, we show conversely how any concrete semisimple rigid tensor C* -category allows one to construct a partial compact quantum group, and we briefly show how the two constructions are inverses of each other.

In the final two sections, we provide some examples of our structures and applications of our main result. In the fourth section, we first consider the construction of a canonical partial compact quantum group from any module C* -category for a semisimple rigid tensor C*-category. We then introduce the notions of Morita, co-Morita and weak Morita equivalence [27] of partial compact quantum groups, and show that two partial compact quantum groups are weakly Morita equivalent if and only if they can be connected by a string of Morita and co-Morita equivalences. In the fifth section, we study in more detail a concrete example of a canonical partial compact quantum group, constructed from an ergodic action of quantum SU (2). In particular, we obtain a partial compact quantum group version of the dynamical quantum SU (2)-group [11,24].

1. Partial compact quantum groups

1.1. Partial algebras

Definition 1.1. An I-partial algebra A is a set I = {k, l, •••}, the object set, together with C-vector spaces kAl, multiplication maps

M = Mk,i,m'- kAi <g> iAm ^ kAm, a <g> b ^ ab

and unit elements lk G kAk such that obvious associativity and unit conditions are satisfied.

We emphasize that 1k = 0 is allowed, in which case for example kAk = {0}. Note that I-partial algebras can be seen as small C-linear categories, but we will use a different notion of morphisms for them than the usual one of functor. Other names for I-partial algebra would be C-algebroid (with object set I) or C[I]-algebra.

By making M the zero map on all other tensor products, we can turn A = ®kil kAl into an associative algebra, the total algebra of A. It is a locally unital algebra by the orthogonal idempotents lk.

For example, for any set I we can define a partial algebra Mati with kMatl = C for all k, l and each Mkl,m scalar multiplication. The associated total algebra is the algebra of all finitely supported matrices based over I. For a general A, one can identify A with finite support I-indexed matrices (akl)k l with akl G kAl, equipped with the natural matrix multiplication.

Working with non-unital algebras necessitates the use of their multiplier algebra [6,43]. Recall that a multiplier m for an algebra A consists of a couple of linear maps

a ^ Lm(a) = ma and a ^ Rm(a) = am such that a(bm) = (ab)m, (ma)b = m(ab) and (am)b = a(mb) for all a, b G A. They form an algebra, the multiplier algebra M(A), under composition for the L-maps and anti-composition for the R-maps. In case A is the total algebra of an /-partial algebra A, the natural homomorphism A ^ M(A) is injective, and M(A) can be identified with matrices (mkl)kl which are rcf in the sense of the following definition.

Definition 1.2. An assignment (k, l) ^ mkl into a set with distinguished zero element is called row-and column-finite (rcf) if it has finite support in either one of the variables when the other variable has been fixed.

When mi G M(A) are such that for each a G A one has mia = 0 = ami for all but a finite set of i, one can define a multiplier mi in the obvious way. One says that the sum i mi converges in the strict topology. We will often use this kind of limit implicitly, for example in the next definition.

Definition 1.3. Let A and B be respectively / and J-partial algebras. A ^-morphism from A to B consists of a map / ^ P(J), the power set of J, and a homomorphism f: A ^ M(B) such that f (lk) = Erev(k) 1r.

Note that a ^-morphism f splits up into linear maps frs: kAl ^ rBs for all r G ¥>(k), s G y(l), which completely determine f. These components satisfy

(a) frf(lk) = ¿rtlr for r, t G <p(k), and

(b) frt(ab) = sev(l) frs(a)fst(b) for all a G kAi, b G iAm, r G <p(k) and t G <f(m),

where the last sum is finite by the rcf property of multipliers. Conversely, if one has a family of linear maps frs with (r, s) ^ frs(a) rcf on 0(k) x 0(l) for each a G kAl, then one can meaningfully impose conditions (a) and (b) on these maps, which then determine a unique ^-morphism f. Note that if the images of ^ are all singletons, we find back the notion of (C-linear) functor.

1.2. Partial coalgebras

The notion of a partial algebra dualizes as follows.

Definition 1.4. An I-partial coalgebra A consists of a set I = {k, l,...}, the object set, together with vector spaces Af, comultiplication maps

At = A( k ): Akm ^ Af < Alm, a ^ a0;i) <g> a^),

and counit maps e = ek: Ak ^ C satisfying obvious coassociativity and counitality conditions.

In the following, we will extend e as the zero functional on Ak when k = l.

1.3. Partial bialgebras

We write 12 for I x I seen as column vectors. To superimpose the notions of partial algebra and partial coalgebra into that of a partial bialgebra, we need the maps

^ I2 ^ PI2 x I2), k )) = {(( k ), ( m ))|l G I},

12 ^ p (7), „(( k )) = { ™ ¡f k = I

We also use the natural tensor product of an I-partial algebra A and J-partial algebra B, which is an I x J-partial algebra A ® B with total algebra A ® B. This corresponds to the usual tensor product of (small) C-linear categories.

Definition 1.5. A partial bialgebra A consists of a set I, the object set, a collection of vector spaces tmAln with I2-partial algebra structure on the , k )A, m) = mAln, and a

V l / ^ n '

^A-homomorphism A ^ A ® A and ^>e-homomorphism e: A ^ Mati whose components turn A^) = mAln into an I x I-partial coalgebra.

Spelled out, this means we have maps and elements

M: kAls ® sAm ^ kAm, Ars: ^ kAS ® kASn, l( k) G kAk, e: kkA\ ^ C

satisfying (co)associativity and (co)unitality, and such that moreover

(a) e(l( k)) = 1,

(b) e(ab) = e(a)e(b) whenever a, b are composable,

(c) All, (1( m)) = kl' 1( k) ® 1( m), and

(d) ArS( ab) = ^21 Art(a)Ats(b) whenever a, b are composable.

Note that this sum in the last entry is finite, as it is implicit in the definition that the applications (r, s) ^ Ars(a) are rcf for each a. We will use the Sweedler notation A(a) = a(1) <g) a(2) for the total comultiplication, and Ars(a) = a(rs;1) ® a(rs;2) for its components.

It will be convenient to consider the multipliers

Ak = £ 1( k) G M(A), Pl = £ 1( k) G M(A). lk

Then by [45, Proposition A.3], there is a unique homomorphism A: M(A) ^ M(A®A) extending A on A and satisfying A(1) = ^k pk < Xk. It follows by elementary calculations that the total objects (A, M, A, e, A(1)) form a regular weak multiplier bialgebra [4, Definition 2.1 and Definition 2.3]. We will call (A, A) the total weak multiplier bialgebra associated to A.

Recall from [4, Section 3] that a regular weak multiplier bialgebra admits four projections nL, nR, nL, nR: A ^ M(A), where for example nL(a) = (e ® id)((a ® 1)A(1)). One computes that for a G mAln, one has

nL (a) = e(a)Xm = e(a)Xk, nL (a) = e(a)Xn = e(a)Xl, nR(a) = e(a)pl = e(a)pn, nR(a) = e(a)pk = e(a)pm.

The base algebra of (A, A) is therefore the algebra Funf (/) of finite support functions on /. By [4, Theorem 3.13], the comultiplication of A is left and right full ('the legs of A(A) span A').

It is of more interest to consider the converse question. If (A, A) is a regular left and right full weak multiplier bialgebra, let us write AL = nL(A) = nL(A) and AR = nR(A) = nR(A) for the base algebras. By [4, Lemma 4.8], the algebra AL is anti-isomorphic to AR by the map a: AL ^ AR sending nL(a) to nR(a). We then refer to AL as the base algebra.

Proposition 1.6. Let (A, A) be a regular left and right full weak multiplier bialgebra whose base algebra is isomorphic to Funf (/) for some set /, and such that moreover ALAR C A. Then (A, A) is the total weak multiplier bialgebra of a uniquely determined partial bialgebra A over /.

The condition ALAR C A is essential, and should be considered as a properness condition. Indeed, this condition can be interpreted as saying that morphism spaces of the 'quantum category' associated to A are compact, which coincides with the notion of properness for a groupoid with discrete object set, cf. [42].

Proof. Write Xk G AL for the function Xk(l) = 5kl, and write a(Xk) = pk G AR. By assumption, l( k) = Xkpl G A. Further A = AAR = AAL = ALA = ARA, cf. the proof of [4, Theorem 3.13]. Hence the l(k) make A into the total algebra of an /2-partial algebra, as AL and AR elementwise commute by [4, Lemma 3.5].

Let us show that A(1) = ^k pk < Xk. By [4, Lemma 3.9], we have (pk < 1)A(a) = (1 ® Xk)A(a) for all a G A. By [4, Lemma 4.10] and the fact that A(1) is an idempotent, we can then write A(1) = ^pk < Xk for some subset /' C /. As by definition nL(A) = Funf(/), we deduce that / = /'. We then have as well that A(l(^)) = l(k) ® l(m) by [4, Lemma 3.3], and it follows that A is a ^a-homomorphism in the sense of Definition 1.5.

For a £ k>Alq and b £ lqAm, we then have e(ab) = e(al(l )b) = e(a)e(b) by [4, Propos ition 2.6.(4)]. The coun itality of e gives 5kll{ lm) = l( ^ )l( D = (e <g> id) (A(l( m))(! ® l( lm 4)) = e(l( k) )l( D for all k, l, m. Summing over m and using Xl = 0, we deduce e(l(k)) = Skl. This shows that e is a -homomorphism.

As A is coassociative and e satisfies the counit property, it is clear that the components of A and e satisfy the conditions for a partial coalgebra, which finishes the proof. □

1.4. Partial Hopf algebras

Definition 1.7. A partial bialgebra A is called a partial Hopf algebra if the total weak multiplier bialgebra (A, A) admits an antipode S: A ^ M(A) in the sense of [4, Theorem 6.8].

By [4, Theorem 6.8, Theorem 6.12 and Proposition 6.13], the antipode is uniquely determined, anti-multiplicative and non-degenerate, and by [4, Corollary 6.16] moreover anti-comultiplicative, A(S(a)) = (S ® S)Aop(a) for all a £ A. We will call S the total antipode of A. The next proposition will show that we have in fact in particular that S(A) C A.

Proposition 1.8. Let A be a partial Hopf algebra. Then S maps tmAln into ¡A^1, and

£a(rS;l)S(ars;2) = e(a)l( k) , £S(a(rs;1) )a(rs;2) = e(a)l( f) , a £ mAf. (L1)

Conversely, if A is a partial bialgebra with maps S: mAln ^ ¡ A^1 satisfying the above identities, then the latter extend by linearity to an antipode of (A, A) and A is a partial Hopf algebra.

Proof. Let A be a partial Hopf algebra. From [4, Lemma 6.14], applied with one of the elements of the form l4 k f, we deduce S (l4 km fal4 lm f) = l4 m fS(a)l4 m f, hence S 4 kmAlnf C ¡A^?. From the identities (6.14) in [4], we obtain ab(1)l(b(2)) = anL(b) for all mm, b £ A. Taking a = ly k f, we find £s b(rs;i)S(brs;2)) = e(b)ly kr f for all b £ mA¡. The other antipode identity in (1.1) follows similarly.

Conversely, assume that A is a partial bialgebra and S is defined on components and satisfies (1.1). Then the linear extension of S satisfies

ba(1)S(a(2)) = bnL(a), S(a(1))a(2)b = nñ(a)b, Va,b £ A. (1.2)

Hence a(i)b(i) <g) a(2)b(2)S(b(3))c = a(1)b(1) ® a(2)nL(b(2))c for all a, b, c £ A. As b(i) <8> nL(b(2)) = A(1)(b ® 1) by an easy computation, identity [4, Theorem 6.8.(2)(vii) holds. In the same way one proves the identity in [4, Theorem 6.8.(2)(viii)]. Finally, [4, Theorem 6.8.(2)(ix)] requires Ylk S(pma)Xm = S(a) for all a £ A, but this is immediate from the condition S( mAln) = ¡Am. n

Lemma 1.9. Let (A, A) be a partial Hopf algebra. Then e o S = e.

Proof. As both e o S and e vanish on tmAln when k = m or l = n, it suffices to check the identity on a G kA\. But then e(S(a)) = e(a(kl;i))e(S(a(kl;2)) = Es e(a(kS;i})e(S(a(kS;2})). By partial multiplicativity of e and (1.1), this equals e (a(ks;1)S(a(ks;2))) = e(a)e(l( k)) = e(a). □

Define now on I the relation k ~ l ^^ l(k) = 0. As e(l(k)) = 1, we have in particular l(k) =0, hence this relation is reflexive. As S(l(^)) = l(l), this relation is symmetric. As All(l(m)) = l(D ® l(m), this relation is transitive. Hence ~ is an equivalence relation.

Definition 1.10. The hyperobject set of a partial Hopf algebra A is the set I/~.

For reasons of technical simplicity, and since it will be sufficient for our purposes, we will later on assume that the hyperobject set is finite. This is no real restriction, as results for infinite hyperobject sets can then easily be derived by inductive limit arguments.

Definition 1.11. A partial Hopf algebra A will be called regular if the antipode S: A ^ A is invertible.

We can characterize regularity in terms of the coopposite partial bialgebra Acop, for which the grading is m(Acop)ln = m A with the same multiplication maps but Ars(a) =

a(rs;2) ® a(rs;1).

Lemma 1.12. A partial Hopf algebra A is regular if and only if the partial bialgebra Acop is a partial Hopf algebra. In this case, S-1 is the antipode of Acop.

Proof. Note that nR/L is the nL/R-map for Acop. Hence, by Proposition 1.8, Acop is a partial Hopf algebra if and only if there exists a map T: A ^ A mapping mAln into nAm and such that ba(2)T(a(1)) = bnR(a) and T(a(2))a(1)b = nL(a)b for all a, b G A.

Now if S is invertible, we see that these identities indeed hold with T = S-1 by applying S to them and using S o nR/L = nL/R. Conversely, if such a T exists, take a G tAln and put e = l(m) and f = l(n). Then, using that we know the behavior of S and T on the local units, we compute on the one hand

ST(ea(1) )S(a(2))a(3)f = ST(ea(1))nR(a(2))f = ST(ea(1)nR(a(2)))f = ST(a),

while on the other

ST(ea(1))S(a(2))a(3)f = S(a(2)T(ea(1) ))a(3)f = S(nR(a(1))T(e))a(2)f = enL(a(1))a(2)f = a.

Hence ST(a) = a for all a G A.

1.5. Invariant integrals

Invariant integrals for /-partial bialgebras over a finite set I were introduced in [14]. A more general definition of invariant integrals for regular left and right full weak multiplier bialgebras was developed in [23], see also [2, Section 3] and the unpublished work [47]. We will base our definition on the characterization obtained in [47, Proposition 2.7], but as in [14] we will assume that our integral has been normalized on the base algebra. The normalization will however be different from the one in [14], where the Dirac functions Ak and pm were assigned weight one, as this normalization does not make sense in case I is infinite.

Definition 1.13. Let A be an /-partial bialgebra. A functional 0: A ^ C is called an

invariant integral if 0( 1 ( k )) = 1 for all k and

(id <0)A(a) = 0 (AfeaAfe) Xk, (0 < id)A(a) = ^ 0 (pmapm) pm

as multipliers in M (A).

It follows from the Larson-Sweedler theorem, [47, Theorem 2.14], that if one moreover assumes 0 to be faithful, in the sense that 0(ab) = 0 for all b (resp. all a) then a = 0 (resp. b = 0), then A is automatically a regular /-partial Hopf algebra. Conversely, we will show in the next section that an invariant integral on a regular /-partial Hopf algebra is automatically faithful.

Note that if 0 is an invariant integral on an /-partial bialgebra, then 0(l( ^ )) = 1 whenever l( ^ ) =0, by applying (id ®0) to Akk(l( ). In particular, also in this case the relation k ~ l l( k ) = 0is an equivalence relation.

An invariant integral will have support on the homogeneous components of the form kmAkm.

Lemma 1.14. Let A be an /-partial bialgebra with invariant integral 0. Then for all a G A and all k, m G /, one has 0(l( ^)a) = 0(al( ^J).

Proof. Cf. the discussion following [47, Proposition 2.7]. Namely, if a G A and s G /, we compute

(id<g>0)((1 <g> As)A(a)) = ps(id<g>0)(A(a)) = (id®0)(A(a))ps = (id®0)(A(a)(1 <g> As)).

As both sides lie in A since (r, s) ^ Ars(a) is rcf, we can apply e to conclude 0(Asa) = 0(aAs). Similarly the identity 0(psa) = 0(aps) can be derived. □

One easily concludes from this that an invariant integral 0 is uniquely determined, since any other invariant integral ^ satisfies

0(a) = ^ (1( kk )) 0(a) = (^ <g> 0)(Akk(a)) = V(a)0(l( £ )) = V>(a), a e hmAhm.

1.6. Partial compact quantum groups

Definition 1.15. A partial *-algebra is a partial algebra A whose total algebra A is equipped with an anti-linear, anti-multiplicative involution a ^ a* with 1 *k = 1 k for all objects k.

This implies that * restricts to anti-linear maps kAi ^ lAk.

Definition 1.16. A partial *-bialgebra is a partial bialgebra A whose underlying partial algebra has been endowed with a partial *-algebra structure such that Ars(a)* = Asr (a*) for all a e tAln. A partial Hopf *-algebra is a partial bialgebra which is at the same time a partial *-bialgebra and a partial Hopf algebra.

Proposition 1.17. An I -partial *-bialgebra A is an I-partial Hopf *-algebra if and only if the weak multiplier *-bialgebra (A, A) is a weak multiplier Hopf *-algebra. In that case, the counit and antipode satisfy e(a*) = e(a) and S(S(a)*)* = a for all a e A.

Note that A is then automatically regular.

Proof. The if and only if part follows immediately from Proposition 1.8, the relation for the counit from uniqueness of the counit [4, Theorem 2.8], and the relation for the antipode from [46, Proposition 4.11]. □

Definition 1.18. A partial compact quantum group consists of a partial Hopf *-algebra A with an invariant integral 0 that is positive in the sense that 0(a*a) > 0 for all a e A. We then write A = P(G), where we refer to G as the partial compact quantum group defined by A, and to A as the algebra of (regular) functions on G.

It will follow from Theorem 2.14 and [14, Theorem 3.3 and Theorem 4.4] that for I finite, a partial compact quantum group is precisely a compact quantum group of face type [14, Definition 4.1]. As the total structure should not be considered compact for I infinite, we have changed the terminology to partial compact quantum group to reflect that only the parts should be considered compact.

1.7. Tensor categories

We assume that the reader is familiar with the basic notions concerning tensor categories - we refer to [28] for an overview. We will always assume that our tensor categories

(C, <) are C-linear and strict. If moreover C is semisimple, then each object has a finite-dimensional endomorphism algebra, and will be isomorphic to a direct sum of irreducible (or, equivalently, simple) objects. We will in general not assume that the unit object 1 of C is simple. A tensor category will be called rigid if each object has a left and right dual object. By tensor functor we will mean a strongly monoidal functor.

If C is semisimple, we know by the Eckmann-Hilton argument that End(1) = C^ for some finite set 1. If an identification with such a set 1 has been made, we will refer to 1 as the hyperobject set of C. To each a G 1 then corresponds a simple subobject la of 1. If a, 3 G 1, we can then construct full subcategories Cap of C consisting of all objects X for which la < X < l^ = X. The collection {Ca/3} can be looked upon as a particular kind of 2-category, or, seeing it as a categorification of the notion of partial algebra, as a partial tensor category. In this way, one could also treat the case where 1 is infinite, in which case the associated total tensor category would be 'non-unital with local units'. However, to be able to stick to more familiar terminology and to avoid some technical points, we will not discuss this more general setting which would bring nothing essentially new to the discussion.

We will also need the more structured notion of tensor C*-category, by which we will understand a tensor category with each Mor(X, Y) a Banach space equipped with an anti-linear map *: Mor(X, Y) ^ Mor(Y, X) satisfying the appropriate submultiplicativity and C*-conditions - see [26] or again [28]. Again, we stress that we do not assume the unit of a tensor C*-category to be simple. In a rigid tensor C*-category, left and right duals are isomorphic, and given an object X we will simply pick a dual object and denote it by X.

2. Representation theory of partial compact quantum groups

2.1. Corepresentations of partial bialgebras

A notion of full comodule for a general weak multiplier bialgebra was introduced in [2, Definition 2.1 and Definition 4.1]. It was then shown in [2, Theorem 5.1] that the category of full comodules forms a (possibly not semisimple) tensor category, and in [2, Theorem 6.2] that the category of finite-dimensional full comodules for a weak multiplier Hopf algebra forms a rigid (but possibly not semisimple) tensor category. However, in many situations the class of finite-dimensional comodules will be too small. We introduce here for partial bialgebras a class of 'intermediate size' comodules, which in the case of partial Hopf algebras with invariant integral will turn out to be large enough. We will however phrase the result in terms of corepresentations in stead of comodules, as this will be more appropriate when discussing matrix coefficients. We then make the connection with the comodules from [2] in Lemma 2.6.

Definition 2.1. Let I be a set. An I x /-graded vector space V = (J)k leI kVi will be called row-and column finite-dimensional (rcfd) if the kVi (resp. ®kei kV.) are finite-dimensional for each k (resp. l) fixed.

We denote by Vect^d the category whose objects are rcfd I x I-graded vector spaces. Morphisms are linear maps T that preserve the grading, and can therefore be written as

T = EIk,iei kTi.

Definition 2.2. Let A be an I-partial bialgebra. A corepresentation X of A on an rcfd I x I-graded vector space V is a family of elements mXln € m,Al„ < Homc( mVn, kVl) satisfying (Apq < id)( kmXln) = (kXlq) ^mXq) 23 and (e < id)^) = Sk,m\n id fc v,.

We use here the standard leg numbering notation, e.g. a23 = 1 << a.

Example 2.3.

1. Equip the vector space C(I) = (J)keI C with the diagonal I x I-grading. Then the family U given by m^l = 5kll5mnl( k^) € m^h is a corepresentation of A on C(I), called the trivial corepresentation.

2. Assume given an rcfd family of subspaces mVn C (J)kl k^h with Apq(mVn) C pVq ® knAn for all indices. Then the elements kXln € kAln ® Homc(mVn, kVi) defined by

kmXln(1 ® b) = A°kP(b) € kmAln <g> kVl for all b € mVn

form a corepresentation X of A on V. Corepresentations of this form will be called regular.

A morphism T between corepresentations (V, X) and (W, Y) of A will be a morphism T from V to W satisfying the intertwiner property (1 ® kTl) mXn = mYn(1 ® mTn). In this way, corepresentations form a category which we will denote Coreprcfd (A).

We next consider the total form of a corepresentation. Let V be an rcfd I x I-graded vector space, and write pkl for the projections on the component kVl. Write End0(V) for the algebra of endomorphisms on V having finite-dimensional support.

Definition 2.4. Let A be a partial bialgebra, and let V be an rcfd I x I-graded vector space. An element X € M(A ® End0(V)) is called a total corepresentation if for all k, l, m, n € I

(1( m) ® id)X(1( n) <8> id) = (1 <g>pkl)X(1 ® Pmn) € A <g> Endo(V), (2.1)

(A < id)(X) = X13X23, (e < id)(X) = idv . (2.2)

Here (e < id)(X) makes sense as a multiplier by (2.1) and the fact that e has support on the k A\.

The following lemma is straightforward.

Lemma 2.5. Let X be a total corepresentation. Then the ^Xln = (l( ^ ) Oid)X(l( Oid) form a corepresentation X of A, and every corepresentation arises in this way from a unique total corepresentation.

We will in the following call X the corepresentation multiplier of X. Recall now the notion of full right comodule for a regular weak multiplier bialgebra (A, A) [2, Definition 2.1 and Definition 4.1]. It consists of a vector space V, equipped with two maps A, p: V O A ^ V O A satisfying certain assumptions. It then follows from [2, Theorem 4.1] that V is a firm bimodule over the base algebra. In particular, if (A, A) is the total weak multiplier bialgebra associated to a partial bialgebra, A, then V becomes in a natural way an I-bigraded vector space. Whenever considering full right comodules, we will consider V with this natural bigrading.

Proposition 2.6. Let A be a partial bialgebra. There is a natural one-to-one correspondence between corepresentations of A and full right comodules for (A, A) on rcfd I-bigraded vector spaces.

Proof. Let (A, p) be a full right comodule of (A, A) on an rcfd vector space V. Let us write mV = ®nmVn etc.

It follows from [2, Lemma 4.2.(1) and (7)] that A maps mVO lA into V.O mA. From [2, Lemma 4.2.(8)] it follows that A has range in ®k kVO kA. As A(vO ab) = A(vO a)(1 Ob) by the multiplier property with respect to p, and as V is rcfd, all this implies that we can define inside the tensor product ^Ah O Homc( mVn, kV.) = Homc( mVn, m,Aln O kVi) an element hmXln by the formula

v ^ (l(O idv)ffA(v O l(n)), v G mVn,

where a is the flip map. Moreover, as A has support on ®n VnO nA by [2, Lemma 4.2.(2)], the maps m.Xn completely determine A.

The defining identity [2, (2.12)] for A, applied to v O l(O l( qn) with v G Vn, leads immediately to the corepresentation identities for the mXln. Finally, [2, Lemma 4.2] and the definition of the right action of the base algebra AR on V implies that for v G kV.,

(e O id)( kX.)v =(id Oe)((1 O l( k ))A(v O l( . )))

= pk(idOe)(A(pkv O ^\))) = pkvpi = v.

We leave it to the reader to check that conversely, each total corepresentation (V, X) leads to a right comodule (V, A, p) by the formulas A(v O a) = aX(a O v), and p(v O a)(pi O 1) = (1 O a)aX(^ ln) O v) for v G Vn. □

If A is a partial bialgebra, the category Coreprcfd (A) is easily seen to be an abelian category with a faithful functor into Vect^d lifting kernels, cokernels and biproducts, cf. [2, Lemma 5.3]. In particular, one can call a corepresentation V irreducible if any morphism T from (resp. into) V either has all Tkl zero or injective (resp. surjective).

Moreover, from [2, Theorem 5.1] we know that the category of all full comodules over (A, A) forms a monoidal category with a strict monoidal imbedding into the tensor category of firm A^-bimodules, that is, I x I-graded vector spaces. The latter tensor product, which we will denote by 0j, can easily be identified concretely as follows:

k(V 0/ W)m = ®l kV 0 ¡Wm.

It follows immediately that Coreprcfd(A) is a tensor subcategory, represented faithfully inside Vect^Cfd. In terms of total corepresentations, the tensor product of X and Y is given by X (J) Y = X12Y13, with associated components

m(X CD Y )Pq = £ (mXn) 12 ( lnYq) 13 , l,n

where the sum is actually finite because of the rcfd condition. The unit is given by the trivial corepresentation (C(/), U).

2.2. Corepresentations of partial Hopf algebras

Assume now that A is a partial Hopf algebra. If V is an rcfd I x I-graded vector space, we will denote by X^ (resp. p^) the projection in End0(V) onto elements with left (resp. right) grading equal to k.

Lemma 2.7. Let (V, X) be a corepresentation of A on an rcfd vector space. Then the

element

X-1 = (S 0 id)(X) € M(A 0 End0(V))

is a generalized inverse of X in the sense that XX-1X = X and X-1XX-1 = X-1. More precisely convergence).

More precisely, one has XX 1 = k Xk <8> X^ and X 1X = 1 pi <g) pV (w.r.t. strict

Proof. This follows immediately from Proposition 1.8 and the corepresentation identity. □

Given a corepresentation X, we then write

m(X-l)ln = (S ® id)( nXm) G kmAln ® Homc( iVk, nVm)

for the components of X-1 .

The following easy lemma will be very useful.

Lemma 2.8. Let A be a partial Hopf algebra. A bigraded map T defines a morphism from (V, X) to (W, Y) if and only if one (and hence both) of the following relations hold:

Y-1(1 <g> T)X = Pn < mTn, Y(1 <g> T)XXk < kTi.

m,n k,l

Proposition 2.9. Let A be a partial Hopf algebra. Then Coreprcfd (A) has left duals. If A is regular, then Coreprcfd (A) also has right duals.

Proof. Let (V, X) be a corepresentation. Denote the dual of vector spaces V and linear maps T by V* and Ttr, respectively, and define the dual of an I x I-graded vector space V = ®kil kV to be the space Vл = ®kJ k(Vл) where k(VA)l = ( V)*. Then using anti-comultiplicativity of S and Lemma 1.9, we see that Vл and the family X given by

mxn := (s ®-trx nxm)

form a corepresentation of A. To see that it is a left dual of X, consider the natural evaluation and coevaluation maps

ev: Vл ®I V ^ C(I), ш <g> v ^ w(v), coev: C(I) ^ V ®I Vл, 5k ^ ^ e(kl) <g> w(lk),

where {е^} and {w(lk)} are a basis and its dual basis for kVi. Note that the right hand sum is finite by the rcfd condition. These maps provide the duality between V and Vл in Vectjf. It then suffices to show that they are also morphisms from the trivial corepresentation to the tensor product representations of X with X. But for example the intertwining property of ev follows from

(1 ® k evk ) kmXln )12( lnXk )1з = (1 <g> k evk) £(S ®-tr)( nXfemb( ^13

l,n l,n

= ôm,q(1 ® mevm)J2(S ® id)( ПХ^1з( nXgk)i3

= \$m,q 1 ( q ) & mevm

= mUk (1 ® mevm).

If A is a regular partial Hopf algebra, it follows that Coreprcfd(A) has right duals since X ^ X is then essentially surjective, with inverse X ^ X" for

mxn := (S-i ®-tr)( nxm). □

Remark 2.10. The previous proposition was proven for general weak multiplier Hopf algebras in [2, Theorem 6.2] with respect to finite-dimensional full comodules.

2.3. Corepresentations of partial Hopf algebras with invariant integral

In the presence of an invariant integral, one can integrate morphisms of bigraded vector spaces to obtain morphisms of corepresentations.

Lemma 2.11. Let (V, X) and (W, Y) be rcfd corepresentations of a partial Hopf algebra A with invariant integral 0. Fix m, n G I, and let T: mVn ^ mVn. Then the families

kTi := (0 <g> id)( n(y-1)m(1 ® T) mxn),

kTi := (0 ® id)( kmvn(1 ® T) ln(X-1£)

form morphisms T and T from (V, X) to (W, Y).

Proof. Viewing T as a linear map V ^ W concentrated at the component at (m, n), we can consider the total forms T = (0 0id)(Y-1(1 0T)X) and T = (0 0id)(Y(1 0T)X-1). We compute

Y-1(1 0 T)X = (0 0 id0 id)((Y-1)23(Y-1)13(1 0 1 0 TX13X23) = ((0 0 id)A 0 id)(Y-1(1 0 T)X)

= £ Pi 0 (0 0 id)((pi 0 1)Y-1(1 0 T)X(pi 0 1)) l

£ Pi 0 kTi.

Hence T is a morphism from X to Y by Lemma 2.8. The assertion for T follows similarly.

Lemma 2.12. Let A be a partial Hopf algebra with invariant integral 0. Let (V, X) be an rcfd corepresentation, and kWi C kV, an invariant family of subspaces. Then there exists an idempotent endomorphism T of (V, X) such that kWi = img kTi for all k, l.

Proof. We can decompose V = © aVp, where for a, ¡3 in the hyperobject set I we write aVp for the direct sum of all kVi with k € a, l € 3. As each aV^ is invariant, we may assume that V = aV^ for some a, 3.

Let Y be the restriction of X to W. Fix n € 3. Let T be a bigraded idempo-tent endomorphism of V with image W, and write

T (m) =

mTn. By , we

obtain endomorphisms T(m) of (V, X). Using column-finiteness of V, we can define T = ^2m T(m). We claim that W is the image of T.

Invariance of W implies (1 O T)X(1 O T) = X(1 O T). Applying (S O id), we get (1 O T)X-1(1 O T) = X-1(1 O T). Then

TT = (0 O id)(X-1(1 O pVT)X(1 O T)) = (0 O id)(X-1(1 O pV)X(1 O T)) = (0 O id)(X-1X(An O T))

= £ 0(l( n ))pVT = £ pVT = T.

As kT sends kVi into kWi, it follows that img T = W, which proves the claim. □

Lemma 2.13. Let A be a partial Hopf algebra with hyperobject set 1, and fix representatives lp G P for each P. If T is a morphism in Coreprcfd(A) and kTip = 0 for all k and 3, then T = 0.

Proof. This follows from the equations in Lemma 2.8. □

Theorem 2.14. Let A be a partial Hopf algebra with invariant integral. Assume that the hyperobject set 1 is finite. Then Coreprcfd (A) is a semisimple tensor category with left duals and hyperobject set 1.

Proof. It is easy to verify that, for each hyperobject a, the space is an invariant subspace of C(I) w.r.t. the trivial representation U, and that the corresponding subrep-resentations

U(a) provide a decomposition of U into irreducible components. By Lemma 2.12 and Proposition 2.9, it now suffices to show that each endomorphism space of Coreprcfd(A) is finite-dimensional.

Choose a representative for each / G 1. Assume that T is an endomorphism of some rcfd corepresentation (V, X) with kTij3 = 0 for all /. From Lemma 2.13, it

follows that T = 0. Hence the map T ^ k p kTi^ is an injective map from End(V, X) into a finite-dimensional space of linear maps. □

In fact, as Coreprcfd(A) is semisimple, it will automatically have right duals as well. This will also follow more concretely from Corollary 2.25 and Proposition 2.9.

For general partial Hopf algebras with invariant integrals, we have a weak form of semisimplicity.

Corollary 2.15. Let A be a partial Hopf algebra with invariant integral. Then every rcfd corepresentation of A decomposes into a (possibly infinite) direct sum of rcfd irreducible corepresentations.

Proof. Any rcfd corepresentation is a (possibly infinite) direct sum of rcfd corepresen-tations with a singleton as left and right hyperobject support. As in Theorem 2.14, we

conclude by Lemma 2.13 that such rcfd corepresentations have a finite-dimensional space of self-intertwiners. The corollary then follows again from Lemma 2.12. □

2.4- Schur orthogonality

Our next goal is to obtain Schur orthogonality for matrix coefficients of corepresentations. We give a little more detail than provided in Hayashi's paper [14].

Given finite-dimensional vector spaces V and W, the dual space of Home(V, W) is linearly spanned by functionals of the form Ufv(T) = (f |Tv), where v G V, f G W*, and (—| —) denotes the natural pairing of W* with W.

Definition 2.16. Let A be a partial bialgebra. The space of matrix coefficients C(X) of an rcfd corepresentation (V, X) is the sum of the subspaces

kmC{X)ln = span {(idmXln) | v G mVn,f G ( kVi)*} Ç kmAln.

As Apq( kmC(X)ln) C pC(X)lq <g> mC(X)n, the kmC(X)ln form a partial coalgebra with respect to A and e. Moreover, for each k, l the I x I-graded vector space feC(X) := (J)mn kC(X)n is rcfd, and the inclusion above shows that it supports a regular corepresentation in the sense of Example 2.3.2.

Lemma 2.17. Let (V, X) be an rcfd corepresentation of a partial bialgebra, and let f G ( kVi)*• Then the family of maps

mTn(/): mVn ^ mC(X)n, w ^ (id®w/w)( kmXln) = (id®f )( mxn(1 ® w)),

is a morphism from X to the regular corepresentation on kC(X)l.

Proof. Denote by Y the regular corepresentation on (\$)mn mC(Xfn- Then for all v G

PmYn (1 ® mTnf )(v)) = (ApP ® fv)( kmXln ) = (id ® id ®f )(( kp X[ )23( PmXqn )l3(1 ® 1 ® v)) = (1 ® pTqif )) PmXn (1 ® v). □ As before, we denote by V* the dual of a vector space V.

Lemma 2.18. Let A be a partial Hopf algebra. Then for any a G ®k,i mAln, the family of subspaces

V(a) = {(id ®f )(Apq(a)): f G ( mAn)*}

supports a regular corepresentation such that a G mVna). If further (W, Y) is an irreducible r a G m Wn.

reducible regular corepresentation, then kWi = kfor all k, l and any non-zero

Proof. Assume that a and

are as above. Taking f = e, one finds a G mVr Next, write Apq(a) = bipq O cipq with linearly independent (cpq)j. Then pVqa = span{bpq : i}, and Ars( pVq(a)) C rVs(a) O pAq as Ks(bpq) O cpq = Ej bjrs O Apq (j) by coassociativity.

If now W is an irreducible regular corepresentation and a G mWn non-zero, then pVq(a) is included in pWq. As a G mV^, it follows by irreducibility and Lemma 2.12 that p Vq(a) = pWq. □

Proposition 2.19. Let A be a partial Hopf algebra with invariant integral. Then the total algebra A is the sum of the matrix coefficients of irreducible rcfd corepresentations.

Proof. Let a G mAin, define pVqa) as in Lemma 2.18 and let X be the regular corepresentation on V(a). Then a = (idOe)(Akp(a)) = (idOe)( kmXln(1 O a)) G lmC(X)in. Decomposing (V(a), X) by Corollary 2.15, we find that a is contained in the sum of matrix coefficients of irreducible rcfd corepresentations. □

Proposition 2.20. Let A be a partial Hopf algebra with invariant integral 0, and let (V, X) and (W, Y) be inequivalent irreducible rcfd corepresentations. Then for all a G

C(X), b gC(Y),

0(S(b)a) = 0(bS(a)) = 0.

Proof. Since 0 vanishes on S( mAn) pAq and on pAjS( mAin) unless (p, q, r, s) = (m, n, k, l), it suffices to prove the assertion for elements of the form a = (id Owf,v)( mXn) and b = (id OwSw)( mY") where f G ( kV)*, v G mVn and g G (mWn)*, w G kWi. Applying Lemma 2.11 to the map T: kVi ^ kW. with T(u) = f (u)w yields morphisms T, T from (V, X) to (W, Y) which are necessarily 0. Inserting the definition of T, we find

0(S(b)a) = 0((S O og,w)( mYn) ■ (id O^f,v)( t^'n))

= (0 O ^g,v)( in(Y-1)km(1 O T) kmXln) = ^g,v( m TTn) = 0.

A similar calculation involving T shows that 0(bS(a)) = 0. □

Corollary 2.21. Let A be a partial Hopf algebra with invariant integral 0. Then 0 = 0oS.

Proof. Assume that a G A lies in an irreducible regular corepresentation which is not a direct summand of the trivial corepresentation. Then 0(a) = 0(S(a)) = 0 by Proposition 2.20. As such a together with the l (m) linearly span A, this proves the corollary. □

For the following theorem, recall from the beginning of the proof of Lemma 2.12 that any rcfd corepresentation (V, X) can be decomposed into a direct sum V = © aVp. Hence if V is irreducible, we have V = aVp for unique a, / in the hyperobject set 1.

We will call a the left and ß the right hyperobject support of V. Recall further that X denotes the dual left corepresentation of an rcfd corepresentation X.

Theorem 2.22. Let A be a partial Hopf algebra with invariant integral ф. Let (V, X) be an irreducible rcfd corepresentation of A with left hyperobject support a and right hyperobject support ß. Then there exists an isomorphism G from (V, X) to (V, X). Moreover, with F denoting the inverse of G, the following hold.

(1) For l G ß and m G a, the numbers da := ^2k TT( kGi) and dp := ^2n Tr( mFn) are non-zero and do not depend on the choice of l or m.

(2) For all к, m G a and l, n G ß,

(ф ® id)( ln(X-1)m kmXln) = da1 Tr( fcGi) idmy„, (ф ® id)( тхПП(Х-1)1) = d- Tr( mFn) idfcVi •

(3) Denote by £kimn the flip map кVi ® mVn ^ mVn ® kV. Then

(ф ® id® id)(( ln(X-1)m)i2( mXln)13) = d-1(id® kGl) о £klmn,

(ф ® id ® id)(( kmXln )i3( n(X-1)m)l2)= d-1( mFn ® id ку> ) О £klmn

Proof. We prove the assertions and equations involving da in (1), (2) and (3) simultaneously; the assertions involving dp follow similarly.

Consider the following endomorphism Fmnkl of mVn ® kVl,

Fmnkl := (ф ® id ® id) (( n(X-1)m)l2( kmXln◦ £mnkl

= (ф ® id ® id) (( n(X-1)km)12£klkl,23 ( m Xln b) -

By applying Lemma 2.11 with respect to the flip map £klkl, we see that the family (Fmnkl)mn is an endomorphism of (V® kVi, X12) and hence Fmnkl = idmvn ® kRl with some kRl G Endc( kVl) not depending on m, n.

On the other hand, since ф = ф о S by Corollary 2.21,

Fmnkl = (ф ® id ® id)((S ® id)( mXln) 12 ( kmXln )1з) ◦ £mnkl

= (ф ® id ® id) (((S ® id)( kmXln ))13((S2 ® id)( mXD)12) О £mnkl

= (ф ® id ® id) (( n (X-1)m)13 (£mnmn) 23 ( m(XЛЛ )n)1^ -

Hence we can again apply Lemma 2.11 and find that the family (Fmnkl)k l is a mor-phism from (mVn®V, X13) to (mVn®V, X13). Now X is also irreducible, since otherwise its double right dual X would split. Hence the space Mor(X, X) is linearly spanned by

a single element G. Therefore Fmnki — mTn O kGi with some mTn G Endc( mVn) not depending on k, l.

We conclude from the above calculations that mTnO kGi = midn O kRi for all indices. This implies that for all m, n with mVn = 0, there exists Xmn G C with kRi = Xmn kGi for all k, l. As however kRi does not depend on m, n, we see Xmn = A G C. In the end, we deduce Fmnki = A(id myn O kGi).

Choose now dual bases (vi)i for kV and (fi)i for ( kV)*. Then

X Tr( kGi) id= £(id )(Fmiki) = (0 O id)(( i(X-1)km) kmXi). i

By Lemma 2.13, we can choose for each l some m G a with mVi =0. Then summing the previous relation over k, the relations J2k( i(X-1)m,) mXl = l( D O idmVl and 0(l( i)) = 1 give X ■ Ek Tr( kGi) = 1. It follows at once that G is not the zero morphism,

and hence X is isomorphic to X. Now all assertions in (1)-(3) concerning da follow. □

Corollary 2.23. Let A be a partial Hopf algebra with invariant integral 0, let (V, X) be an irreducible corepresentation of A, let F be an isomorphism from (V, X) to (V, X) and G = F-1, and let a = (idOwf,v)( kmXln) and b = (id Owgw)( "pX?) for f G ( kV)*, v G mVn, g G (mVn)*, w G V Then

0(S(b)a) ■ £Tr(rGn) = (g|v)(f |Gw), 0(aS(b)) ^Tr(raFs) = (g|Fv)(f |w).

Proof. Apply O&f,v to the formulas in Theorem 2.22.(c). □

Corollary 2.24. Let A be a partial Hopf algebra with invariant integral, and let ((V(x), Xx))xei be a maximal family of mutually non-isomorphic irreducible rcfd corep-resentations of A. Denote the regular corepresentation on kC(Xx)1 by kY^. Then there exists a linear isomorphism

( kV(X))* ^ Mor(Xx, kYX)

assigning to each f G ( kV^x))* the morphism T(f) of Lemma 2.17, and with inverse the map T ^ e o kTi. Furthermore, the map

0 0 (( kY(x))* O mVnx)) ^ A, f O w M- T(f)(w)

xGl k,i,m,n

is a linear isomorphism.

Proof. The injectivity of the first map follows from Corollary 2.23. Its surjectivity follows immediately by checking that the map T M e o kTi is an inverse.

The injectivity of the second map follows again from Corollary 2.23, and its surjectivity from Proposition 2.19. □

In particular, it follows from the previous two corollaries that an invariant integral 0 is faithful, i.e. 0(ab) = 0 or 0(ba) = 0 for all b implies a = 0. This can also be proven more directly along the lines of [44, Proposition 3.4].

The following corollary generalizes [25, Theorem 3.3] and [14, Corollary 3.6].

Corollary 2.25. Let A be a partial Hopf algebra with invariant integral 0. Then A is regular.

Proof. Injectivity of S follows from Corollary 2.23. As further X = X for any irreducible corepresentation X, it follows that S2(^C(X)ln) C kmC(X)n, hence S is surjective by Proposition 2.19 and finite dimensionality of each mC(X)ln. □

2.5. Unitary corepresentations of partial compact quantum groups

Let us write B(H, G) for the space of bounded morphisms between Hilbert spaces H and G.

Definition 2.26. Let G be a partial compact quantum group. A corepresentation X of P(G) on a rcfd collection of Hilbert spaces kHl is called unitary if m(X-1)ln = (n^m)* as elements inside m,P(G)ln <8> B( lHk, nHm). We also refer to X as a unitary representation of G.

For example, viewing C« as a direct sum of the trivial Hilbert spaces C, the trivial corepresentation U on Ci1) is unitary.

It is easily seen that the tensor product of corepresentations lifts to a tensor product of unitary corepresentations. We hence obtain a tensor C*-category RepM rcfd(G) = CorepM rcfd(P(G)) of unitary corepresentations, where intertwiners are bounded bigraded maps T: H^K commuting with the corepresentations. Our aim is to show that, in case the hyperobject set I is finite, it is a semisimple rigid tensor C*-category.

In the following, we use the physicist convention that inner products on Hilbert spaces are anti-linear in their first argument.

Lemma 2.27. Let A define a partial compact quantum group with positive invariant integral 0, and let mVn C (J)k l mAln define a regular corepresentation X. Then with respect to the inner product given by (a|b) := 0(a*b) each kVl is a Hilbert space and X unitary.

Proof. Let a G mVn, b G mVn' and define Homc( mVn, mVn>) ^ C by T ^ (b|Ta). Then

£(idOwM)(( kmXln,)* kmXln)) = £(idO0)(A£?(b)*A£?(a)) k k

= £(0 ® id)(Aik(b*)Aki(a)) k

= (0 O id)(An(b*a)) = 0(b*a)l( ln) = 5n>n 1( ln) O (b|a>.

Hence Ek(mXln')* ^n = \$n,n'l( O id mvn, and X is unitary by definition of the generalized inverse of a corepresentation. □

Proposition 2.28. Let A define a partial compact quantum group. Then every rcfd corepresentation of A is isomorphic to a unitary corepresentation.

Proof. By Theorem 2.14 and Corollary 2.24, every corepresentation is isomorphic to a direct sum of irreducible regular corepresentations, which are unitary by Lemma 2.27. □

From Theorem 2.14, we deduce the following corollary.

Corollary 2.29. Let G be a partial compact quantum group with finite hyperobject set I. Then RepM rcfd(G) is a semisimple rigid tensor C*-category with hyperobject set I.

Proof. Write A = P(G). We can immediately derive the corollary from Theorem 2.14 once we know that the forgetful functor F: CorepM rcfd(A) ^ Coreprcfd (A) is an equivalence of categories.

Clearly F is faithful, and by Proposition 2.28 it is essentially surjective. To see that it is full, it is sufficient to show that F(X) = F(Y) for irreducibles X, Y iff X = Y. But if T: F (X) ^ F (Y) is an isomorphism, it follows from the unitarity of X and Y that also the component-wise adjoint T*: F(Y) ^ F(X) is an isomorphism, and then also |T| = (T*T)1/2 is an isomorphism from F(X) to F(X). It follows that if T = U|T|, then U is a unitary intertwiner from F(X) to F(Y), hence U is an intertwiner from X to Y

in Corep„rcfd(A). □

2.6. Schur orthogonality for partial compact quantum groups

Proposition 2.30. Let G be a partial compact quantum group and let (H, X) be an irreducible unitary rcfd representation of G. Then one can find an isomorphism F in Reprcfd(G) between (H, X) and (H, X) such that each kFi is positive.

Proof. By Proposition 2.28, there exists an isomorphism T: X ^ Y for some unitary corepresentation Y on H*, so that in total form (1 OT)X = Y(1 OT). We apply SO—tr and -* O-*tr, respectively to find X (1oTtr) = (1oTtr)Y and (1OT * tr)X = Y(1®T *tr).

Combining both equations, we find X(1 O TtrT*tr) = (1 O TtrT* tr)X. Thus, we can take F := TtrT*tr. □

The Schur orthogonality relations in Corollary 2.23 can be rewritten using the involution. If v G kHl, v' G mHn and uvv< (T) = (v|Tv'}, then

S ((id ®^,v )( tK )) = (id ®wv,v )( ?(X-X ))

= (id ®Wv,v' )(( mxm = (id ®^v,v )( txn*.

This equation and Corollary 2.23 imply the following corollary.

Corollary 2.31. Let G be a partial compact quantum group with associated partial Hopf *-algebra A and positive invariant integral 0. Let (H, X) be an irreducible unitary rcfd representation of G, let F be a positive isomorphism from (H, X) to (H, X) in Coreprcfd(A) and G = F-1, and let a = (id®wVjV/)(m,Xln) and b = (id®www/)(m,Xln), where v, w G kHi and v', w' G mHn. Then

(w|v/)(v|Gw/) (w|Fv/)(v|w/)

0(b a) = Er Tr( PGn) , 0(ab ) = Es Tr( mFs) •

Remark 2.32. In fact, Proposition 2.30 and Corollary 2.31 show the following. Let A be a partial Hopf * -algebra admitting an invariant integral 0, which a priori we do not assume to be positive. Suppose however that each irreducible corepresentation of A is equivalent to a unitary corepresentation. Then 0 is necessarily positive.

3. Tannaka-Kreïn-Woronowicz duality for partial compact quantum groups

In the previous section, we showed how any partial compact quantum group with finite hyperobject set I gave rise to a semisimple rigid tensor C*-category with hyperobject set 1 and a faithful morphism into the tensor C*-category of rcfd Hilbert spaces. In this section we reverse this construction, and show that the two structures are in duality with each other. We first deal with the purely algebraic setting without C*-structures.

Fix a (strict) semisimple tensor category C with (necessarily finite) hyperobject set I, and fix a faithful tensor functor F : C^ Vect^d with product and unit constraints i and ¡i. Then F(1a) = l2(Ia) for some non-empty subset Ia Ç I, and {Ia} is a partition of I. We write k' = a if k G Ia. We write Fkl(X) = kF(X)l for k, l G I and X gC, so that each Fkl is a C-linear functor from C into the category Vectfd of finite-dimensional vector spaces.

For X, Y gC, we write the inclusion maps associated to i as

¿X?: Fki(X) ® Fim(Y) ^ 0rFkr(X) ® Frm(Y) = k (F (X ) F (Y ))m ^^ Fkm(X ® Y ),

and we write the associated projection maps as

Fkm(X ® Y) ^ Fki(X) ® Fm(Y).

We choose a set I parametrizing a maximal family of mutually inequivalent irreducible objects {ua}aex in C. We assume that the ua include the unit objects 1a for a G 1, so that we may identify 1 CI. For a GI, there exist unique Xa, pa G 1 with ua G C\aPa. For a, ¡3 G 1 fixed, we write Iap for the set of all a GI with Xa = a and pa = Note that ua O ub = 0 if pa = Xb. When a, b, c G I, we write c < a ■ b if Mor(uc, ua O ub) = {0}. Note that with a, b fixed, there is only a finite set of c with c < a ■ b. We also use this notation for multiple products.

Recall that we write V* for the dual of a (finite-dimensional) vector space V.

Definition 3.1. For a GI and k, l, m, n G I, define vector spaces

kmAln(a) = Home (Fmn(ua),Fkl(ua ))*,

and write = ©aeiL,Aln(a), A(a) = ®k,i,m,nmAln(a), A = ®k,i,m,nt,Aln.

Note that mAln(a) = 0 unless k = m' = Xa and l' = n' = pa. We further write A = { mAln | k, l, m, n}.

Definition 3.2. For r, s G I, we define Ars: m Aln ^ k Als O rmAsn as the direct sum over a of the duals of the composition

Home(Frs(ua),FH(ua)) O Home(Fmn(ua),Frs(ua)) ^ Home(Fmn(ua),FH(ua)), sending x O y to x o y.

Lemma 3.3. The couple (A, A) is an I x I-partial coalgebra with counit map e: kA\(a) ^ C sending f to f (idFkl(Ua)). Moreover, for each fixed f G mAln(a), the matrix (Ars(f ))rs is rcf.

Proof. Coassociativity and counitality are immediate by duality, as the Homc(i1mn(ua), Fki(ua)) form a partial algebra with units idFfci(„a) for each fixed a. The rcf condition follows from the fact that the total F(ua) is rcfd. □

In the next step, we define a partial algebra structure on A. First note that we can identify

Nat(Fm„,Ffei) = J^Homc (Fmn(ua),Fki(ua)),

where Nat(Fmn, Fkl) denotes the space of natural transformations from Fmn to Fkl. Similarly, we can identify

Nat(Fmn ® Fpq,Fki ® Frs) = Hornc(Fmn(u6) ® Fpq(uc),Fki(ub) <g> Frs(uc)),

where Fki ® FPS: C xC^ Vectfd sends (X, Y) to Fki(X) ® FPS(Y). As such, there is a natural pairing of these Nat-spaces with resp. tmAln and tmAln ® PpAsq. For example, tmAln can be identified with the subspace of functionals on Nat(Fmn, Fkl) of finite support with respect to I.

Definition 3.4. We define a product map

M: kAq ® lqAm ^ kAm, (f • g)(x) = (f ® g)(Als(x)) for x G Nat(Frt,Fkm), where Alq(x) is the natural transformation

Aq (x): Fpq ® Fst ^ Fkl ® Flm, Aq (x)x, y = ◦ xx0y o 1%\$ for X, Y gC

Note that indeed f • g has finite support as a functional on Nat(FPt, Fkm): if f is supported at b GI and g at c GI, then f • g has support in the finite set of a GI with a < b • c, since if x is a natural transformation with support outside this set, one has xUb®uc = 0, and hence any of the (A S(x)) u =0.

Lemma 3.5. The above product maps turn (A, M) into an I x I-partial algebra. Proof. We can extend the map (Als ® id) on Nat(FPt, Fkm) ® Nat(FtU, Fmn) to a map (AS ® id): Nat(FPt ® Ftu,Fkm ® Fmn) ^ Nat(FPs ® Fst ® Ftu,FM ® Flm ® Fmn),

S ® id)(x)X,Y,Z = (nXkY) ® idFmn(Z)) xX0Y,Z (l-xY ® idFtu(Z)) •

By finite support, we then have that

((f • g) • h)(x) = (f ® g ® h)((AS ® id)^m(x)), Vf G kAls,g G lqAm,h G mAn,x G Nat(FPu,Fkn)^

Similarly, ((f • g) • h)(x) = (f ® g ® h)((id®Am)AlS(x)). The associativity then follows from the 2-cocycle condition for the i- and n-maps.

By a similar argument, one sees that the units are given by l(k) G kAk(1a), which for a = k' = l' correspond to 1 in the canonical identifications

k Ak(a) = Homc(Fn(1a),Fkk (1a))* = Homc(C, C)* = C, and which are zero otherwise. □

Proposition 3.6. The partial algebra and coalgebra structures on A define a partial bialgebra structure on A.

Proof. We only show multiplicativity, which means showing that for each x G Nat(Fuw, Fkm) and y G Nat(Fri, Fuw), one has pointwise that Ais(x o y) = Ev Aiv(x) o AV(y). This follows from the fact that J2v iXuY"^njuY"^ = idu(F(X)®if(y))w, where we again note that the left hand side sum is in fact finite. □

Lemma 3.7. Define 0: A ^ C as the functional which is zero on ^^^(a) with a = 1k', and which coincides with the canonical identification m,Am,(1k') — C on the unit component for k! = m'. Then the functional 0 is an invariant integral.

Proof. The normalization condition 0(l(k)) = 1 is immediate by construction. Let 0m be the natural transformation from Fmm to Fkk which has support on direct sums of 1k' with itself, and with (0,m)tk, = 1 for k' = m' and 0 otherwise. Then for f G m.Am, we have 0(f) = f (0m). The invariance of 0 then follows from the easy verification that for x G Nat(Frr, Fkl) one has for example x o 0rm = Jkjll( (x)0m. □

Let us further impose for the rest of this section that C also admits left (and hence right) duality. The following lemma just writes out how the tensor functor F preserves duality.

Lemma 3.8. For all k, l and X GC, the maps

coevX := ◦ Fkk (coevx): C ^ FU(X) <g> Fik(XX), evX := Fu(evx) o if^: Fik(X) <g> Fki(X) ^ C

define a duality between Fki(X) and Fik(X).

Proposition 3.9. The partial bialgebra A is a regular partial Hopf algebra with I as its hyperobject set.

Proof. The statement concerning the hyperobject set I is clear by construction of the units l( k). By Corollary 2.25, it is now sufficient to prove that A is a partial Hopf algebra.

For any x G Nat(Fmn, Fki), let us define S(x) G Nat(Fik, Fnm) by

S(x)X = (id <£> evX) o (id <8> id) o (coevXm <8> id).

Then the assignment S dualizes to maps S: mAin ^ nAm by S(f)(x) = f(S(x)). We claim that S is an antipode for A.

Let us check for example the formula Er f(nr;i)S(f(nr;2)) = Skme(f )l( for f G mAii. By duality, this is equivalent to the pointwise identity of natural transformations Er ^n (id ®S)Air (x) = 4,ml( k) (x) id Fki for x G Nat(Fnn, Fkm), where Mn and (id <S>S) are dual to respectively Anr and id ®S.

Let us fix X G C. Then for any x G Nat(Fnr, Fki), y € Nat(Frn, Fim), we have

(^M? (id ®S)(x < y)) = (id << evm^ (xx << yX << id)(coevfr < id). For any x G Nat(Fnn, Fkm), we therefore have

(M? (id (x))^ = (id << evml X^^X^f j3 < id)(coevf << id).

We sum over r , use naturality of x, and obtain ^ (M? (id ®S)Alr (x)) = (id <g> evml )(nfm) xX0f Fnn(coevx) << id)

= Sk,m 1( k) (x) (id < evml )(nXXm)Fmm (coevx) < id)

= Sk,m 1 (? )(x)( id < evml)(coevm1 < id) = ¿k,™1! ?) (x)idFkl ( x) . □

Assume now that C is a semisimple rigid tensor C*-category, and F a *-functor from C to (Hilbfdj/x/. Let us show that A, as constructed above, becomes a partial Hopf *-algebra with positive invariant integral. In the following definition, we borrow the notation used in the proof of Proposition 3.9, and we write the (left and right) dual of an object X as X.

Definition 3.10. We define *: kmAln ^ lnAkm by the formula f*(x) = f(S(x)*) for x G Nat( F„m, Flk).

Lemma 3.11. The operation * is an anti-linear, anti-multiplicative, comultiplicative involution.

Proof. Anti-linearity is clear. Comultiplicativity follows from the fact that (xy)* = y*x* and S(xy) = S(y)S(x) for natural transformations. To see anti-multiplicativity of *, note first that, since S is anti-multiplicative for A, the map S is anti-comultiplicative on natural transformations. Now as (¿f^™3)* = nXk™ by assumption, we also have Als(x)* = As(x*), which proves anti-multiplicativity of * on A. Finally, involutivity follows from the involutivity of x ^ S'(x)*, which is a consequence of the fact that one can choose evf = (coevf )* and coevf = (evf )*. □

Proposition 3.12. With the above *-structure, (A, A) defines a partial compact quantum group.

Proof. The only thing which is left to prove is that our invariant integral 0 is a positive. We will use the notation from the proof of Lemma 3.7. Now it is easily seen from the

definition of 0 that the mAin(a) are all mutually orthogonal. It is furthermore clear that 0(f *) = 0(f), since S(0m) = (0m,)* = 0m. Hence it suffices to prove that the hermitian form (f\g) = 0(f*g) on mAin(a) is positive-definite.

Let us write f(x) = fx"). By definition, 0(f *g) = (f < g)((S < id)Akm 0)).

Assume that f (x) = (v'\xav) and g(x) = (w'\xaw) for v, w G Fmn(ua) and v', w' G Fki(ua). Then f (x) = (v\xav'), and using the expression for S as in Proposition 3.9 we find that

0(f *g) = (v < w'\(evkai)23 (Akm(0inh,a)24(coevmn )l2(v' < w)).

However, up to a positive non-zero scalar, which we may assume to be 1 by proper rescaling, we have Am(0in)a,a = (evjjl)*(evkl). Hence

0(f *g) = (v < w'\(evki)23((evki)* (ev^))24(coevmn) 12 (v' < w)) = (v < w'\(evki)23(evki)*4(w < v'))

= (v\w)(evkai \v')2)(evaki\w')2)*,

where ev^ \z)2 denotes the map y ^ evJki(y < z). This clearly defines a positive definite inner product on kmAln(a) = Fmn(ua) < Fki(ua)*. □

For G an /-partial compact quantum group with finite hyperobject set 1, let us write Fg for the forgetful functor Repu rcfd(G) ^ Hilb^d.

Theorem 3.13. Fix a set / and a finite set 1. Then the assignment G ^ (Repu rcfd(G), Fg ) is (up to isomorphism/equivalence) a one-to-one correspondence between /-partial compact quantum groups with hyperobject set 1 and semisimple rigid tensor C*-categories C with hyperobject set 1 and faithful tensor *-functor into Hilb^f[.

Proof. By Corollary 2.29, Repurcfd(G) is a semisimple rigid tensor C*-category with hyperobject set 1, and Fg a faithful tensor *-functor into Hilb^f[. Conversely, the results of this section assign to any semisimple rigid tensor C* -category with hyperobject set 1 and faithful tensor *-functor into Hilb^d an /-partial compact quantum group with finite hyperobject set 1. Let us now show that these two maps are inverses of each other, up to isomorphism/equivalence.

Fix now A = P(G), and let B be the partial Hopf *-algebra with invariant integral constructed from Corepu rcfd(A) with its natural forgetful functor. Then we have a map B ^ A which piecewise goes from mBln(a) = Hom( ■mV^, kv/a))* to hmAln(a) sending f to (id<f)(Xa), where the (V(a), ^*a) run over a maximal family of non-equivalent irreducible unitary corepresentations of A. It is easy to check from the definition of B that this map is a morphism of partial Hopf *-algebras. By Corollary 2.24, it is bijective.

Conversely, let C be a semisimple rigid tensor C*-category with hyperobject set I and faithful tensor *-functor F into Hilb^Xa . Let A be the associated partial Hopf *-algebra. For each irreducible ua G C, let V(a) = F( ua), and

km(Xa)ln = £ e* < ei G kmAln < Homc(Fmn(ua),Fkl(ua)), i

where ei is a basis of Homc(Fmn(ua), Fkl(ua)) and e* a dual basis. From the definition of A it easily follows that each Xa is a unitary corepresentation for A. Clearly, Xa is irreducible. As the matrix coefficients of the Xa span A, it follows that the Xa form a maximal class of non-isomorphic unitary corepresentations of A. Hence we can find a unique equivalence C 4 CoreprcfdM(A) sending X to (F(X), Xx) and such that ua 4 Xa. From the definitions of the coproduct and product in A, it is readily verified that the natural morphisms X™'*: Fkl(X) < Flm(Y) 4 Fkm(X < Y) turn it into a monoidal equivalence.

4. Examples

4.1. Hayashi's canonical partial compact quantum groups

Let C be a semisimple rigid tensor C*-category. A semisimple module C*-category D consists of a semisimple C*-category D and a bifunctor C xD 4 D with natural coherence maps such that the obvious module axioms are satisfied [33,28].

Choose a labeling I for a distinguished maximal set {ua} of mutually non-isomorphic irreducible objects of D. Then for a, b GI, we can define

F(X) = ®a,tFab(X), Fab(X) = Hom(ua,X < u6), X gC,

where Fab(X) is a Hilbert space (possibly zero) for the inner product (f\g) = f *g. It is easy to check that one obtains in a natural way a tensor *-functor F from C to Hilb^Xd1, where the rcfd condition follows from Hom(ua, X<ub) = Hom(X<ua, ub), by Frobenius reciprocity. It is not necessarily faithful, as some 1a may act as the zero functor, but using duality one sees that the tensor functor will be faithful on the full tensor C*-subcategory of all X with X < 1a = 1a < X = 0 whenever F(1a) = 0. The associated partial Hopf *-algebra A(c,d) will be called the canonical partial compact quantum group associated with (C, D).

For example, given a faithful tensor *-functor F: C 4 Hilbf and defining D = Hilbfd as the category of /-graded finite-dimensional Hilbert spaces V = ®k kV with

k (X < V) = 0lFkl(X) < V, X gC,V g Hilbd, we get back the reconstruction obtained in the previous section.

If C is a semisimple rigid tensor C*-category, one can take D = C endowed with the module structure coming from the tensor product of C. The associated partial Hopf *-algebra Ac coincides with Hayashi's construction [16] in case C has only finitely many irreducible object classes.

As an example coming from compact quantum group theory, let G be a compact quantum group with ergodic action on a unital C*-algebra C (X). Then the collection of finitely generated G-equivariant C (X)-Hilbert modules forms a semisimple module C*-category over Repu (G), cf. [9].

4.2. Morita equivalence

Definition 4.1. Two partial compact quantum groups G and H with finite hyperobject set are said to be Morita equivalent if there exists a monoidal *-equivalence Repu rcfd(G) ^

Repu,rcfd(H ).

In particular, if G and H are Morita equivalent they have the same hyperobject set, but they need not share the same object set.

Definition 4.2. A linking partial compact quantum group consists of a partial compact quantum group G defined by a partial Hopf *-algebra A over a set / with a distinguished partition / = /1 U /2 such that the idempotents l( j) = Eiei- l(k) G M(A) are central, and such that for each r G /i, there exists s G /i+1 such that l( SS) =0 (with the indices i considered modulo 2).

If A defines a linking partial compact quantum group G, we can split the total algebra A into four component algebras Aj = Al(i) = l(i) A. It is readily verified that for equal indices, the Ai together with all Ars with r, s G /i define themselves partial compact quantum groups Gi, called the corner partial compact quantum groups of G. It is clear from the conditions on a linking partial compact quantum group that the partial compact quantum groups G,G1 and G2 all share the same hyperobject set.

Proposition 4.3. Two partial compact quantum groups with finite hyperobject set are Morita equivalent iff they arise as the corners of a linking partial compact quantum group.

Proof. Suppose first that G1 and G2 are Morita equivalent partial compact quantum groups with associated partial Hopf *-algebras A1 and A2 over respective sets /1 and /2. Then we may identify their corepresentation categories with the same abstract tensor C*-category C based over their common hyperobject set 1. This C comes endowed with two forgetful functors Fi to Hilb^X^ corresponding to the respective Ai.

With / = /1 U /2, we can combine the Fi into a global (faithful) tensor *-functor F: C ^ HilbIcxfI, with F(X) = F1(X) © F2(X). Let A be the associated partial Hopf *-algebra constructed from the Tannaka-Krein-Woronowicz reconstruction procedure.

From the precise form of this reconstruction, it follows immediately that mAln = 0

j) = E keii,leij !( k

if either k, l or m, n do not lie in the same /i. Hence the l( j) = E keI- lei- l(k) are

central.

Moreover, fix k G /i and any l G /i+1 with k' = l'. Then Nat(Fll, Fkk) = {0}. It follows that l(k) =0. Hence A defines a linking partial compact quantum group. It is clear that A1 and A2 are the corners of A.

Conversely, suppose that A1 and A2 arise from the corners of a linking partial compact quantum group defined by A with invariant integral 0. We will show that the associated partial compact quantum groups G and Gl are Morita equivalent. Then by symmetry G and G2 are Morita equivalent, and hence also G1 and G2.

For (V, X) G Rep„rcfd(G), let F(V, X) = (W, Y) be the pair obtained from (V, X) by restricting all indices to those belonging to /1. It is immediate that (W, Y) is a unitary corepresentation of A1, and that the functor F is a tensor *-functor from RepM rcfd(G) to RepM rcfd(G1). What remains to show is that F is an equivalence of categories, i.e. that F is faithful, full and essentially surjective.

By assumption, the hyperobject set of a linking partial compact group coincides with the hyperobject sets of its corners. Hence, using the assumed rigidity, we obtain that F is faithful since End(1) = End(F (1)).

To complete the proof, it is sufficient to show that F induces a bijection between isomorphism classes of irreducible unitary corepresentations of A and of A1. Note that by Theorem 2.14 and Lemma 2.17, each such class can be represented by a restriction of the regular corepresentation of A or A1 , respectively.

So, let (W, Y) be an irreducible restriction of the regular corepresentation of A1. Pick a non-zero a G mWn, define pVq(a) C (J)kl kAlq as in (2.18) and form the regular corepresentation (V(a), X) of A. Then pVqa) = pW(a for all p, q G /1 by Lemma 2.18 2. and hence F(V, X) = (W, Y). Since F is faithful, (V, X) must be irreducible.

Conversely, let (V, X) be an irreducible restriction of the regular corepresentation of A. Since F is faithful, there exist k, l G /1 such that kVl = 0. Applying Corollary 2.24, we may assume that pVq C kAlq for some k, l G /1 and all p, q G /. But then F(V, X) is a restriction of the regular corepresentation of A1. If F(V, X) would decompose into a direct sum of several irreducible corepresentations, then the same would be true for (V, X) by the argument above. Thus, F(V, X) is irreducible.

Finally, assume that (V, X) and (W, Y) are inequivalent irreducible unitary corepresentations of A. Then 0(C(V, X)*C(W, Y)) = 0 by Corollary 2.31. Since 0 is faithful by Corollary 2.24, C(V, X) n C(W, Y) = 0, and hence C(F(V, X)) n C(F(W, Y)) = 0. So F(V, X) and F(W, Y) are inequivalent. □

If G1 and G2 are Morita equivalent compact quantum groups, the total partial compact quantum group coincides with the co-groupoid G constructed in [1].

4.3. Weak Morita equivalence

Definition 4.4. A semisimple rigid linking tensor C*-category consists of a semisimple rigid tensor C*-category C with a distinguished partition 1 = 11U12 of its hyperobject set such that for each a G 11, there exists ¡3 G 12 with Cap = {0}.

The corners Ci of C are the full semisimple rigid tensor C*-subcategories of objects X with X = li <g) X <g) li, where li = ©ae1a.

The following notion is essentially the same as the one by M. Muger [27].

Definition 4.5. Two semisimple rigid tensor C*-categories C1 and C2 over respective sets 11 and 12 are called Morita equivalent if there exists a semisimple rigid linking tensor C*-category C over the set 1 = 11 U 12 whose corners are isomorphic to C1 and C2.

We say two partial compact quantum groups G1 and G2 with finite hyperobject set are weakly Morita equivalent if their representation categories Repurcfd(Gi) are Morita equivalent.

One can prove directly that this is indeed an equivalence relation, but it will follow indirectly from the discussion below.

The following notion is dual to that of linking partial compact quantum group.

Definition 4.6. A co-linking partial compact quantum group consists of a partial compact quantum group G defined by a partial Hopf *-algebra A over an index set I, together with a distinguished partition I = I1 U I2 such that l(£) = 0 whenever k G Ii and l G Ii+i, and such that for each k G Ii; there exists l G Ii+i with £A\ = 0.

It is again easy to see that if we restrict all indices of a co-linking partial compact quantum group to one of the distinguished sets /i, we obtain a partial compact quantum group Ai which we will call a corner. If we write ei = Ek leI. l( l), we can decompose the total algebra A into components Aij = eiAej, and correspondingly write A in matrix

notation A = ( 11 ,12), where Aii = Ai. A21 A22

Lemma 4.7. If A is a co-linking partial compact quantum group, then ij^Ajk — ik.

Proof. It suffices to show A12A21 = A11. Take k G /1, and pick l G /2 with kAl = {0}. Then in particular, we can find an a G kAl with e(a) = 1. Hence for any m G /1, we have l( k) = l( m )a(1)S(a(2}) G A12A21. Hence this latter space contains all local units of A11. As it is a right An-module, it follows that it is in fact equal to A11. □

It follows that An and A22 are Morita equivalent algebras, where for non-unital algebras one can define Morita equivalence by asking for the existence of a (non-unital) linking algebra satisfying the conclusion of the previous lemma.

Definition 4.8. We call two partial compact quantum groups co-Morita equivalent if there exists a co-linking partial compact quantum group having these partial compact quantum groups as its corners.

Lemma 4.9. Co-Morita equivalence is an equivalence relation.

Proof. The idea is standard, and consists in concretely building the appropriate colinking partial compact quantum groups. Let us illustrate this for transitivity. In the proof, we write ~ for the relation of co-Morita equivalence.

Let G1, G2 and G3 be three partial compact quantum groups with G1 ~ G2 and G2 ~ G3.

in the upper left and lower right 2 by 2 corners the two co-linking partial compact quantum groups between resp. the partial compact quantum groups G1 and G2, and G2 and G3. For example, A13 is constructed as A12 ®a22 A23. It is straightforward to define a regular weak multiplier Hopf *-algebra structure on A{12 3} satisfying the conditions of Proposition 1.6 and restricting to the given structures on the 2 by 2 corners.

Let now 0 be the functional which is zero on the off-diagonal entries Aij and which coincides with the invariant positive integrals on the Aii. Then it is readily checked that 0 is invariant. To show that 0 is positive, we invoke Remark 2.32. Indeed, any irreducible corepresentation of A{12 3} has coefficients in a single Aij. For those i, j with \i — j\ < 1, we know that the corepresentation is unitarizable by restricting to a corner 2 x 2-block. If however the corepresentation X has coefficients living in (say) A13, it follows from the identity A12A23 = A13 that the corepresentation is a direct summand of a product Y © Z of corepresentations with coefficients in respectively A12 and A23. This proves unitarizability of X. It follows from Remark 2.32 that 0 is positive, and hence A{12 3} defines a partial compact quantum group.

We claim that the subspace A{1j3} (in the obvious notation) defines a co-linking compact quantum group between G1 and G3. In fact, it is clear that the A11 and A33 are corners of A{13}, and that l(k) = 0 for k, l not both in I1 and I3. To finish the proof, it is sufficient to show now that for each k G I1, there exists l G I3 with kAl = 0, as the other case follows by symmetry using the antipode. But there exists m G I2 with kAm = {0}, and l G I3 with mAl = {0}. As in the discussion following Definition 4.6, this implies that there exists a G kAm and b G mAl with e(a) = e(b) = 1. Hence e(ab) = 1, showing k Al = {0}. □

Proposition 4.10. Assume that two partial compact quantum groups G1 and G2 with finite hyperobject set are co-Morita equivalent. Then they are weakly Morita equivalent.

Proof. Consider the corepresentation category C of a co-linking partial compact quantum group A over I = I1 UI2. Let I 4 1 be the partition of I along the hyperobject set. Then by the defining property of a co-linking partial compact quantum group, also 1 =

Then we can build a 3 by 3 matrix algebra A{123}

Ii U 12 with 1i = ^(Ij) is a partition. In particular, writing again li = ®aei 1a, we have that 1i®C®1i = RepM rcfd(Gi), since any (irreducible) unitary rcfd corepresentation of Ai is automatically also a(n irreducible) corepresentation of A.

Fix now a G Ii and k G a. As A is co-linking, there exists l G I2 with £A\ = {0}. By Lemma 2.18 there exists a non-zero regular unitary corepresentation supported inside ®m,nmAln. If then l G Iß with ß G 12, it follows that Caß = 0. By symmetry, we also have that for each a G 12 there exists ß G Ii with Caß = {0}. This proves that the C forms a linking partial tensor C*-category over I = I U 12. □

Proposition 4.11. Let C be a semisimple rigid linking tensor C'-category over I = Ii U I2• Let I be a set parametrizing a maximal family of non-equivalent irreducible unitary rcfd corepresentations C, and let I = Ii UI2 be the partition of I corresponding to the one of I• Then the associated canonical partial compact quantum group is a co-linking partial compact quantum group over I = li Ul2.

To be clear, to a Gl one assigns the unique a G 1 such that la ® ua = ua, and this provides the corresponding partition of I = li Ul2 = Uae 1 la.

Proof. Let A = Ac define the canonical partial compact quantum group with object set I. A fortiori, l( = 0 if a and b are not both in li or l2.

Fix now a Gla for some a G Ij. Pick ß G Ii+i with Caß = {0}, and let (V, X) be a non-zero irreducible unitary rcfd corepresentation inside Caß. Applying Lemma 2.13 with respect to the identity morphism, we get that there exists b Glß with aVb = {0}. As (e <g) id) aXjb = id avb, we find that aAb = 0. This proves that A defines a co-linking partial compact quantum group.

Note that the corners of the canonical partial compact quantum group associated with the semisimple rigid linking tensor C*-category are not the canonical partial compact quantum groups associated to the corners of the linking tensor C*-category, the reason being that the canonical construction is based only on left tensor multiplication and does not 'cut down on the right'. Rather, the two corner constructions will be related by a Morita equivalence.

Theorem 4.12. Two partial compact quantum groups Gi and G2 with finite hyperobject set are weakly Morita equivalent if and only if they are connected by a string of Morita and co-Morita equivalences•

Proof. Clearly if two partial compact quantum groups are Morita equivalent, they are weakly Morita equivalent. By Proposition 4.10, the same is true for co-Morita equivalence. This proves one direction of the theorem.

Conversely, assume Gi and G2 are weakly Morita equivalent. Let C be a semisimple rigid linking tensor C*-category between RepMrcfd(G1) and RepM rcfd(G2). Then the Gi are Morita equivalent with the corners of the canonical partial compact quantum group

associated to C. But Proposition 4.11 shows that these corners are co-Morita equivalent. □

5. Partial compact quantum groups from reciprocal random walks

5.1. Reciprocal random walks and the Temperley-Lieb category

In this section, we investigate a special class of partial compact quantum groups constructed from t-reciprocal random walks [9]. We first recall this notion, slightly changing the terminology for the sake of convenience.

Definition 5.1. Let t G Mo. A t-reciprocal random walk consists of a quadruple (r, w, sgn, i) where r = (r(0), r(1), s, t) is a graph with source and target maps s and t, where w is a weight function w: r(1) 4 M+ and sgn a sign function sgn: r(1) 4 {±1}, and where i is an involution e 4 e on r(1) interchanging source and target, satisfying for all e the weight reciprocality w(e)w(e) = 1, the sign reciprocality sgn(e) sgn(e) = sgn(t), and the random walk property J2s(e)=v |7|w (e) = 1 for all v G T(0).

By [9, Proposition 3.1], there is a uniform bound on the number of edges leaving from any given vertex v, i.e. r has a finite degree. For examples of t-reciprocal random walks, we refer to [9].

Let now 0 < \q\ < 1, and let Tq be the Temperley-Lieb C*-tensor category, which is the universal tensor C*-category with irreducible unit and duality, generated by a single self-adjoint object X and duality morphism R: 1 4 X < X satisfying

R*R = \q\ + \q\ 1, (R* < idx)(idx <R) = — sgn(q) idx .

Then if r = (r, w, sgn, i) is a —(q + q--^-reciprocal random walk, we have a *-functor Fr from Tq into Hilb^d with I = r(0), by sending X to the bigraded Hilbert space Hr = l2(r(1)), where the r(0) -bigrading is given by 5e G s(e)Hit(e), and R to the morphism

Rr: l2(r(0)) 4Hr ®r(G> Hr, Rr\$v = £ sgn(e)yW(ë)Se <g> Së.

e,s(e)=v

Note that Hr is rcfd as r has finite degree. Up to equivalence, Fr only depends upon the isomorphism class of (r, w), and is independent of the chosen involution or sign structure. Conversely, every tensor *-functor from Tq into Hilbjf for some set I arises in this way [8].

5.2. Partial compact quantum groups from reciprocal random walks

Let r = (r, w, sgn, i) be a — (q + q -^-reciprocal random walk. Let us denote by A(r) the I-partial compact quantum group associated to the functor Fr by the Tannaka-

Krein-Woronowicz reconstruction result. Our aim is to give a direct representation of the associated algebra A(r) by generators and relations. We will write rvw C r(1) for the set of edges with source v and target w.

Theorem 5.2. The *-algebra A(r) is the universal *-algebra generated by self-adjoint orthogonal idempotents l( W) for v, w G r(0) and elements (uej)ef er(i) where the uej G

SsU)A(r)Kf) satisfy u*f = sgn(e) sgn(f)v//wJ)uz, f and

£ uleugj = SeJl( tWe) ), Vw G r( 0), e, f G r( 1), (5.1)

t (g)=w

£ ue,gufgg = SeJl( s(ve) ) Vv G r(0), e, f G r(1). (5.2)

S(g)=v

The partial Hopf *-algebra structure is given by Avw(uej) = E s(g)=v ueg <8> ugj,

t(g)=w

£(ue, f ) = Se , f and S(ue , f ) = uf, e.

Note that the sums in (5.1) and (5.2) are in fact finite, as r has finite degree.

Proof. Let (H, V) be the generating unitary corepresentation of A(r) on H = l2(r(1)). Then V decomposes into parts ^Vi = Ee f Ve, f <8> Ee, f G m,Aln <8> B( mHn, kHi), where the Ee, j are the natural matrix units and with the sum over all e with s(e) = k, t(e) = l and all f with s(f) = m, t(f) = n. By construction V defines a unitary corepresentation of A(r), hence the relations (5.1) and (5.2) are satisfied for the ve,f. Now as Rr is an intertwiner between the trivial representation on C(r( )) = ®ver(o) C and V (T)r(o) V, we have for all v G r(0) that

£ veJ v g, h <8> ((EeJ <g> Eg h)RrSv )=£ l( W ) ® RrSv , (5.3)

e,f,g,heT(1) w

t(f)=s(h),t(e)=s(g)

£ sgn(k^w(k) (ve,kvgk ® Se <g> Sg

e,g,k t(e) = s(g),s(k)=v

£ sgn(k)Vw(k) (i(w) ® Sk ® Sk).

w,k s(k)=w

S° if t(e) = s(g) = Z we have Ek,s(k)=v sgn(kVw(k)ve,kv g,k = Se,g sgn(eVw(e)1{ ^ ) . Multiplying to the left with ve l and summing over all e with t(e) = z, we see from (5.1)

that also v* f = sgn(e) sgn(f )y WWfv^ f holds. Hence the ve, f satisfy the universal relations in the statement of the theorem. The formulas for comultiplication, counit and antipode then follow immediately from the fact that V is a unitary corepresentation.

Let us now a priori denote by B(r) the *-algebra determined by the relations (5.1), (5.2) and the relation for the adjoint as above, and write B(r) for the associated r(0) x r(0)-partial *-algebra induced by the local units l( W). Write A(l( W)) = Ezer(o) l(v) < l(W) and A(ue,f) = Eser<1) ue,g < ug,f, which makes sense in M(B(r) < B(r)) as the degree of r is finite. Then an easy computation shows that J2t(g)=w A(uge)* A(ugj) = 5ej A(l( tWe))). Similarly, the analogue of (5.2) holds for A(uej). As also the relation for the adjoint holds trivially for A(uej), it follows that we can define a *-algebra homomorphism A: B(r) 4 M(B(r) < B(r)) sending ueff to A(ue, f) and l(W) to A(l(W)). Cutting down, we obtain maps Avw: rB(r)sz 4 VB(r)W < VB(r)W which are easily seen to satisfy Definition 1.5. Moreover, the AVW are coassociative as they are coassociative on generators.

Let now Ev W be the matrix units for l 2(r(0)) . Then one verifies again directly from the defining relations of B(r) that one can define a *-homomorphism e: B(r) 4 B(l2(r(0))) sending l (W) to Sv W ev,v and ue,f to Sej es(e),t(e). We can hence define a map e: B(r) 4 C such that e(x) = e(x)evW for all x G vB(r)W, and which is zero elsewhere. Clearly it defines a morphism on the partial algebra B(r). As e satisfies the counit condition on generators, it follows by partial multiplicativity that it satisfies the counit condition on the whole of B(r), i.e. B(r) is a partial *-bialgebra.

It is clear now that the uef define a unitary corepresentation U of B(r) on Hr. Moreover, from (5.1) and the formula for u* f we can deduce that Rr: Cr(o) 4 Hr <r(o) Hr is a morphism from C(r(0)) to U ©r(o) U in CorepM rcfd(B(r)), cf. (5.3). From the universal property of Tq, it then follows that we have a tensor *-functor Gr: Tq 4 CorepM rcfd(B(r)) with Gr(X) = U. On the other hand, as we have a A-preserving *-homomorphism B(r) 4 A(r) by the universal property of B(r), we have a strongly monoidal *-functor Hr: Coreprcfd u(B(r)) 4 Corepu(A(r)) = Tq which is inverse to Gr. Since the commutation relations of A(r) are completely determined by the morphism spaces of Tq, it follows that we have a *-homomorphism A(r) 4 B(r) sending ve f to ue f. This proves the theorem.

We remark that for finite graphs with their canonical weights coming from the Perron-Frobenius eigenvalues ([8, Section 3.1]), these partial compact quantum groups were considered in [14, Section 6].

5.3. Partial compact quantum groups from homogeneous reciprocal random walks

Let us now consider a particular class of 'homogeneous' —(q + q--^-reciprocal random walks. Namely, assume that there exists a finite set T partitioning r(1) = Uar^1^ such that for each a G T and v G r(0), there exists a unique ea(v) G r^1^ with source v. Write av for the range of ea(v). Assume moreover that T has an involution a 4 a such that

ea(v) = ea(av). Then for each a, the map v ^ av is a bijection on r(0) with inverse v ^ av. In particular, also for each w G r(0) there exists a unique fw(a) G r^1^ with target w.

Let us further denote wa(v) = w(ea(v)) and sgna(v) = sgn(ea(v)). Let again A(r) be the total * -algebra of the associated partial compact quantum group. Using Theorem 5.2, we have the following presentation of A(r): it is generated by self-adjoint mutually orthogonal idempotents l( w) and elements (uab)vw := uea(v),eb(v) for a, b G T and

v, w G r(0) with defining relations (ua,b)*v,w = (ua:b)av,bw and

^ ^ (ua,b) av,w (ua,c) av,z Sw,z Sb,cl( bw) , ^ ^ (ub,a)w,v (uc,a) z,v Sb,cSw,z l ( v ) . aET aET

The element (uab)vw lives inside the component vwA(r)6w.

Let us now consider M(A(r)), the multiplier algebra of A(r). For a function f on r(0) x r(0), write f (X, p) = £vw f (v, w)l( w) G M(A(r)). Similarly, for a function f on r(0) we write f (X) = £v,w /(v)l( VJ and f (p) = £v,w f (w)l( VJ. We then write for example f (aX, p) for the element corresponding to the function (v, w) ^ f (av, w).

We can further form in M(A(r)) the elements ua b = w(ua,b)v,w. Then u = (ua b) is a unitary m x m matrix for m = #T. Moreover,

a,b ua,b

Yb(p) Ya(X) '

where Ya(v) = sgna(v)y^wa(v). We then have the following commutation relations between functions on r(0) x r(0) and the entries of u:

f (X, p)ua,b = uabf (aX, ap), (5.5)

where f (aX, bp) is given by (v, w) ^ f (av, bw). Further, A(ua,b) = A(1) J2c(uac < uc,b). Note that the * -algebra generated by the ua^b is no longer a weak Hopf * -algebra when r(0) is infinite, but rather one can turn it into a Hopf * -algebroid.

Remark 5.3. The weak multiplier Hopf algebra A(r) is related to the free orthogonal dynamical quantum groups introduced in [41] as follows. Denote by G the free group generated by the elements of T subject to the relation a = a-1 for all a G T. By assumption on r, the formula (af )(v) := f (av) defines a left action of G on Fun(r(0)). Denote by C C Fun(r(0)) the unital subalgebra generated by all Ya and their inverses and translates under G, write the elements of T C G as a tuple in the form V = (a1, a1,..., an, an), and define a V xV matrix F with values in C by Fa^b := Sb^a^a. Then the free orthogonal dynamical quantum group AC(V, F, F) introduced in [41] is the universal unital *-algebra generated by a copy of C <g> C and the entries of a unitary V x V-matrix v = (va,b) satisfying

va,b(f < g) = (af < bg)va,b, (aFaa < 1)v* b = va,b(1 < Fb -b)

for all f, g G C and a, b gV. The second equation can be rewritten as v* ^ = va, b(Y—1 < Yb). Comparing with (5.4) and (5.5), we see that there exists a *-homomorphism

~ ( f ® g 4 f (A)g(p),

AC(V, F, F) 4 M(A(r)), f g u mph

{ va,b 4 ua,b.

The two quantum groupoids are related by an analogue of the unital base changes considered for dynamical quantum groups in [41, Proposition 2.1.12]. Indeed, Theorem 5.2 shows that A(r) is the image of A'C(V, F, F) under a non-unital base change from C to Funf (r(0)) along the natural map C 4 M(Funf (r(0))).

5.4. Dynamical quantum SU(2) as a partial compact quantum group

As a particular example, take 0 < \q\ < 1 and x > 0, and consider the graph with

ri0) = x\q\Z and = {(y, z) \ y/z G {\q\, \q\-1}}. Endow r with the involution (y, z) 4

(z, y), the weight w(y, z) = z+zy-1 and the sign sgn(y, z) = aM if y/z = \q\M, where a+ = 1

and a- = — sgn(q). Consider further the set T = {+, —} with the non-trivial involution,

and label the edges (y, z) with ¡j if y/z = \q\M. Write F(y) = \q\ 1 —, and put

F i/2(o— 1) 1

a = f 1/2 (\—1)u— and ¡3 = f 1 /2(\—1)u—+. Then our relations for the uf v are equivalent

to the commutation relations

a/3 = qF (p — 1) ¡a aft* = qF (\)fi*a (5.6)

aa* + F (A)/= 1, a* a + q—2F (p — 1) —1/*3 = 1, (5.7)

F (p — 1) —1 aa* + 33* = F (A — 1) —1, F (A)a*a + q—233* = F (p), (5.8)

f (A)g(p)a = af (A + 1)g(p +1), f (A)g(p)3 = 3f (A +1)g(p — 1). (5.9)

These are precisely the commutation relations for the dynamical quantum SU(2)-group as in [24, Definition 2.6], except that the precise value of F has been changed by a shift in the parameter domain. The (total) coproduct on Ax also agrees with the one on the dynamical quantum SU(2)-group. Note that the case of q a root of unity case was considered in [20, Section 5], see also [18,10] for generalizations to higher rank (in resp. the unitary and non-unitary case).

Acknowledgments

K. De Commer was supported by the FWO grant G.0251.15N, and T. Timmermann was supported by the SFB 878 of the DFG. We thank M. Soleimani Malekan and an anonymous referee for comments on the paper. We thank A. Van Daele for providing details concerning [47].

References

J. Bichon, Hopf—Galois objects and cogroupoids, Rev. Un. Mat. Argentina 55 (2) (2014) 11—69. G. Böhm, Comodules over weak multiplier bialgebras, Internat. J. Math. 25 (5) (2014) 1450037 (57 pages).

G. Böhm, S. Caenepeel, K. Janssen, Weak bialgebras and monoidal categories, Comm. Algebra 39 (12) (2011) 4584-4607.

G. Böhm, J. Gómez-Torrecillas, E. López-Centella, Weak multiplier bialgebras, Trans. Amer. Math. Soc. (2015), in press, arXiv:1306.1466.

G. Böhm, F. Nill, K. Szlachányi, Weak Hopf algebras. I. Integral theory and C*-structure, J. Algebra 221 (2) (1999) 385-438.

J. Dauns, Multiplier rings and primitive ideals, Trans. Amer. Math. Soc. 145 (1969) 125-158. K. De Commer, T. Timmermann, Dynamical quantum su(2) groups on the operator algebra level, in preparation.

K. De Commer, M. Yamashita, Tannaka-Krein duality for compact quantum homogeneous spaces. II. Classification of quantum homogeneous spaces for quantum SU(2), J. Reine Angew. Math. (2015), in press, arXiv:1212.3413.

K. De Commer, M. Yamashita, Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ. 28 (31) (2013) 1099-1138.

P. Etingof, D. Nikshych, Dynamical quantum groups at roots of 1, Duke Math. J. 108 (1) (2001) 135-168.

P. Etingof, A. Varchenko, Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Comm. Math. Phys. 196 (3) (1998) 591-640.

P.H. Hai, Tannaka-Krein duality for Hopf algebroids, Israel J. Math. 167 (2008) 193-225. T. Hayashi, Quantum group symmetry of partition functions of IRF models and its application to Jones' index theory, Comm. Math. Phys. 157 (2) (1993) 331-345.

T. Hayashi, Compact quantum groups of face type, Publ. Res. Inst. Math. Sci. 32 (2) (1996) 351-369. T. Hayashi, Face algebras. I. A generalization of quantum group theory, J. Math. Soc. Japan 50 (2) (1998) 293-315.

T. Hayashi, A canonical Tanaka duality for finite semisimple tensor categories, arXiv:math/9904073. T. Hayashi, Face algebras and tensor categories, in: Representation Theory and Noncommutative Harmonic Analysis, Kyoto, 1998, Sürikaisekikenkyüsho Kokyüroku 1082 (1999) 34-47 (in Japanese). T. Hayashi, Face algebras and unitarity of su(n)¿-TQFT, Comm. Math. Phys. 203 (1) (1999) 211-247.

T. Hayashi, Galois quantum groups of IIi-subfactors, Tohoku Math. J. (2) 51 (3) (1999) 365-389. T. Hayashi, A brief introduction to face algebras, in: New Trends in Hopf Algebra Theory, La Falda, 1999, in: Contemp. Math., vol. 267, Amer. Math. Soc., Providence, RI, 2000, pp. 161-176. T. Hayashi, Realization of amenable bicategories and weak amenability of fusion algebras, J. Math. Soc. Japan 53 (1) (2001) 147-174.

A. Joyal, R. Street, An introduction to Tannaka duality and quantum groups, in: Category Theory, Como, 1990, in: Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 413-492.

B.-J. Kahng, A. Van Daele, The Larson-Sweedler theorem for weak multiplier Hopf algebras, arXiv:1406.0299.

E. Koelink, H. Rosengren, Harmonic analysis on the SU(2) dynamical quantum group, Acta Appl. Math. 69 (2) (2001) 163-220.

R.G. Larson, Characters of Hopf algebras, J. Algebra 17 (1971) 352-368. R. Longo, J.E. Roberts, A theory of dimension, K-Theory 11 (2) (1997) 103-159. M. Müger, From subfactors to categories and topology I. Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Algebra 180 (2003) 81-157.

M. Müger, Tensor categories: a selective guided tour, Rev. Un. Mat. Argentina 51 (1) (2010) 95-163. S. Neshveyev, Duality theory for nonergodic actions, arXiv:1303.6207.

S. Neshveyev, L. Tuset, Compact Quantum Groups and Their Representation Categories, Cours Spécialisés (Specialized Courses), vol. 20, Société Mathématique de France, Paris, 2013. D. Nikshych, L. Vainerman, Finite quantum groupoids and their applications, in: New Directions in Hopf Algebras, in: Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 211-262.

F. Nill, Axioms for weak bialgebras, arXiv:math/9805104.

V. Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2) (2003) 177-206.

[34] H. Pfeiffer, Tannaka—Krein reconstruction and a characterization of modular tensor categories, J. Algebra 321 (12) (2009) 3714-3763.

[35] P. Schauenburg, Tannaka Duality for Arbitrary Hopf Algebras, Algebra Berichte (Algebra Reports), vol. 66, Verlag Reinhard Fischer, Munich, 1992.

[36] P. Schauenburg, Face algebras are Xr-bialgebras, in: Rings, Hopf Algebras, and Brauer Groups, Antwerp/Brussels, 1996, in: Lecture Notes in Pure and Appl. Math., vol. 197, Dekker, New York, 1998, pp. 275-285.

[37] P. Schauenburg, Weak Hopf algebras and quantum groupoids, in: Noncommutative Geometry and Quantum Groups, Warsaw, 2001, in: Banach Center Publ., vol. 61, Polish Acad. Sci, Warsaw, 2003, pp. 171-188.

[38] J.V. Stokman, Vertex-IRF transformations, dynamical quantum groups and harmonic analysis, Indag. Math. (N.S.) 14 (3-4) (2003) 545-570.

[39] K. Szlachanyi, Fiber functors, monoidal sites and Tannaka duality for bialgebroids, arXiv:0907.1578.

[40] M. Takeuchi, Groups of algebras over a 0 a, J. Math. Soc. Japan 29 (3) (1977) 459-492.

[41] T. Timmermann, Free dynamical quantum groups and the dynamical quantum group SU^(2), in: P.M. Soltan, W. Pusz (Eds.), Operator Algebras and Quantum Groups, in: Banach Center Publ., vol. 98, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2012, pp. 311-341.

[42] J.-L. Tu, Non-Hausdorff groupoids, proper actions and k-theory, Doc. Math. 9 (2004) 565-597, (electronic).

[43] A. Van Daele, Multiplier Hopf algebras, Trans. Amer. Math. Soc. 342 (2) (1994) 917-932.

[44] A. Van Daele, An algebraic framework for group duality, Adv. Math. 140 (2) (1998) 323-366.

[45] A. Van Daele, S. Wang, Weak multiplier Hopf algebras. Preliminaries, motivation and basic examples, in: P.M. Soltan, W. Pusz (Eds.), Operator Algebras and Quantum Groups, in: Banach Center Publ., vol. 98, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2012, pp. 367-415.

[46] A. Van Daele, S. Wang, Weak multiplier Hopf algebras. The main theory, arXiv:1210.4395.

[47] A. Van Daele, S. Wang, Weak multiplier Hopf algebras III. Integrals and duality, preprint, University of Leuven (Belgium) and Southeast University of Nanjing, China.

[48] S.L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted su(n) groups, Invent. Math. 93 (1) (1988) 35-76.