Introduction to compact (matrix) quantum groups and Banica Speicher (easy) quantum groups

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 127, No. 5, November 2017, pp Introduction to compact (matrix) quantum groups and Banica Speicher (easy) quantum groups MORITZ WEBER Saarland University, Saarbrücken, Germany MS received 14 July 2017; published online 27 November 2017 Abstract. This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5 24 January We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz s quantum version of the Tannaka Krein theorem. Building on this, we define Banica Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica Speicher quantum groups and we list some open problems. Keywords. Compact quantum groups; compact matrix quantum groups; easy quantum groups; Banica Speicher quantum groups; noncrossing partitions; categories of partitions; tensor categories; Tannaka Krein duality. Mathematics Subject Classification. 20G42; 05A18; 46LXX. 1. Introduction The study of symmetries in mathematics is almost as old as mathematics itself. From the 19th century onwards, symmetries are mostly modelled by actions of groups. However, modern mathematics requires an extension of the symmetry concept to highly noncommutative situations. This was the birth of quantum groups in the 1980 s; in the ICM 1986 in Berkely, the notion quantum group was coined by Drinfeld (see [82, Preface]), one of the pioneers together with Jimbo. See also the preface of Klimyk and Schmüdgen s book for more on the origins of quantum groups in quantum physics and why they should serve as the concept of symmetry in physics [84, Preface]. Also Woronowicz had applications to physics in mind when he introduced topological quantum groups in In [132, Introduction], he writes that in the existing theory (in physics), it is known that the symmetry described by the considered group is broken referring to elementary particle physics. We will follow his approach to quantum groups This is a survey article on the emerging subject of Banica Speicher quantum groups, based on a first and extensive set of notes produced by Soumyashant Nayak of a lecture series in IMSc, Chennai during January 05 24, Indian Academy of Sciences 881

2 882 Moritz Weber based on the concept of non-commutative function algebras by Gelfand Naimark, using C -algebras as our underlying algebras. The following illustrates Gelfand Naimark s philosophy of (topological) quantum spaces. Topology Non-comm. topol. X comp. space C(X) cont. fcts. AC -algebra identif. (comm. C -alg.) fg = gf (non-comm.). Now, what are symmetries of such quantum spaces? In the same spirit, we should have Comp. groups Comp. quantum groups. G comp. group C(G) cont. fcts. AC -algebra. G G G C(G) C(G G) : A A A In the first half of these lecture notes, we give the basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz s quantum version of the Tannaka Krein theorem. See [86,87,95] for older surveys on compact quantum groups, or [98,111] for more recent books. We do not cover the more general concept of locally compact quantum groups of Kustermans and Vaes [88]. Moreover, we neglect algebraic quantum groups (which are based on the theory of Hopf algebras), see [82,84,94] for more on this, or [111] for some links and similarities between algebraic and topological perspectives on quantum groups. In the second half, we give an introduction to Banica Speicher quantum groups [31], also called easy quantum groups. They were defined in 2009 as a class of compact matrix quantum groups determined by the combinatorics of set partitions. The construction relies on Woronowicz s Tannaka Krein theorem, a quantum version of Schur Weyl duality. His theorem basically states that compact matrix quantum groups are in one-to-one correspondence to certain tensor categories. The idea behind Banica Speicher quantum groups is now to define a class of combinatorial objects which behaves like a tensor category and to which we may actually associate one which thus yields a quantum group by Tannaka Krein duality. Thus, Banica Speicher quantum groups are of a very combinatorial nature. To give a feeling for them, let us be slightly more precise. Consider a partition p of the finite set {1,...,k, 1,...,l } into disjoint subsets. We represent such partitions pictorially using lines to represent the disjoint subsets. For instance, with k = 2 and l = 4, p ={{1}, {2, 2 }, {1 }, {3, 4 }} is represented as We associate a linear map T p : (C n ) k (C n ) l P (2, 4). to such a partition p P(k, l). We then define operations on partitions, which match nicely via the assignment p T p with tensor category like operations on the maps T p (such as forming the tensor product, the composition or the involution of linear maps). A class

3 CMQGs and Banica Speicher quantum groups 883 of partitions which is closed under these operations is called a category of partitions; it thus induces a tensor category via p T p ; and it thus induces a compact matrix quantum group via Tannaka Krein, a Banica Speicher quantum group. Summarizing Combinatorics Comp. QG C categ. of part. R C tensor categ. G BS QG p partition p T p T p linear map TK duality We end the lectures by sketching the classification of Banica Speicher quantum groups and listing some applications. Moreover, we survey the state-of-the-art on Banica Speicher quantum groups and we list several open problems. See [31,103,128] or the chapter [118, Ch. Easy quantum groups] for more on Banica Speicher quantum groups. These lecture notes are a transcript of a series of eight lectures, 90 min each, held at IMSc, Chennai, India from 5 24 January For some of the proofs, details are left out, as we tried to focus more on the motivation of definitions and concepts rather than on complete proofs. We assume the reader to be familiar with the basics of operator algebras, in particular, with the theory of C -algebras. 2. Definition and examples of CMQGs Example 2.1 (Warmup: groups as symmetries). The study of symmetries arising in mathematics is an important tool in order to learn about the geometry of the considered objects (see for instance, the fundamental group in algebraic topology). We are interested in the viewpoint of groups as symmetries via actions on some spaces. Let us take a look at a few examples: (a) For the finite set X n := {x 1, x 2,...,x n } of n points, let Aut(X n ) be the set of bijective maps f : X n X n. The group Aut(X n ) is isomorphic to the symmetric group S n which naturally acts on X n via permutations. (b) The group of isometries of the sphere { } n S n 1 := x R n xi 2 = 1 R n i=1 is defined by Iso(S n 1 ) := {A : S n 1 S n 1 Ax, Ay = x, y }. It is isomorphic to the orthogonal group O n. (c) The cube may be viewed as a graph with eight vertices, twelve edges. Its automorphism group consists of bijections α : such that two vertices α(v i ) and α(v j ) are connected if and only if v i and v j are connected. The automorphism group is the wreath product Z 2 S 3 := Z 3 2 S 3 of Z 2 with the symmetric group S 3, where Z 2 := Z/2Z. More generally, for the n-hypercube, the automorphism group is Z 2 S n. Below we explicitly check the case n = 2. The automorphisms of the square (on four vertices) are

4 884 Moritz Weber e α β σ αβσ αβ ασ βσ The group (Z 2 Z 2 ) S 2 = α, β, σ α 2 = β 2 = e,αβ = βα,σ 2 = e,σασ = β,σβσ = α has elements e,α,β,σ,αβσ,αβ,ασ,βσ and the above automorphisms fulfill the relations of (Z 2 Z 2 ) S 2. Note that the graph below has the same symmetry group Z 2 S n as the n-hypercube. (2n vertices) Reminder 2.2 (Basic facts in operator algebras). We want to study symmetries in an operator algebraic context. To that end we review some basic ideas in operator algebras. They are a gateway to interpretations of what a symmetry should be in the non-commutative context. Philosophy behind non-commutative function algebras A compact Hausdorff space X gives rise to a commutative unital C -algebra, namely C(X), the space of complex-valued continuous functions on X. Conversely, a commutative unital C -algebra A is *-isomorphic to C(X), where X := Spec(A) := {φ : A C φ is a *-homomorphism,φ = 0} is a compact Hausdorff space (with the weak- topology). Hence, we may identify compact topological spaces with commutative unital C -algebras. Therefore, non-commutative C - algebras may be viewed as non-commutative function algebras over quantum spaces, and the theory of C -algebras turns into a non-commutative topology. See also [77, Ch. 1]. Similarly a measure space (X,μ)may be uniquely identified with a commutative von Neumann algebra, namely L (X,μ), the space of measurable bounded functions on X acting by multiplication on the Hilbert space L 2 (X,μ). Thus, the study of von Neumann algebras may be viewed as non-commutative measure theory. Two equivalent definitions of C -algebras C -algebras may be defined concretely as norm closed *-subalgebras of B(H), thesetof bounded linear operators on a Hilbert space H; or abstractly as Banach algebras with an involution * such that a a = a 2. The equivalence of the two definitions follows from the GNS construction.

5 CMQGs and Banica Speicher quantum groups 885 Definition of universal C -algebras The abstract definition of a C -algebra enables us to construct C -algebras in a purely abstract way: as universal C -algebras. A universal C -algebra is prescribed by a set of generators and relations which are realizable as bounded operators on a Hilbert space and enforce a uniform bound on the norm of the generators. We define a universal C -algebra as follows: Let E ={x i i I } be a set of generators. Let P(E) be the set of non-commutative polynomials in x i, xi. Let R P(E) be a set of relations. Let I(R) P(E) be the ideal generated by the set of relations R. Let A(E, R) := P(E)/I(R) be the quotient of P(E) by I(R); itistheuniversal -algebra generated by E and R. Let x :=sup{p(x) p be a C -seminorm on A(E, R)}; here p is a C -seminorm, if p(λy) = λ p(x), p(x + y) p(x)+ p(y), p(xy) p(x)p(y) and p(x x) = p(x) 2. If for all x A(E, R), wehave x <, then we define C (E, R), theuniversal C -algebra generated by E and R as the completion of A(E, R)/{x x =0} in the norm. Universal property of universal C -algebras More important than the actual definition of a universal C -algebra is its universal property. If B is a C -algebra such that the elements {y i B i I } satisfy the relations R, then there is a *-homomorphism, ϕ : C (E, R) B such that ϕ(x i ) = y i. Examples. (i) The universal C -algebra C (u, 1 u u = uu = 1) is isomorphic to the algebra of continuous functions C(S 1 ) on the sphere S 1 C. (ii) A universal C -algebra C (x x = x ) does not exist as x is not finite. (iii) A universal C -algebra C (x, y xy yx = 1) does not exist as no two bounded operators x, y can satisfy the relation xy yx = 1. Motivation 2.3 (Symmetries of quantum spaces?). Coming back to viewing C -algebras as function algebras over quantum spaces what are symmetries of quantum spaces? Let us take a look at the classical case first, at symmetries of topological spaces, given by actions of groups. Let X be a compact Hausdorff space, (G, ) be a compact group acting on X.Inviewof the machinery of Reminder 2.2, the first step is to pass to the algebras of functions (which are commutative C -algebras). Indeed, an action α : G X X yields a *-homomorphism on the dual level (i.e. the level of function algebras) α : C(X) C(G X) = C(G) C(X) by composition. However, G is more than a topological space, it comes with a group law :G G G (s, t) s t.

6 886 Moritz Weber Dualization of the group law gives us the following map: : C(G) C(G G) = C(G) C(G) f ((s, t) f (s t)). Here, we used the isomorphism C(G G) = C(G) C(G) (s, t) f (s)g(t) f g. Performing the second step of the machinery of Reminder 2.2, we shall now replace C(G) and C(X) by possibly non-commutative C -algebras A and B in order to have a noncommutative analog of a quantum space and its symmetry. Moreover, the dualization of the group law suggests that a quantum group should come with some map : A A A. Note that, as C(G) is a nuclear C -algebra, we did not have to bother with what precisely means in this context, but in the general case, we will be interested in the minimal tensor product and may abuse notation by using to mean min. See also Definition 2.8 for more on actions. DEFINITION 2.4 (Compact quantum group (CQG)) A compact quantum group (CQG) is a unital C -algebra A together with a unital *-homomorphism (the co-multiplication) : A A min A, such that ( id) = (id ) (co-associativity), and (A)(1 A) and (A)(A 1) are respectively dense in A min A.Here (A)(1 A) := span{ (a)(1 b) a, b A} and (A)(A 1) := span{ (a)(b 1) a, b A}. We also write A = C(G) in the spirit of Reminder 2.2 and we write G for the CQG. See [132,136] for the original definition. Remark 2.5 (CQGs generalize compact groups). (a) A quantum group is not a group! (b) If G is a compact group, then C(G) is a unital C -algebra and is co-associative. Indeed, we use associativity of G in order to show: ( id) ( f )(s, t, u) = ( f )((st), u) = f ((st)u) = f (s(tu)) = (id ) ( f )(s, t, u).

7 CMQGs and Banica Speicher quantum groups 887 We verify it for ( f ) = f 1 f 2 first and then it follows by taking linear combinations. The density condition of Definition 2.4 may also be verified easily. We infer that every compact group is a CQG. (c) Conversely, if (A, )is a CQG such that A is commutative, then A is isomorphic to C(G) with G := Spec( A), and : C(G) C(G) C(G) = C(G G) yields a map m : G G G by m : Spec(A A) Spec(A) ϕ ϕ. It is a group law, hence G is a compact semigroup (co-associativity of implies associativity of m). Now, what is the density condition in Definition 2.4 for? Fact 1: If G is a compact semigroup such that the linear spaces (C(G))(1 C(G)) and (C(G))(C(G) 1) are respectively dense in C(G) C(G), then G has cancellation property (i.e. st = su t = u). Fact 2: If G is a compact semigroup with cancellation property, then G is a group. Hence, the density condition characterizes the step from semigroup to group and we infer that G is a compact group. (d) Summarizing, we conclude that CQGs generalize compact groups. Example 2.6 (CQGs coming from group algebras). In order to see that CQGs are an honest generalization of compact groups, we shall come up with examples of CQGs (A, )such that A is non-commutative. For doing so, let G be a discrete group. (a) Recall the construction of the group C -algebra associated to G. The space CG := α g g finite linear combinations,α g C g G is a *-algebra by ( )( αg g βh h) := α g β h gh and ( ) αg g := ᾱg g 1. The abstract completion of CG with respect to the norm x :=sup{ π(x) π : CG B(H)} yields a C -algebra denoted by Cmax (G). It is isomorphic to the universal C -algebra C (u g, g G u g unitary, u g u h = u gh, u g = u g 1). As an example, verify that C (Z) = C(S 1 ), see also the example in Reminder 2.2. The instance of actually being a norm may be proven using the faithful map λ of item (b) below.

8 888 Moritz Weber With : C max (G) C max (G) C max (G) given by the extension of (u g ) := u g u g to C max (G), we have that (C max (G), ) is a CQG. If G is non-abelian, then C max (G) is non-commutative. (b) Likewise the reduced group C -algebra of G gives rise to a CQG. Recall that the left regular representation of G, λ : CG B(l 2 (G)) is defined by the linear extension of λ(g)(δ h ) := δ gh, where (δ h ) h G is an orthonormal basis of the Hilbert space l 2 (G).Asλ is faithful on CG, we know that λ(cg)) B(l 2 (G))) is isomorphic to CG. We may thus define C red (G) := λ(cg) B(l2 (G))) as a concrete completion of CG, the reduced group C -algebra of G. Since the norm in item (a) above is given by the supremum over all representations, there is a natural map φ : Cmax (G) C red (G) and the comultiplication map red : Cred (G) C red (G) C red (G) λ(g) λ(g) λ(g) factorizes through φ, i.e. we have red φ = (φ φ). Hence, also (C red (G), red) is a CQG. We refer to [111, Ex ] for more details. Motivation 2.7 (Dualizing actions of groups). Recall from Motivation 2.3 how to dualize the action of a group. Let α : G X X be an action of a compact group G on a compact Hausdorff topological space X. Thus, we obtain α : C(X) C(G X) = C(G) C(X) f f α. It satisfies (id α) α = ( id α) (follows from g(hx) = (gh)x). DEFINITION 2.8 ((Co-)action of a CQG) An action (also called co-action) ofacqg(a, ) on a C -algebra B is a unital *- homomorphism α : B A min B such that

9 CMQGs and Banica Speicher quantum groups 889 (i) (id α) α = ( id) α, (ii) α(b)(a 1) is linearly dense in A min B. We may differentiate between left and right action. Some authors require slightly different structures (like, modelling e x = x), see for instance [124, Definition 2.1]. Example 2.9 (Quantum permutation group S n + ). The examples in Example 2.6 are CQGs coming from classical groups. Let us now come to an honest quantum example. Let X n := {x 1,, x n } be a finite set of points. Then, its automorphism group Aut(X n ) is exactly the permutation group S n, see Example 2.1. Now,ifweviewX n as a quantum space what is its quantum symmetry group? The first step is to dualize the set X n and we obtain ) C(X n ) = C (p n 1,...,p n projections p i = 1. Since the p i form a basis, any action of a CQG (A, )on C(X n ) is of the form α : C(X n ) C(X n ) A n p j p i a ij i=1 for some elements a ij A. These elements a ij need to satisfy several relations: α : C(X n ) C(X n ) A n p j p i a ij i=1 for some elements a ij A. These elements a ij need to satisfy several relations: i=1 α(p j ) = α(p j ) a ij = a ij α(p j ) = α(p j ) 2 i p i a ij = k,i p i p k a ij a kj = i p i a 2 ij a ij = a 2 ij α is unital 1 1 = α(1) = α j p j = i, j p i a ij = i p i j a ij j a ij = 1. This led Wang [124] in 1998 to the definition C(S + n ) := A s(n) := C ( u ij, 1 i, j n u ij projections, k u ik = k ) u kj = 1. We may equip this C -algebra with a comultiplication by putting (u ij ) := u ij := u ik u kj. k

10 890 Moritz Weber Using the orthogonality of the projections u ik and u il for k = l (can be deduced from the fact that k u ik = 1), we check u 2 ij = k,l u ik u il u kj u lj = k u ik u kj = u ij and u ik = k k u kj = 1 1. Thus, by the universal property of C(S n + ),themap is a *-homomorphism from C(S+ n ) to C(S n + ) C(S+ n ) indeed. Moreover, it is co-associative due to ( id) (u ij ) = k (u ik ) u kj = k,l = l u il u lk u kj u il (u lj ) = (id ) (u ij ). The density condition holds true, since (u ij )(1 u mj ) = k u ik u kj u mj = u im u mj implies u im 1 = j u im u mj (A)(1 A), fromwhichwemayinfera 1 (A)(1 A), see also more details in Theorem 4.6. Similarly, we have 1 u mj (A)(1 A) and thus A A = (A)(1 A). This shows that (C(S n + ), ) is a CQG; actually, it is the quantum automorphism group of X n,see[124]. Thus, in the category of CQGs, the space X n has more automorphisms: Its automorphism group is S n in the category of groups, whereas it is S n + in the category of CQGs. We may see that C(S n + ) is non-commutative for n 4 due to the following argument for n = 4. Consider a C -algebra B generated by two non-commuting projections p, q. By the universal property of the universal C -algebra C(S 4 + ), one may consider a *-homomorphism from C(S 4 + ) to B, which sends (u ij) to the respective entries of the matrix below: p 1 p p p q 1 q q q As B is non-commutative, C(S + 4 ) must also be non-commutative. Thus S 4 = S + 4.

11 CMQGs and Banica Speicher quantum groups 891 Let us finish this example by remarking that C(S n ) is the abelinization of C(S n + ), i.e.: C(S n + )/ u iju kl u kl u ij = C(S n ). This can be proven by representing the symmetric group S n M n (C) as permutation matrices and then checking that the coordinate functions u ij : S n C g g ij satisfy the relations of C(S n + ). Using Stone Weierstraß s theorem, we infer that we have a surjection from C(S n + ) to C(S n) sending u ij to ũ ij. Moreover, the characters φ on C(S n + )/ u iju kl u kl u ij correspond exactly to permutation matrices and vice versa. Thus Spec(C(S n + )/ u iju kl u kl u ij ) = S n. Comparing the comultiplication on C(S n ) arising as in Motivation 2.3, and making use of the fact that the multiplication of s, t S n is simply given by matrix multiplication, we infer (ũ ij )(s, t) =ũ ij (st) = ũ ik (s)ũ kj (t). k Under the natural isomorphism C(S n S n ) = C(S n ) C(S n ) as in Motivation 2.3, this amounts to ( ) (ũ ij )(s, t) = ũ ik ũ kj (s, t), k in perfect analogy to the comultiplication map defined on C(S + n ). This proves that S n S + n in the sense of the following definition. DEFINITION 2.10 (Quantum subgroup) (A, A ) is a quantum subgroup of (B, B ) if there is a surjection ϕ : B A such that A ϕ = (ϕ ϕ) B. DEFINITION 2.11 (Compact matrix quantum group (CMQG)) Given n N,acompact matrix quantum group (CMQG) is defined as a pair (A, u) where A is a unital C -algebra which is generated by u ij A, 1 i, j n, the entries of the matrix u, u = (u ij ) and u t = (u ji ) are invertible, and the map : A A min A defined by (u ij ) = k u ik u kj is a *- homomorphism. Again, we write G for the CMQG in the sense of Remark 2.2 with C(G) = A a possibly noncommutative C -algebra, see [132]. Remark 2.12 (CMQGs are CQGs). In Corollary 4.7, we will see that every CMQG is a CQG. In fact, the introduction of CMQGs predates one of CQGs. In 1987, Woronowicz defined CMQGs under the name of compact matrix pseudogroups [132]; in 1995, he defined CQGs [136].

12 892 Moritz Weber Example 2.13 (Orthogonal quantum group O + n ). In the same way we stepped from S n to S + n, we may define a quantum version of the orthogonal group O n M n (C). Note that C(O n ) can be written as a universal C -algebra, namely: C(O n ) = C (u ij, 1 i, j n u ij = uij, u = (u ij) is orthogonal, u ij commute). We may express the fact that u is orthogonal by the following relations: u ik u jk = u ki u kj = δ ij. k k In 1995, Wang [122] defined the free orthogonal quantum group O + n by. C(O + n ) := A o(n) := C (u ij, 1 i, j n u ij = u ij, u = (u ij) is orthogonal). It is easy to see that O n + is a CMQG containing O n, S n + and S n as quantum subgroups. Hence, there are more quantum (orthogonal) rotations than classical ones. One can show that O n + is the quantum isometry group of the quantum sphere, [26, Theorem 7.2]. Wang and van Daele [115] (seealso[124, Appendix] and the definition by Banica [2]) also defined deformed versions of O n + by C(O + (Q)) := A o (Q) := C (u ij, 1 i, j n u unitary and u = QūQ 1 ). Here Q GL n (C) such that Q Q = c1 n, c R and ū := (u ij ). Moreover, u ij = u ij in general. For Q = 1 n,wehaveo + (Q) = O + n. Example 2.14 (Unitary quantum group U n + ). The free unitary quantum group U n + CMQG given by is the C(U + n ) := A u(n) := C (u ij, 1 i, j n u, u t unitary). We may also write it as C(U + n ) = C (u ij, 1 i, j n u, ū unitary) revealing that C(O n + ) is nothing but the quotient of C(U n + ) by u =ū. Thus, O+ n U n +. Moreover, the abelinization of C(U n + ) is C(U n), sou n U n +. Note that the requirement of u t being unitary is automatic for commuting u ij, but not for non-commuting ones. In fact, C (u ij, 1 i, j n u unitary) does not giverisetoacmqgasu t fails to be invertible [122, Example 4.1].

13 CMQGs and Banica Speicher quantum groups 893 Again, there is a deformed version given by C(U + (Q)) := C (u ij, 1 i, j n u unitary and QūQ 1 unitary), see [115,122]. Example 2.15 (Quantum special unitary group SU q (2)). Historically, Woronowicz s quantum version of SU(2) was the first example of a CQG. Recall SU(2) := {u M 2 (C) u unitary, det(u) = 1} M 2 (C). We may write SU(2) as {( ) a c SU(2) = a, c C, a 2 + c 2 = 1}. c ā For q [ 1, 1]\{0}, we define SU q (2) as the CMQG given by ( ) ) C (α, γ α qγ γ α is a unitary. Thus, its comultiplication is given by (α) = α α qγ γ, (γ) = γ α + α γ. Moreover, we have [111, section 6.2] ( ) SU q (2) = O q 1, 0 see [133]. See also [134] for higher dimensional versions SU q (n). Example 2.16 (Hyperoctahedral quantum group H n + ). Recall from Example 2.1(c) that the hyperoctahedral group H n = Z 2 S n = σ S n, a 1,...,a n a 2 i = e, a i a j = a j a i,σa i σ 1 = a σ(i) is the symmetry group of the n-hypercube and a certain graph. How to obtain a quantum version of it? We first represent it as a matrix group: H n = {(uij ) orthogonal u ik u jk = u ki u kj = 0fori = j} O n M n (C), 1... a i a i :=

14 894 Moritz Weber To do so, check that {(u ij )...} is a group indeed containing all matrices a i and all permutation matrices σ ; moreover, check that u ik u jk = 0 implies that at most one entry per row is nonzero; together with i u2 ik = 1, this implies that this nonzero entry is either 1 or 1; hence the group {(u ij )...} consists of all permutation matrices where any entry 1 may be replaced by 1; all these matrices may be constructed in a 1,...,a n,σ. This reveals that we should define the hyperoctahedral quantum group H n + as the CMQG given by C(H n + ) := C ( u ij u ij = uij, u orthogonal, u iku jk = u ki u kj = 0, i = j ). Note that the crucial step was to find a representation of H n as a matrix group in terms of algebraic relations on the entries u ij. One can prove that H n + is the quantum symmetry group of the graph as in Example 2.1(c); however, it is not the quantum symmetry group of the n-hypercube. Actually, when Bichon [37] introduced H n + in 2004, he also gave a definition of a free wreath product with S n + and he proved: H n + = Z 2 S n +. Finally, we have S n + H n + O+ n and H n H n +. See also [15] for more on H n +. Remark 2.17 (Locally compact quantum groups). There is a generalization of CQG to locally compact quantum groups given by Kustermans and Vaes [88]. Remark 2.18 (Sources for examples of CQGs). The main approaches to obtain examples of CQGs are: (i) Liberation ( let drop commutativity fg = gf ), like S n +, O+ n, U n +,see[31,122,124, 128] amongst others. (ii) Deformation ( deform commutativity: fg = qgf, q C ), like SU q (2), O ( Q), U + (Q),see[115,133] amongst others. (iii) Quantum isometry groups ( quantum group of Riemannian isometries on a noncommutative Riemannian manifold à la Connes s non-commutative geometry ), see [26,35] amongst others. In 5, we will elaborate more on the so-called Banica Speicher quantum groups which are CMQGs arising from (i). 3. The Haar state Motivation 3.1 (Dualizing the Haar measure on groups). One very striking feature of Woronowicz s definition of a CQG is the existence of a Haar state dual to the Haar integration for groups. Let us first consider the classical case. Let G be a compact group. By Riesz s theorem, there is a unique left-invariant Haar measure μ G on G such that for all t G and all f C(G): f (st) dμ G (s) = f (s) dμ G (s). G G

15 CMQGs and Banica Speicher quantum groups 895 Hence, we have a positive linear functional h : C(G) C h G f dμ G which remains invariant under left-translation: (h id) ( f )(t) = ( f )(s, t) dμ G (s) G = f (st) dμ G (s) = f (s) dμ G (s) = h( f )1 C(G) (t). Having formulated the features of the Haar integration in quantum terms, i.e. as properties of the algebra C(G) rather than as a characterization making use of elements g G, we may now proceed to the quantum case and prove an analog of the Haar integration. Theorem 3.2 (Existence of a Haar state). Let (A, )be a compact quantum group. Then there is a unique state (the Haar state ) such that h : A C (id h) = (h id) = h 1 A. Proof. Define g h := (g h), for g, h A (the space of continuous linear functionals on A) and a h := (id h) (a), for a A, h A. Uniqueness: Ifh is another such state, then h (a) = h(h (a) 1 A ) = h((id h ) (a)) = (h h ) (a) = h ((h id) (a)) = h(a). Check h((id h ) = h h for (a) = a 1 a 2 and extend via linearity. Existence: Forω A positive, we define K ω := {ρ A ρ is a state,ρ ω = ω ρ = ω(1 A )ρ}. Then K ω is a closed subset (in the weak-* topology) of (A ) + 1 := {ρ A positive ρ 1} and hence it is compact. We will prove the existence of a Haar state in the following three steps:

16 896 Moritz Weber (1) K ω =. (2) i=1 n K ω i =, for all positive ω i A, n N. (3) By Cantor s intersection principle and (2), we may find a state h ω (A ) + K ω =. It satisfies (id h) = (h id) = h. Proof of (1). Let ω A be a state. Define ω n := n 1 nk=1 ω k where one inductively defines ω (n+1) = ω (ω n ) and ω 1 = ω. As(A ) 1 is compact, (ω n ) n N has an accumulation point ρ. Since ω ω n ω n = n 1 ω (n+1) ω n 2, we have ω ρ = ρ. Likewise ρ = ρ ω. This settles K ω =. Proof of (2). Let ω 1,ω 2 A be positive and define ω := ω 1 + ω 2. We only need to prove K ω K ω1. Then = K ω K ω1 K ω2 and iteratively, = K n i=1 ω i i=1 n K ω i. Assume that ω is a state. Let ρ K ω and define L ρ ω := {q A A (ρ ω)(q q) = 0}. Then L ρ ω L ρ ω1 since ω 1 ω. Moreover L ρ ω1 Ker(ρ ω 1 ) by Cauchy Schwarz. For ψ L : A A defined by ψ L (a) := (id ρ) (a) ρ(a)1, one can show that (id ψ L )( (A)) L ρ ω. Thus, keeping in mind that L ρ ω is a left ideal, we have Hence 1 ψ L (A) (A 1)(id ψ L ) (A) (A 1)L ρ ω Ker(ρ ω 1 ). 0 = (ρ ω 1 )(1 ψ L (a)) = (ω 1 ρ)( (a)) ω 1 (1)ρ(a). Thus ρ is in K ω1. Proof of (3). For h in ω (A ) + K ω,wehaveh ω = ω h = h for all states ω in A.For a A, letb := h(a) (id h)( (a)) A. Then ω(b) = 0 for all states ω in A. Thus b = 0. We refer to [132, Thm. 4.2],[136, Thm. 2.3],[114] and [111, Thm ] for details regarding this proof. Remark 3.3 (Haar state not faithful). The Haar state is not always faithful. For instance, consider the CQG (Cmax (G), ) given by a discrete group G, see also Example 2.6. The Haar state h max on Cmax (G) is given by h max(u g ) = δ g,e, since (id h max )( (u g ) = u g h max (u g ) = u e δ g,e = 1δ g,e = h max (u g ). It is a tracial state i.e. h max (ab) = h max (ba) for all a, b in C max (G). The Haar state h red on (C red (G), ) is given by h red(x) = xδ e,δ e as can be verified directly. Now, the natural map φ : C max (G) C red (G) satisfies h max = h red φ. Thus h max is not faithful in the case when φ is not an isomorphism (i.e. when G is not amenable). However, h red is always faithful on C red (G). DEFINITION 3.4 (Reduced version of a CQG) Let G = (A, )be a compact quantum group and h be its Haar state. The reduced version of G is given by (A red, red ) where A red := π h (A) B(H h ) (by the GNS construction

17 CMQGs and Banica Speicher quantum groups 897 with respect to the Haar state h) and red (π h (x)) := (π h π h )( (x)). Onemayalso associate to G the enveloping von Neumann algebra L(G) of A red. PROPOSITION 3.5 (Haar state of reduced version) The comultiplication red is well-defined and the Haar state on (A red, red ) is given by h red (π h (x)) := h(x). It is faithful. Proof. Let π h (x) = 0. Then h(x x) = 0. But then (h h)( (x x)) = h((id h)( (x x))) = h(h(x x)1) = h(x x) = 0. Thus (x) Ker(π h h ) = Ker(π h π h ). We conclude that red is well-defined. Moreover, (id h red ) red (π h (x)) = (id h red )(π h π h )( (x)) = π h (id h)( (x)) = π h (h(x)1) = h(x) = h red (π h (x)), seealso[98, Cor ]. DEFINITION 3.6 (Kac type) A compact quantum group is said to be of Kac type if the Haar state is a trace. Example 3.7 (Kac type CQGs). From Remark 3.3, we infer that (Cmax (G), ) and (Cred (G), ) are both of Kac type. See also Proposition 4.12 and Example 4.13 for more on Kac type CQGs. PROPOSITION 3.8 (Free product of CQGs) Let (A, A ) and (B, B ) be CQGs. Then the free product is a CQG given by (A C B, ), where is the extension of A and B arising from the universal property of the unital free product A C BoftheC -algebras A and B. The Haar state h on the free product is given by the free product (in the sense of free probability) of the Haar states h A and h B of (A, A ) and (B, B ), i.e. (i) h i A = h A, (ii) h i B = h B, (iii) For all c 1,, c n A B A C B such that h A (c i ) = 0 or h B (c i ) = 0 respectively and c i, c i+1 come from different algebras, we have: h(c 1 c n ) = 0. Moreover, if (A, A ) and (B, B ) are CMQGs, so is their free product, see [122] and [111, Sect. 6.3].

18 898 Moritz Weber PROPOSITION 3.9 (Tensor product) Let (A, A ) and (B, B ) be CQGs. Then the tensor product is a CQG given by (A max B,ρ ( A B )), where ρ : (A min A) max (B min B) (A max B) min (A max B). Its Haar state is h A h B and it is a CMQG if the original ones are, see [123] and [111, Sect. 6.3]. 4. Representation theory and Tannaka Krein Motivation 4.1 (Representations of groups). Let G be a compact group and let U : G B(H ) be a finite dimensional representation, i.e. dim H = m and U is continuous. Then U C(G, M m (C)) = C(G) M m (C), i.e. U = u αβ e αβ. α,β We have U(gh) = u αβ (gh) e αβ = (u αβ )(g, h) e αβ α,β α,β and U(g)U(h) = α,β γ u αγ (g)u γβ (h) e αβ. Since U(gh) = U(g)U(h), by comparing the coefficients we have (u αβ )(g, h) = γ u αγ (g)u γβ (h) = ( uαγ u γβ ) (g, h). Note that we used the isomorphism C(G G) = C(G) C(G) in the last equality above. This motivates the following definition. DEFINITION 4.2 (Representations of CQGs) Let A be a unital C -algebra with a unital *-homomorphism : A A min A.A(finite dimensional) representation of (A, )is a matrix u = (u αβ ) M m (A) such that (u αβ ) = γ u αγ u γβ.

19 CMQGs and Banica Speicher quantum groups 899 If the matrix has an inverse, then we call u non-degenerate and if it is unitary, we call it a unitary representation.if(a, )is a CQG, this defines (finite dimensional) representations for CQGs. See also [95, Def. 3.5, Def. 5.1] for more on representations, including infinite dimensional ones. DEFINITION 4.3 (Tensor products of representations) Let u,vbe two representations of a CQG (A, ), i.e. u = e αβ u αβ M n(u) (C) A and Then we define v = e γδ v γδ M n(v) (C) A. (a) u v := e αβ e γδ u αβ v γδ M n(u) (C) M n(v) (C) A. (b) u := e αβ u αβ. DEFINITION 4.4 (Equivalence, irreducibility, intertwiners) Let u B(H u ) A and v B(H v ) A be representations of a CQG (A, ). (a) If a linear operator T B(H u, H v ) is such that Tu = vt, then T is called an intertwiner. (b) The representations u and v are called equivalent, if there exists an intertwiner T B(H u, H v ) which is invertible. (c) The representation v is called irreducible, if every intertwiner Tu = ut is of the form T = λ id. PROPOSITION 4.5 (Irreducibility properties) (a) If u is an irreducible representation, then u is irreducible. (b) If u is an irreducible unitary representation, then u is equivalent to a unitary representation. See [95, Lemma 6.9, Proposition 6.10]. Theorem 4.6 (Generation by coefficients of representations). Let A be a unital C - algebra with a unital *-homomorphism : A A min A. (a) If A is generated (as a normed algebra) by the matrix elements of the nondegenerate finite dimensional representations, then (A, )is a CQG. (b) If A is generated (as a C -algebra) by elements (u αβ ) such that (u αβ ) = u αγ u γβ and u = (u αβ ), u t = (u βα ) are invertible, then (A, )is a CQG.

20 900 Moritz Weber Proof. We prove (a) first. Note that it suffices to check the co-associativity of on the generators, but we have already checked co-associativity, see Example 2.9.Nowlet(w αβ ) be the inverse of some non-degenerate, finite dimensional representation (u αβ ). Then (u αγ )(1 w γβ ) = u αɛ u ɛγ w γβ γ γ,ɛ = ɛ u αɛ δ ɛβ = u αβ 1 (A)(1 A). holds. Next we show that a 1, b 1 (A)(1 A) implies ab 1 (A)(1 A). We holds take a 1 = γ (a 1,γ )(1 a 2,γ ) and b 1 = ɛ (b 1,ɛ )(1 b 2,ɛ ) for some a 1,γ, a 2,γ, b 1,γ, b 2,γ A. Then we have ab 1 = γ (a 1,γ )(1 a 2,γ )(b 1) = γ (a 1,γ )(b 1)(1 a 2,γ ) = γ,ɛ (a 1,γ b 1,ɛ )(1 b 2,ɛ a 2,γ ) and hence A 1 (A)(1 A) A A. Thus, if x y A A, we write x 1 = i (a i )(1 b i ) and we compute ( ) x y = (x 1)(1 y) = i (a i )(1 b i ) (1 y) = i (a i )(1 b i y) (A)(1 A). This shows that A A = (A)(1 A). Similarly we have that (A)(1 A) = A A. This proves the density condition.

21 CMQGs and Banica Speicher quantum groups 901 For part (b), note that u = (u t ) is invertible since u t is, and (u ij ) = (u ij) = k u ik u kj. Thus u, u are non-degenerate representations which generate A and we may use (a) to reach the conclusion. See also [95, Sect. 3]. COROLLARY 4.7 (CMQGs are CQGs) Every compact matrix quantum group is a compact quantum group. Proof. This is a immediate consequence of Theorem 4.6(b). Theorem 4.8 (Decomposition of representations). (a) Every non-degenerate finite dimensional representation v is equivalent to a unitary representation. (b) Every unitary representation decomposes into a direct sum of irreducible finite dimensional representations. (c) The right regular representation contains all irreducible unitary representations. Proof. For (a), let v be a non-degenerate finite dimensional representation and put y := (id h)(v v) M m (C). Note that v v is invertible, since v is invertible and as a positive, invertible element v v is strictly positive, i.e v v>ε 1forsomeε>0. Then y 0 and in particular, y ε 1, since the Haar state h is positive. Thus ω := (y 1 2 1)v(y 1 2 1) is a unitary representation equivalent to v. In order to check for instance ω ω = 1, verify y 1 = (id h id)(id )(v v) = v (y 1)v. As for (b), we consider the C -algebra of intertwiners D = Hom (v, v). We choose a maximal family (p n ) n of pairwise orthogonal, minimal projections in D. Then one shows, using the fact that D acts non-degenerately on H, that H = p n H. Finally the restriction of v to p n H is irreducible, since p n H is a minimal invariant subspace. For (c), we refer to [95].

22 902 Moritz Weber DEFINITION 4.9 (Hopf *-algebras) Let A be a unital algebra. We denote by m the multiplication map m : A A A, a b ab and by η : C A the embedding λ λ 1 A. A Hopf *-algebra consists of a unital algebra A together with (a) a comultiplication : A A A; (b) a homomorphism ε : A C with (ε id) = id = (id ε), called the counit; (c) a linear map S : A A with m (S id) = η ε = m (id S), called the antipode or coinverse; The counit and the antipode dualize the idea of a neutral element resp. an inverse. Theorem 4.10 (Algebraic picture of CQGs). Let (A, )be a CQG. Let A 0 be the subspace of A spanned by the matrix elements of all finite dimensional, unitary representations. Then (a) A 0 A is a dense *-algebra. (b) (A 0 ) A 0 A 0 (algebraic tensor product). (c) (A 0, 0 A0 ) is a Hopf *-algebra. (d) The Haar state h A0 is faithful on A 0. Proof. First we prove (a). For the density, one can show that the GNS-construction with respect to the Haar state yields a unitary representation δ h (sometimes called the regular representation) such that span(δ hi, j ) = A. By Theorem 4.8(b), this representation is equivalent to a direct sum of irreducible representation. Hence the linear span of all finite dimensional unitary representations is dense in A. Next we show that A 0 is an algebra. Take and u = e ij u ij M n(u) (C) A v = e kl v kl M n(v) (C) A.

23 CMQGs and Banica Speicher quantum groups 903 Then u v = e ij e kl u ij v kl M n(u) (C) M n(v) (C) A. This means, if u ij,v kl are the coefficients of the representation, their product are the coefficients of the tensor product of the two representations. Showing that A 0 is *-closed can be done as in Theoreom 4.6(b). Next we prove (b). It suffices to check that all monomials u α 1 i 1 j 1...u α k i k j k A 0 are mapped to A 0 A 0.Wehave ( ) ( ) ( ) u α 1 i 1 j 1...u α k i k j k = u α 1 i 1 j 1... u α k i k j k = i 1 γ 1...u α k i k γ k u α 1 γ 1 j 1...u α k γ 1,...,γ k u α 1 where each summand is contained in A 0 A 0. For (c), we define the counit by γ k j k, ε : A 0 C, ε(u α ij ) = δ ij and the antipode S : A 0 A 0, S(u α ij ) = (uα ji ), where uij α are the coefficients of irreducible representations. It can be shown that they form a basis for A 0, thus ε and S are well-defined. Since A 0 is spanned by all entries uij α of unitary representations u α, it suffices to check the properties of the counit, resp. antipode. We have (ε id)( (u α ij )) = k ε(u α ik )uα kj = k δ ik u α kj = uα ij and ( ( )) ( )) m (S id) ( (uij α ) = m (S id) uik α uα kj k = k (u α ik ) u α kj = δ i, j 1 = η(ε(uij α )), and analogously the right counterparts. For part (d), we refer to [111, Sect. 5].

24 904 Moritz Weber Remark 4.11 (Subtleties of the algebraic picture). (a) The counit ε and antipode S are uniquely determined. (b) Theorem 4.10 shows that we have a canonical algebraic quantum group sitting inside our CQG. Conversely, starting with a Hopf*-algebra (A 0, ), we may associate a CQG (A, )to it such that A 0 A in the sense of the previous theorem. (c) In general, the antipode on the C -algebra A need not be bounded. Thus, A might fail to be a Hopf *-algebra itself. However, we will always find a dense Hopf *-algebra by Theorem See also [98, Sect. 1.6] and [113]. PROPOSITION 4.12 (Characterisations of Kac type CQGs) Let (A, ) be a compact quantum group and let (A 0, 0 ) be the Hopf *-algebra, Sthe antipode on A 0. Then the following statements are equivalent: (i) The Haar state is a trace (i.e. (A, )is of Kac type in the sense of Definition 3.6). (ii) S 2 = id. (iii) S is *-preserving. A proof can be found in [98, Sect ], see also [84, Section 11.3]. Example 4.13 (Kac type and non-kac type). (a) S n +, O+ n, U n + are of Kac type using ε : A C, ε(u ij ) = δ ij, S : A A, S(u ij ) = u ji. These maps exist by the universal properties of C(S n + ), C(O+ n ) and C(U n + ) respectively. (b) SU q (2), O n + (Q), U n + (Q) are not of Kac type, if Q Q = 1. ε : A C, ε(u ij ) = δ ij S(u ij ) = u ji, S(u ij ) = (Q Q) il 1 u ml (Q Q) mj. One sees that S is not *-preserving, hence these CQGs fail to be of Kac type. In fact, this was one of the main motivations for Woronowicz to introduce SU q (2). Inhis philosophy, Kac type CQGs are closer to the classical setting note for instance that CQGs arising as (Cred (G), ) are always of Kac type, see Example 3.7. Remark 4.14 (Dual of a CQG). We may define the dual of a CQG (generalizing Pontryagin duality). Let (A, ) beacqg,{u α : α I } mutually inequivalent unitary irreducible representations, A 0 A as before. Let B 0 be the space of all linear functionals on A given by x h(ax), a A 0. Then B 0 is a subalgebra of A and isomorphic to the algebraic direct sum α I M n(α) (C). We may define a comultiplication on B 0 by using the multiplier algebra of the algebraic tensor product B 0 B 0. The completion yields a discrete quantum group which is by definition the dual quantum group. The Pontryagin duality tells us that a locally compact abelian group G may be reconstructed from the information contained in its dual group of characters Ĝ. The classical Tannaka Krein duality is an extension to compact non-abelian groups with the role of Ĝ being replaced by its category of finite dimensional unitary representations. See also [63].

25 CMQGs and Banica Speicher quantum groups 905 DEFINITION 4.15 (W*-category) Let R be a set of objects equipped with a binary operation :R R R. Let{H r } r R be a family of finite dimensional Hilbert spaces and for any r, s R, letmor(r, s) be a linear subspace of B(H r, H s ). Then (R,, {H r } r R, {Mor(r, s)} r,s R ) is called a concrete monoidal W*-category if the following conditions hold: (i) For any r R, the identity operator id r acting on H r belongs to Mor(r, r). (ii) If a Mor(r, r ), b Mor(r, r ), then ba Mor(r, r ). (iii) For any r, s R and a Mor(r, s) we have a Mor(s, r). (iv) If H r = H s and id r Mor(r, s), then r = s. (v) If a Mor(r, r ) and b Mor(s, s ), then a b Mor(r s, r s ), for any r, s, r, s R. (vi) For any p, r, s R: (p r) s = p (r s). (vii) There exists 1 R such that H 1 = C, 1v = v1 = v. A concrete monoidal W*-category (R, {H r } r R, {Mor(r, s)} r,s R ) is called complete if the following additional conditions hold: (viii) For r R and any unitary v : H r K, where K is a Hilbert space, there exists s R such that H s = K and v Mor(r, s). (ix) For r R and any orthogonal projection p Mor(r, r), there exists s R such that H s = ph r and i Mor(s, r), where i is the embedding H s H r. (x) For r, r R, there exists s R such that H s = H r H r and the canonical embeddings H r H r H r and H r H r H r belong to Mor(r, s) and Mor(r, s) respectively. An element r R is said to be the complex conjugate of r R if there is an invertible anti-linear map j : H r H r such that the map defined by t j : C H r H r t j (1) = i e i j (e i ) is in Mor(1, r r), and the map t j : H r H r C defined by t j (e h e i ) = j 1 (e h ), e i is in Mor(r r, 1). A finite subset Q of R is said to generate the W*-category, if for any s R, there are morphisms b k Mor(q (k) 1 q n (k) k, s), k = 1, 2,...,m for some q (k) 1, q(k) 2,...,q(k) n k Q such that k b kbk = id s Mor(s, s).

26 906 Moritz Weber PROPOSITION 4.16 (Representations of a CQG form a W*-category) Let G = (A, u) be a CMQG. Denote by Rep G the set of all finite dimensional unitary representations of G and Mor(v, w) := {T : H v H w linear maps T v = wt } the space of intertwiners, where v B(H v ) A and w B(H w ) A. Then (Rep G,, {H r } r Rep G, {Mor(r, s)} r,s Rep G ) forms a complete concrete monoidal W*- category. Furthermore for all v Rep G, there is v Rep G and {u, u} generates Rep G. Proof. The proof is rather straightforward. Check for instance (v): T i v i = w i T i for i = 1, 2 (T 1 T 2 )(v 1 v 2 ) = (w 1 w 2 )(T 1 T 2 ). See also [134, Thm.1.2]. DEFINITION 4.17 (Model of a W*-category) Let (R,, {H r } r R, {Mor(r, s)} r,s R ) be a concrete monoidal W*-category generated by { f, f }. (a) Let B be a unital C -algebra and {v r } r R be a family of unitaries, i.e. v r B(H r ) B. Then M = (B, {v r } r R ) is called a model of R if (i) v r s = v r v s. (ii) v r (t 1) = (t 1)v s for any r, s R and t Mor(s, r). (b) Let B be a unital C -algebra and v be a unitary element of B(H r ) B. We say that (B,v) is an R-admissible pair if there exists a model M = (B, {v r } r R ) such that v f = v. Theorem 4.18 (Tannaka Krein for CMQGs). Let R = (R,, {H r } r R, {Mor(r, s)} r,s R ) be a concrete monoidal W*-category such that { f, f } generates R. Then there is a CMQG G = (A, u) such that R = Rep G, where R is the completion of R (in a natural sense). Moreover, G is universal in the following sense: if G = (B,v) is a CMQG such that R Rep G, there is a homomorphism A B, u v. Sketch of the proof. At first, we assume that R is a complete concrete monoidal W*- category. Let R irr be a complete set of mutually non-equivalent irreducible elements of R with 1 R R irr (there is such a set, because R is complete). Set A := α R irr B(H α ), where B(H α ) is the space of linear functionals defined on B(H α ) for every α R irr. For α R irr and ρ B(H α ) the corresponding element of A will be denoted by u α ρ. Therefore every a A can be written as finite sum a = α R irr u α ρ.

27 CMQGs and Banica Speicher quantum groups 907 The embedding B(H α ) A,ρ u α ρ is linear, hence there exists a unique uα B(H α ) A such that (ρ id)u α = u α ρ for any ρ B(H α ). One can show that A equipped with the multiplication :A A A, u α ρ uβ σ := (ρ σ id)u αβ is a unital algebra, where u 1 R is the unit of A. Now we introduce a *-algebra structure on A. Any element of R admits complex conjugation and the complex conjugation of any α R irr is irreducible. Let j α : H α H α be the corresponding invertible antilinear mapping. Set m j := j α mj 1 α for α R irr, m B(H α ). For any ρ B(H α ),letρ j be the linear functional on B(H α ) such that ρ j (m j ) = ρ(m) for all m B(H α ). Equipped with the antilinear involution :A A,(u α ρ ) := u α, A is a *-algebra. ρ j Let A be the space of all linear functionals defined on A. There exists h A such that (id h)u α = { 1, if α = 1, 0, otherwise. One can show that h is faithful on A. Therefore π h is faithful and for the C -seminorm p(x) := π h (x), we get p(x) = 0forx = 0. This yields that. :=sup{p(x) pc seminorm on A} is a norm on A. The completion A := A. of A with respect to this norm is a unital C -algebra. Define u := u f, then G = (A, u) is a CMQG with R = Rep G. Assume that (B,v)is another R-admissible pair. Let M = (B,(v r ) r R ) be the model of R such that v f = v. Define φ M : A B,φ M (u α ρ ) = (ρ id)vα. The extension φ M : A B is a *-homomorphism with (id φ M )u = v. See [134]. Remark 4.19 (Beauty of the Tannaka Krein theorem). Let us highlight two main aspects of Woronowicz s Tannaka Krein duality. Firstly, note that Proposition 4.16 and Theorem 4.18 are actually dual to each other: Given a CMQG, its representation theory forms a W*- category, but the converse is also true any W*-category yields a CMQG. Thus, we have a way of producing CMQGs by specifying their representation theory.

28 908 Moritz Weber The second aspect is more subtle. Note that for Theorem 4.18, we did not require the W*-category R to be complete. Thus, we do not need to specify the whole representation theory of a CMQG, we only need to come up with parts of it. This is a drastic reduction of complexity from which we will profit in the next chapter. See also [110] (or rather the appendix [108] in the arxiv version of this article) for more on the interpretation of the Tannaka ikrein theorem. 5. From Tannaka Krein to Banica Speicher quantum groups DEFINITION 5.1 (Categories of partitions) (a) A partition p P(k, l) is a decomposition of k + l points (k of which are upper, l lower ) into disjoint subsets called the blocks. We illustrate examples of partitions pictorially below: P (2, 4), P (4, 1). The set of all partitions {P(k, l) : k, l N 0 } is denoted by P. (b) Let p P(k, l), q P(k, l ), then their tensor product p q P(k + k, l + l ) is the partition obtained by placing p and q side-by-side. Pictorially, = ( P (2, 4))) ( P (4, 1)) ( P (6, 5)) (c) Let p P(k, l), q P(l, m),thecomposition qp P(k, m) is the partition obtained by aligning the lower points of p right above the upper points of q and ignoring the l middle points so obtained. Certain loops/blocks may appear purely in the middle with no connections to the points in the upper or lower ends of the aligned partitions and are removed. They are referred to as removed blocks. Pictorially ( P (2, 4)) = ( P (2, 1)). ( P (4, 1))