Proc. Indian Acad. Sci. (Math. Sci.) Vol. 127, No. 5, November 2017, pp. 881–933. https://doi.org/10.1007/s12044-017-0362-3

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

MORITZ WEBER

Saarland University, Saarbrücken, Germany E-mail: [email protected]

MS received 14 July 2017; published online 27 November 2017

Abstract. This is a transcript of a series of eight lectures, 90min each, held at IMSc Chennai, India from 5–24 January 2015. 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. How- ever, modern mathematics requires an extension of the symmetry concept to highly non- commutative 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 1987. 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, 2015.

© Indian Academy of Sciences 881 882 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 phi- losophy 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 → GC(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 correspon- dence 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 1 2 ∈ P (2, 4). 1 234

We associate a linear map n ⊗k n ⊗l Tp : (C ) → (C ) to such a partition p ∈ P(k, l). We then define operations on partitions, which match nicely via the assignment p → Tp with tensor category like operations on the maps Tp (such as forming the tensor product, the composition or the involution of linear maps). A class 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 → Tp; 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.  RC tensor categ. ←→ G BS QG p partition p → Tp Tp 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, 90min each, held at IMSc, Chennai, India from 5–24 January 2015. 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 math- ematics 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 Xn := {x1, x2,...,xn} of n points, let Aut(Xn) be the set of bijective maps f : Xn → Xn. The group Aut(Xn) is isomorphic to the symmetric group Sn which naturally acts on Xn via permutations. (b) The group of isometries of the sphere   n n−1 := ∈ Rn | 2 = ⊆ Rn S x xi 1 i=1 is defined by − − − Iso(Sn 1) := {A : Sn 1 → Sn 1 | Ax, Ay = x, y }.

It is isomorphic to the orthogonal group On. (c) The cube may be viewed as a graph with eight vertices, twelve edges. Its auto-

morphism group consists of bijections α : → such that two vertices α(vi ) and α(vj ) are connected if and only if vi and v j are connected. The automorphism Z := Z⊕3  Z group is the wreath product 2 S3 2 S3 of 2 with the symmetric group S3, where Z2 := Z/2Z. More generally, for the n-hypercube, the automorphism group is Z2 Sn. Below we explicitly check the case n = 2. The automorphisms of the square (on four vertices) are 884 Moritz Weber

eαβ σ

αβσ αβ ασ βσ

The group 2 2 2 (Z2 ⊕ Z2)  S2 = α, β, σ | α = β = e,αβ = βα,σ = e,σασ = β,σβσ = α

has elements e,α,β,σ,αβσ,αβ,ασ,βσ and the above automorphisms fulfill the relations of (Z2 ⊕ Z2)  S2. Note that the graph below has the same symmetry group Z2 Sn 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 L2(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=a2. The equivalence of the two definitions follows from the GNS construction. 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 ={xi | i ∈ I } be a set of generators. • ( ) , ∗ Let P E be the set of non-commutative polynomials in xi 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),wehavex < ∞, 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 {yi ∈ B | i ∈ I } satisfy the relations R, then ∗ there is a *-homomorphism, ϕ : C (E, R) → B such that ϕ(xi ) = yi . Examples. (i) The universal C∗-algebra C∗(u, 1 | u∗u = uu∗ = 1) is isomorphic to the algebra of continuous functions C(S1) on the sphere S1 ⊂ 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. 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 non- commutative 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 ⊗ pre- cisely 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). CMQGs and Banica–Speicher quantum groups 887

We verify it for ( f ) = f1 ⊗ f2 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 can- cellation 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. C∗ G (a) Recall the construction⎧ of the group -algebra associated to ⎫. The space ⎨ ⎬ C := α ,α ∈ C G ⎩ g g finite linear combinations g ⎭ g∈G is a *-algebra by    αg g βhh := αgβh gh and  ∗  −1 αg g := α¯ g g . 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 ug g G ug unitary uguh ugh ug ug−1 As an example, verify that C∗(Z) = C(S1), 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. 888 Moritz Weber

With  : ∗ ( ) → ∗ ( ) ⊗ ∗ ( ) Cmax G Cmax G Cmax G given by the extension of

(ug) := ug ⊗ ug ∗ ( ) ( ∗ ( ), ) ∗ ( ) to Cmax G , we have that Cmax G is a CQG. If G is non-abelian, then Cmax 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( 2(G)) is defined by the linear extension of

λ(g)(δh) := δgh, 2 where (δh)h∈G is an orthonormal basis of the Hilbert space (G).Asλ is faithful on CG, we know that λ(CG)) ⊆ B( 2(G))) is isomorphic to CG. We may thus define ∗ ( ) := λ(C ) ⊆ B( 2( ))) Cred G G 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 repre- φ : ∗ ( ) → ∗ ( ) sentations, there is a natural map Cmax G Cred G and the comultiplication map

 : ∗ ( ) → ∗ ( ) ⊗ ∗ ( ) red Cred G Cred G Cred G λ(g) → λ(g) ⊗ λ(g)

factorizes through φ, i.e. we have

red ◦ φ = (φ ⊗ φ) ◦ . ( ∗ ( ),  ) Hence, also Cred G red is a CQG.

We refer to [111, Ex. 5.1.2] 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 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 dif- ferent structures (like, modelling e · x = x), see for instance [124, Definition 2.1].

+ Example 2.9 (Quantum permutation group Sn ). The examples in Example 2.6 are CQGs coming from classical groups. Let us now come to an honest quantum example.

Let Xn := {x1, ··· , xn} be a finite set of points. Then, its automorphism group Aut(Xn) is exactly the permutation group Sn, see Example 2.1.Now,ifweviewXn as a quantum space – what is its quantum symmetry group? The first step is to dualize the set Xn and   we obtain  n ∼ ∗  C(Xn) = C p1,...,pn projections  pi = 1 . i=1 Since the pi form a basis, any action of a CQG (A,)on C(Xn) is of the form

α : C(Xn) → C(Xn) ⊗ A n p j → pi ⊗ aij i=1 for some elements aij ∈ A. These elements aij need to satisfy several relations:

α : C(Xn) → C(Xn) ⊗ A n p j → pi ⊗ aij i=1 for some elements aij ∈ A. These elements aij need to satisfy several relations:

∗ ∗ α(p j ) = α(p j ) ⇒aij = a  ij   α( ) = α( )2 ⇒ ⊗ = ⊗ = ⊗ 2 p j p j pi aij pi pk aijakj pi aij i k,i i ⇒a = a2 ij ij ⎛ ⎞ ⎛ ⎞     ⎝ ⎠ ⎝ ⎠ α is unital ⇒1 ⊗ 1 = α(1) = α p j = pi ⊗ aij = pi ⊗ aij j i, j i j  ⇒ aij = 1. j

This led Wang [124] in 1998 to the definition    ( +) := ( ) := ∗ , ≤ , ≤ | , = = . C Sn As n C uij 1 i j n uij projections uik ukj 1 k k ∗  We may equip this C -algebra with a comultiplication by putting ( ) :=  := ⊗ . uij uij uik ukj k 890 Moritz Weber

Using the orthogonality of the projections uik and uil for k = l (can be deduced from the = fact that k uik 1), we check   2 = ⊗ = ⊗ =  uij uikuil ukjulj uik ukj uij k,l k and    =  = ⊗ . uik ukj 1 1 k k ( +)  ( +) Thus, by the universal property of C Sn ,themap is a *-homomorphism from C Sn ( +) ⊗ ( +) to C Sn C Sn indeed. Moreover, it is co-associative due to  ( ⊗ id) ◦ (uij) = (uik) ⊗ ukj k = uil ⊗ ulk ⊗ ukj , k l = uil ⊗ (ulj) l = (id ⊗ ) ◦ (uij).

The density condition holds true, since  (uij)(1 ⊗ umj) = uik ⊗ ukjumj = uim ⊗ umj k implies  uim ⊗ 1 = uim ⊗ umj ∈ (A)(1 ⊗ A), j fromwhichwemayinferA ⊗ 1 ⊂ (A)(1 ⊗ A), see also more details in Theorem 4.6. Similarly, we have 1 ⊗ umj ∈ (A)(1 ⊗ A) and thus A ⊗ A = (A)(1 ⊗ A). ( ( +), ) This shows that C Sn is a CQG; actually, it is the quantum automorphism group of Xn,see[124]. Thus, in the category of CQGs, the space Xn has more automorphisms: + Its automorphism group is Sn in the category of groups, whereas it is Sn in the category of CQGs. ( +) ≥ We may see that C Sn is non-commutative for n 4 due to the following argu- ment 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 S4 , one may consider a ( +) ( ) *-homomorphism from C S4 to B, which sends uij to the respective entries of the matrix below: ⎛ ⎞ p 1 − p 00 ⎜ − ⎟ ⎜ 1 pp 00⎟ . ⎝ 00q 1 − q ⎠ 001− qq

( +) = + As B is non-commutative, C S4 must also be non-commutative. Thus S4 S4 . CMQGs and Banica–Speicher quantum groups 891

( ) ( +) Let us finish this example by remarking that C Sn is the abelinization of C Sn , i.e.: ( +)/ − ∼ ( ). C Sn uijukl ukluij = C Sn

This can be proven by representing the symmetric group Sn ⊂ Mn(C) as permutation matrices and then checking that the coordinate functions

uij : Sn → C g → gij ( +) satisfy the relations of C Sn . Using Stone–Weierstraß’s theorem, we infer that we have ( +) ( ) ˜ φ a surjection from C Sn to C Sn sending uij to uij. Moreover, the characters on ( +)/ − C Sn uijukl ukluij correspond exactly to permutation matrices and vice versa. Thus ( ( +)/ − ) ∼ . Spec C Sn uijukl ukluij = Sn

Comparing the comultiplication on C(Sn) arising as in Motivation 2.3, and making use of the fact that the multiplication of s, t ∈ Sn is simply given by matrix multiplication, we infer  (u˜ij)(s, t) =˜uij(st) = u˜ik(s)u˜kj(t). k ∼ Under the natural isomorphism C(Sn × Sn) = C(Sn) ⊗ C(Sn) as in Motivation 2.3,this amounts to   (u˜ij)(s, t) = u˜ik ⊗˜ukj (s, t), k ( +) ⊂ + in perfect analogy to the comultiplication map defined on C Sn . This proves that Sn Sn 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 uij ∈ A, 1 ≤ i, j ≤ n, the entries of the matrix u, t • u = (uij) and u = (u ji) are invertible,  •  : → ⊗ ( ) = ⊗ and the map A A min A defined by uij k uik ukj 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]. 892 Moritz Weber

+ Example 2.13 (Orthogonal quantum group On ). In the same way we stepped from Sn to + ⊆ (C) Sn , we may define a quantum version of the orthogonal group On Mn . Note that ∗ C(On) can be written as a universal C -algebra, namely:

( ) = ∗( , ≤ , ≤ | = ∗ , = ( ) , C On C uij 1 i j n uij uij u uij is orthogonal uij commute).

We may express the fact that u is orthogonal by the following relations:   uiku jk = ukiukj = δij. k k

+ In 1995, Wang [122] defined the free orthogonal quantum group On by. ( +) := ( ) := ∗( , ≤ , ≤ | = ∗ , = ( ) ). C On Ao n C uij 1 i j n uij uij u uij is orthogonal

+ , + It is easy to see that On is a CMQG containing On Sn and Sn as quantum subgroups. Hence, there are more quantum (orthogonal) rotations than classical ones. One can show + that On 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 On by

+ ∗ −1 C(O (Q)) := Ao(Q) := C (uij, 1 ≤ i, j ≤ n|u unitary and u = QuQ¯ ).

∈ (C) ¯ = , ∈ R ¯ := ( ∗ ) = ∗ Here Q GLn such that QQ c1n c and u uij . Moreover, uij uij in = +( ) = + general. For Q 1n,wehaveO Q On .

+ + Example 2.14 (Unitary quantum group Un ). The free unitary quantum group Un is the CMQG given by

( +) := ( ) := ∗( , ≤ , ≤ | , t ). C Un Au n C uij 1 i j n u u unitary

We may also write it as

( +) = ∗( , ≤ , ≤ | , ¯ ) C Un C uij 1 i j n u u unitary ( +) ( +) =¯ + ⊂ + revealing that C On is nothing but the quotient of C Un by u u. Thus, On Un . ( +) ( ) ⊂ + Moreover, the abelinization of C Un is C Un ,soUn Un . Note that the requirement t of u being unitary is automatic for commuting uij, but not for non-commuting ones. In fact,

∗ C (uij, 1 ≤ i, j ≤ n|u unitary) does not giverisetoaCMQGasut fails to be invertible [122, Example 4.1]. CMQGs and Banica–Speicher quantum groups 893

Again, there is a deformed version given by + ∗ −1 C(U (Q)) := C (uij, 1 ≤ i, j ≤ n|u unitary and QuQ¯ unitary), see [115,122].

Example 2.15 (Quantum special unitary group SUq (2)). Historically, Woronowicz’s quantum version of SU(2) was the first example of a CQG. Recall

SU(2) := {u ∈ M2(C)|u unitary, det(u) = 1}⊆M2(C).

We may write SU(2) as     a −¯c  SU(2) =  a, c ∈ C, |a|2 +|c|2 = 1 . c a¯

For q ∈[−1, 1]\{0}, we define SUq (2) as the CMQG given by       α − γ ∗ ∗ α, γ  q . C  γα∗ 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]   + 01 SU (2) =∼ O , q −q−1 0 see [133]. See also [134] for higher dimensional versions SUq (n).

+ Example 2.16 (Hyperoctahedral quantum group Hn ). Recall from Example 2.1(c) that the hyperoctahedral group

= Z = σ ∈ , ,..., | 2 = , = ,σ σ −1 = Hn 2 Sn Sn a1 an ai e ai a j a j ai ai aσ(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 =∼ {(u ) orthogonal|u u = u u = 0fori = j}⊂O ⊂ M (C), n ij ⎛ ik jk ⎞ki kj n n 1 ⎜ ⎟ ⎜ .. ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ 1 ⎟  ⎜ ⎟ ai → a := ⎜ −1 ⎟ . i ⎜ ⎟ ⎜ 1 ⎟ ⎜ . ⎟ ⎝ .. ⎠ 1 894 Moritz Weber

{( )| ...}  To do so, check that uij is a group indeed containing all matrices ai and all per- mutation matrices σ; moreover, check that uiku jk = 0 implies that at most one entry per 2 = row is nonzero; together with i uik 1, this implies that this nonzero entry is either 1 or −1; hence the group {(uij) | ...} consists of all permutation matrices where any entry −  ,...,  ,σ 1 may be replaced by 1; all these matrices may be constructed in a1 an .

+ This reveals that we should define the hyperoctahedral quantum group Hn as the CMQG given by   ( +) := ∗ | = ∗ , , = = , = . C Hn C uij uij uij u orthogonal uiku jk ukiukj 0 i j

Note that the crucial step was to find a representation of Hn as a matrix group in terms + of algebraic relations on the entries uij. One can prove that Hn 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 Hn in 2004, he also gave a + definition of a free wreath product ∗ with Sn and he proved: + = Z +. Hn 2 ∗ Sn Finally, we have + ⊂ + ⊂ + ⊂ +. Sn Hn On and Hn Hn + See also [15] for more on Hn .

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 Sn On Un ,see[31,122,124, 128] amongst others. ( (ii) Deformation (‘deform commutativity: fg = qgf, q ∈ C’), like SUq (2), O Q), U +(Q),see[115,133] amongst others. (iii) Quantum isometry groups (‘quantum group of Riemannian isometries on a non- commutative 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 inte- gration 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 μG on G such that for all t ∈ G and all f ∈ C(G):

f (st) dμG (s) = f (s) dμG (s). G G CMQGs and Banica–Speicher quantum groups 895

Hence, we have a positive linear functional h : C(G) → C 

h → f dμG G which remains invariant under left-translation: 

(h ⊗ id) ◦ ( f )(t) = ( f )(s, t) dμG (s) G

= f (st) dμG (s) 

= f (s) dμG (s)

= h( f )1C(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’) h : A → C such that

(id ⊗ h) ◦  = (h ⊗ id) ◦  = h · 1A.

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) · 1A) = h((id ⊗ h ) ◦ (a))   = (h ⊗ h ) ◦ (a) = h ((h ⊗ id) ◦ (a)) = h(a).

  Check h((id ⊗ h ) = h ⊗ h for (a) = a1 ⊗ a2 and extend via linearity. Existence:Forω ∈ A positive, we define

 Kω := {ρ ∈ A |ρ is a state,ρ∗ ω = ω ∗ ρ = ω(1A)ρ}.

Then Kω is a closed subset (in the weak-* topology) of

( )+ := {ρ ∈  |ρ≤ } A 1 A positive 1 and hence it is compact. We will prove the existence of a Haar state in the following three steps: 896 Moritz Weber

(1) Kω = ∅. ∩n = ∅, ω ∈ , ∈ N (2) i=1 Kωi for all positive i A n . (3) By Cantor’s intersection principle and (2), we may find a state h ∈∩ω∈(A)+ Kω = ∅. It satisfies (id ⊗ h) ◦  = (h ⊗ id) ◦  = h.  ω ∈  ω := 1 n ω∗k Proof of (1). Let A be a state. Define n n k=1 where one inductively ∗(n+1) ∗n ∗1  defines ω = ω ∗ (ω ) and ω = ω.As(A )1 is compact, (ωn)n∈N has an accu- ρ ω ∗ ω − ω = 1 ω∗(n+1) − ω≤ 2 , ω ∗ ρ = ρ mulation point . Since n n n n we have . Likewise ρ = ρ ∗ ω. This settles Kω =∅.  Proof of (2). Let ω1,ω2 ∈ A be positive and define ω := ω1 + ω2. We only need to prove  n Kω ⊆ Kω . Then ∅ = Kω ⊆ Kω ∩ Kω and iteratively, ∅ = K n ω ⊆∩ Kω . 1 1 2 i=1 i i=1 i Assume that ω is a state. Let ρ ∈ Kω and define

∗ Lρ⊗ω := {q ∈ A ⊗ A|(ρ ⊗ ω)(q q) = 0}. ⊆ ω ≤ ω ⊆ (ρ ⊗ ω ) Then Lρ⊗ω Lρ⊗ω1 since 1 . Moreover Lρ⊗ω1 Ker 1 by Cauchy– Schwarz. For ψL : A → A defined by ψL (a) := (id ⊗ρ)(a) − ρ(a)1, one can show that (id ⊗ψL )((A)) ⊆ Lρ⊗ω. Thus, keeping in mind that Lρ⊗ω is a left ideal, we have

1 ⊗ ψL (A) ⊆ (A ⊗ 1)(id ⊗ψL )(A) ⊆ (A ⊗ 1)Lρ⊗ω ⊆ Ker(ρ ⊗ ω1).

Hence

0 = (ρ ⊗ ω1)(1 ⊗ ψL (a)) = (ω1 ⊗ ρ)((a)) − ω1(1)ρ(a). ρ Thus is in Kω1 .  Proof of (3). For h in ∩ω∈(A)+ Kω,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. 5.1.6] 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 ∗ ( ) ( ) = δ Haar state hmax on Cmax G is given by hmax ug g,e, since

(id ⊗hmax)((ug) = ughmax(ug) = ueδg,e = 1δg,e = hmax(ug). ( ) = ( ) , ∗ ( ) It is a tracial state i.e. hmax ab hmax ba for all a b in Cmax G . ∗ ( ),  ( ) = δ ,δ The Haar state hred on (Cred G ) is given by hred x x e e as can be verified φ : ∗ ( ) → ∗ ( ) = ◦ φ directly. Now, the natural map Cmax G Cred G satisfies hmax hred . Thus hmax is not faithful in the case when φ is not an isomorphism (i.e. when G is not amenable). ∗ ( ) However, hred is always faithful on Cred 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 (Ared,red) where Ared := πh(A) ⊆ B(Hh) (by the GNS construction CMQGs and Banica–Speicher quantum groups 897 with respect to the Haar state h) and red(πh(x)) := (πh ⊗ πh)((x)).Onemayalso associate to G the enveloping von Neumann algebra L(G) of Ared.

PROPOSITION 3.5 (Haar state of reduced version)

The comultiplication red is well-defined and the Haar state on (Ared,red) is given by hred(πh(x)) := h(x). It is faithful.

∗ Proof. Let πh(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(πh⊗h) = Ker(πh ⊗ πh). We conclude that red is well-defined. More- over,

(id ⊗hred)red(πh(x)) = (id ⊗hred)(πh ⊗ πh)((x)) = πh(id ⊗h)((x)) = πh(h(x)1) = h(x)

= hred(πh(x)), seealso[98, Cor. 1.7.4]. 

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 ◦ iB = h B, (iii) For all c 1, ··· , cn ∈ A∪B ⊂ A∗C B such that h A(ci ) = 0 or h B(ci ) = 0 respectively and ci , ci+1 come from different algebras, we have: h(c1 ···cn) = 0.

Moreover, if (A,A) and (B,B) are CMQGs, so is their free product, see [122] and [111, Sect. 6.3]. 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, Mm(C)) = C(G) ⊗ Mm(C), i.e.  U = uαβ ⊗ eαβ . α,β

We have   U(gh) = uαβ (gh) ⊗ eαβ = (uαβ )(g, h) ⊗ eαβ α,β α,β and ⎛ ⎞   ⎝ ⎠ U(g)U(h) = uαγ (g)uγβ(h) ⊗ eαβ . α,β γ

Since U(gh) = U(g)U(h), by comparing the coefficients we have   (uαβ )(g, h) = uαγ (g)uγβ(h) = uαγ ⊗ uγβ (g, h). γ

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 = (uαβ ) ∈ Mm(A) such that  (uαβ ) = uαγ ⊗ uγβ. γ 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 = eαβ ⊗ uαβ ∈ Mn(u)(C) ⊗ A and  v = eγδ ⊗ vγδ ∈ Mn(v)(C) ⊗ A.

Then we define  ⊗ v := ⊗ ⊗ v ∈ (C) ⊗ (C) ⊗ (a) u  eαβ eγδ uαβ γδ Mn(u) Mn(v) A. ∗ (b) u := eαβ ⊗ uαβ .

DEFINITION 4.4 (Equivalence, irreducibility, intertwiners)

Let u ∈ B(Hu) ⊗ A and v ∈ B(Hv) ⊗ A be representations of a CQG (A,).

(a) If a linear operator T ∈ B(Hu, Hv) 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(Hu, Hv) 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 repre- sentation. 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 (uαβ ) such that  (uαβ ) = uαγ ⊗ uγβ

and u = (uαβ ), ut = (uβα) are invertible, then (A,)is a CQG. 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(wαβ ) be the inverse of some non-degenerate, finite dimensional representation (uαβ ). Then   (uαγ )(1 ⊗ wγβ) = uα ⊗ uγ wγβ γ γ,  = uα ⊗ δβ  = uαβ ⊗ 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 = (a1,γ )(1 ⊗ a2,γ ) γ and  b ⊗ 1 = (b1,)(1 ⊗ b2,)  for some a1,γ , a2,γ , b1,γ , b2,γ ∈ A. Then we have  ab ⊗ 1 = (a1,γ )(1 ⊗ a2,γ )(b ⊗ 1) γ  = (a1,γ )(b ⊗ 1)(1 ⊗ a2,γ ) γ  = (a1,γ b1,)(1 ⊗ b2,a2,γ ) γ, and hence A ⊗ 1 ⊆ (A)(1 ⊗ A) ⊆ A ⊗ A. Thus, if x ⊗ y ∈ A ⊗ A, we write  x ⊗ 1 = (ai )(1 ⊗ bi ) i and we compute   x ⊗ y = (x ⊗ 1)(1 ⊗ y) = (ai )(1 ⊗ bi ) (1 ⊗ y) i = (ai )(1 ⊗ bi y) ∈ (A)(1 ⊗ A). i

This shows that A ⊗ A = (A)(1 ⊗ A). Similarly we have that (A)(1 ⊗ A) = A ⊗ A. This proves the density condition. CMQGs and Banica–Speicher quantum groups 901

For part (b), note that u = (ut )∗ is invertible since ut is, and  ( ∗ ) = ( )∗ = ∗ ⊗ ∗ . uij uij uik ukj k

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 dimen- sional 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) ∈ Mm(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

1 − 1 ω := (y 2 ⊗ 1)v(y 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 (pn)n of pairwise orthogonal, minimal projections in D. Then one shows, using the fact that D acts non-degenerately on H, that  ∼ H = pn H.

Finally the restriction of v to pn H is irreducible, since pn H is a minimal invariant subspace. For (c), we refer to [95].  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 · 1A.

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 A0 be the sub- space of A spanned by the matrix elements of all finite dimensional, unitary representations. Then

(a) A0 ⊆ A is a dense *-algebra. (b) (A0) ⊆ A0 ⊗ A0 (algebraic tensor product). ( , | ) (c) A0 0 A0 is a Hopf *-algebra. | (d) The Haar state h A0 is faithful on A0.

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 δh (sometimes called the regular representation) such that

(δ ) = . span hi, j A

By Theorem 4.8(b), this representation is equivalent to a direct sum of irreducible repre- sentation. Hence the linear span of all finite dimensional unitary representations is dense in A. Next we show that A0 is an algebra. Take  u = eij ⊗ uij ∈ Mn(u)(C) ⊗ A and  v = ekl ⊗ vkl ∈ Mn(v)(C) ⊗ A. CMQGs and Banica–Speicher quantum groups 903

Then  u ⊗ v = eij ⊗ ekl ⊗ uijvkl ∈ Mn(u)(C) ⊗ Mn(v)(C) ⊗ A.

This means, if uij,vkl are the coefficients of the representation, their product are the coefficients of the tensor product of the two representations. Showing that A0 is *-closed can be done as in Theoreom 4.6(b). Next we prove (b). It suffices to check that all monomials

α α u 1 ...u k ∈ A i1 j1 ik jk 0 are mapped to A0 ⊗ A0.Wehave

α α α α  1 ... k =  1 ... k ui j ui j ui j ui j 1 1 k k  1 1 k k α α α α = u 1 ...u k ⊗ u 1 ...u k , i1γ1 ik γk γ1 j1 γk jk γ1,...,γk where each summand is contained in A0 ⊗ A0. For (c), we define the counit by

ε : → C,ε(α ) = δ A0 uij ij and the antipode

: → , ( α ) = ( α )∗, S A0 A0 S uij u ji

α where uij are the coefficients of irreducible representations. It can be shown that they form ε α a basis for A0, thus and S are well-defined. Since A0 is spanned by all entries uij of unitary representations uα, it suffices to check the properties of the counit, resp. antipode. We have   (ε ⊗ )(( α )) = ε( α ) α = δ α = α id uij uik ukj ikukj uij k k and   ( ⊗ ) ( α ) = ( ⊗ ) α ⊗ α m S id uij m S id uik ukj  k = ( α )∗ α uik ukj k = δi, j · 1 = η(ε( α )), uij and analogously the right counterparts. For part (d), we refer to [111, Sect. 5].  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 (A0,), we may associate a CQG (A,)to it such that A0 ⊆ 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 4.10. 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 (A0,0) be the Hopf *-algebra, Sthe antipode on A0. 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) S2 = id. (iii) S is *-preserving. A proof can be found in [98, Sect. 1.7.7], see also [84, Section 11.3].

Example 4.13 (Kac type and non-Kac type). +, +, + (a) Sn On Un are of Kac type using ε : A → C,ε(uij) = δij, : → , ( ) = ∗ . S A A S uij u ji ( +), ( +) ( +) These maps exist by the universal properties of C Sn C On and C Un respec- tively. ( ), +( ), +( ) ∗ = (b) SUq 2 On Q Un Q are not of Kac type, if Q Q 1. ε : A → C,ε(u ) = δ ij ij ( ) = ∗ , ( ∗ ) = ( ∗ )−1 ( ∗ ) . S uij u ji S uij Q Q il uml 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 SUq (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,{uα : α ∈ I } mutually inequivalent unitary irreducible representations, A0 ⊆ A as before. Let B0 be the space of all linear functionals on A given  by x → h(ax), a ∈ A0. Then B0 is a subalgebra of A and isomorphic to the algebraic direct sum ⊕α∈I Mn(α)(C). We may define a comultiplication on B0 by using the multiplier algebra of the algebraic tensor product B0 ⊗ B0. 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 recon- structed from the information contained in its dual group of characters G . The classical Tannaka–Krein duality is an extension to compact non-abelian groups with the role of G being replaced by its category of finite dimensional unitary representations. See also [63]. 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{Hr }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(Hr , Hs). Then (R, ·, {Hr }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 idr acting on Hr 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 Hr = Hs and idr ∈ 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 H1 = C, 1v = v1 = v.

A concrete monoidal W*-category (R, {Hr }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 : Hr → K , where K is a Hilbert space, there exists s ∈ R such that Hs = K and v ∈ Mor(r, s). (ix) For r ∈ R and any orthogonal projection p ∈ Mor(r, r), there exists s ∈ R such that Hs = pHr and i ∈ Mor(s, r), where i is the embedding Hs → Hr .  (x) For r, r ∈ R, there exists s ∈ R such that Hs = Hr ⊕ Hr and the canonical embeddings Hr → Hr ⊕ Hr and Hr → Hr ⊕ Hr 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 : Hr → Hr such that the map

t j : C → Hr ⊗ Hr defined by  t j (1) = ei ⊗ j(ei ) i is in Mor(1, r · r), and the map

t j : Hr ⊗ Hr → C defined by

−1 t j (eh ⊗ ei ) = j (eh), ei 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 ∈ ( (k) ··· (k), ), = , ,..., (k), (k),..., (k) ∈ morphismsbk Mor q1 qnk s k 1 2 m for some q1 q2 qnk Q ∗ = ∈ ( , ) such that k bkbk ids Mor s s . 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 : Hv → Hw linear maps | T v = wT } the space of intertwiners, where v ∈ B(Hv) ⊗ A and w ∈ B(Hw) ⊗ A. Then (Rep G, ⊗, {Hr }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):

Ti vi = wi Ti for i = 1, 2 ⇒ (T1 ⊗ T2)(v1 ⊗ v2) = (w1 ⊗ w2)(T1 ⊗ T2). See also [134,Thm.1.2]. 

DEFINITION 4.17 (Model of a W*-category)

Let (R, ·, {Hr }r∈R, {Mor(r, s)}r,s∈R) be a concrete monoidal W*-category generated by { f, f }. ∗ r r (a) Let B be a unital C -algebra and {v }r∈R be a family of unitaries, i.e. v ∈ B(Hr )⊗B. r Then M = (B, {v }r∈R) is called a model of R if (i) vr·s = vr ⊗ vs. (ii) vr (t ⊗ 1) = (t ⊗ 1)vs 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(Hr ) ⊗ B. We say that r (B,v) is an R-admissible pair if there exists a model M = (B, {v }r∈R) such that v f = v.

Theorem 4.18 (Tannaka–Krein for CMQGs). Let R = (R, ·, {Hr }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 Rirr be a complete set of mutually non-equivalent irreducible elements of R with 1R ∈ Rirr (there is such a set, because R is complete). Set   A := B(Hα) ,

α∈Rirr

 where B(Hα) is the space of linear functionals defined on B(Hα) for every α ∈ Rirr.  α For α ∈ Rirr and ρ ∈ B(Hα) the corresponding element of A will be denoted by uρ . Therefore every a ∈ A can be written as finite sum  α a = uρ .

α∈Rirr CMQGs and Banica–Speicher quantum groups 907

 α α The embedding B(Hα) → A,ρ → uρ is linear, hence there exists a unique u ∈ B(Hα) ⊗ A such that

α α (ρ ⊗ id)u = uρ

 for any ρ ∈ B(Hα) . One can show that A equipped with the multiplication ·:A × A → α β αβ 1 A, uρ · uσ := (ρ ⊗ σ ⊗ id)u is a unital algebra, where u R is the unit of A. Now we introduce a *-algebra structure on A. Any element of R admits complex con- jugation and the complex conjugation of any α ∈ Rirr is irreducible. Let jα : Hα → Hα be the corresponding invertible antilinear mapping. Set

j −1 m := jαmjα

 j for α ∈ Rirr, m ∈ B(Hα). For any ρ ∈ B(Hα) ,letρ be the linear functional on B(Hα) such that ρ j (m j ) = ρ(m) for all m ∈ B(Hα). Equipped with the antilinear involution ∗:A → A,( α)∗ := α , A uρ uρ j is a *-algebra. Let A be the space of all linear functionals defined on A. There exists h ∈ A such that  α 1, if α = 1, (id ⊗h)u = 0, otherwise.

∗ One can show that h is faithful on A. Therefore πh is faithful and for the C -seminorm p(x) := πh(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. r Assume that (B,v)is another R-admissible pair. Let M = (B,(v )r∈R) be the model f α α of R such that v = v. Define φM : A → B,φM (uρ ) = (ρ ⊗ id)v . The extension

φM : A → B is a *-homomorphism with

(id ⊗φM )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. 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:

1 2 ∈ P (2, 4), 1 234

1234 ∈ P (4, 1). 1

The set of all partitions {P(k, l) : k, l ∈ N0} 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)) CMQGs and Banica–Speicher quantum groups 909

(d) If p ∈ P(k, l), then the involution p∗ ∈ P(l, k) is the partition obtained by turning p upside down. Pictorially,

∗ = (∈ P (4, 1)) (∈ P (1, 4))

(e) A subset C ⊆ P (consisting of sets C(k, l) ⊆ P(k, l))isacategory of partitions if it is closed under (b), (c), (d) and if the identity partition |∈P(1, 1) and the pair partition ∈ P(0, 2) are both in C . See [31].

Example 5.2 (Categories of partitions).

(a) P is a category of partitions. (b) The set NC of all non-crossing partitions (i.e. the lines linking the various points in the partition may be drawn in such a way that they do not cross) is a category of

partitions. The partition ∈ P(3, 2) is an example of a non-crossing partition whereas the partition ∈ P(2, 2) is an example of a partition with crossings. (c) The set P2 of all pair partitions (all blocks are of size two) is a category of partitions. (d) The set NC2 := P2∩NC of all non-crossing pair partitions is a category of partitions.

DEFINITION 5.3 (Linear maps associated to partitions)

Let p ∈ P(k, l) and let n ∈ N be a fixed natural number. Define

n ⊗k n ⊗l Tp : (C ) → (C ) as the linear map given by  ( ⊗···⊗ ) := δ ( , ) ⊗···⊗ , Tp ei1 eik p i j e j1 e jl j1,··· , jl

k l where i = (i1,...,ik) ∈{1, 2,...,n} and j = ( j1,..., jl ) ∈{1, 2,...,n} are multi- n ndices and (ei )i=1,...,n is the canonical orthonormal basis of C . Moreover, δp(i, j) is defined as follows: Label the upper points in the partition p with i1,...,ik and the lower points with j1,..., jl . Then δp(i, j) = 1 if the partition p connects only equal-valued indices, and 0 otherwise. For example, consider the partition p := ∈ P(4, 1).If i = (1, 3, 4, 4), j = (1), then δp(i, j) = 0 as can be seen from the picture below:

1344

1 910 Moritz Weber

Similarly if i = (1, 1, 4, 4), j = (1), then δp(i, j) = 1 as can be seen below:

1144

1

Example 5.4 (Linear maps associated to partitions). (a) For the partition p = ∈ P(2, 2),wehave ( ⊗ ) = ⊗ . Tp ei1 ei2 ei2 ei1

This is because δp(i, j) = 1 if and only if j can be obtained by flipping the indices in i. This may be deduced from the picture below: i1 i2

j j 1 2. (b) For p =|∈ P(1, 1), we note that Tp is the identity map.

PROPOSITION 5.5 (Operations on partitions pass to linear maps) If p, q are partitions in P, then

(a) Tp ⊗ Tq = Tp⊗q . α(q,p) (b) Tq Tp = n Tqp, where α(q, p) denotes the number of removed blocks in the composition. Note that p and q need to have appropriate sizes in order to perform the composition. ∗ (c) (Tp) = Tp∗

Proof. (a) Let p ∈ P(k, l) and q ∈ P(k, l). Since the partition p⊗q is obtained by concatenating     p and q,wehaveδp⊗q (i ⊗i , j ⊗ j ) = δp(i, j) · δq (i , j ) (where the suggestive notation i ⊗ i denotes the concatenation of the labelling sets i and i.) Unwrapping the definitions of Tp, Tq , Tp ⊗ Tq , Tp⊗q , we see that

( ⊗ )( ⊗···⊗ ⊗  ⊗···⊗  ) Tp Tq ei1 eik ei ei 1 k = ( ⊗···⊗ ) ⊗ (  ⊗···⊗  ) Tp ei1 eik Tq ei ei  1 k = δ ( , )( ⊗···⊗ ) ⊗ δ ( , )(  ⊗···⊗  ) p i j e j1 e jl q i j e j e j 1 l ,  j j = δ ( , ) · δ ( , )( ⊗···⊗ ⊗  ⊗···⊗  ) p i j q i j e j1 e jl e j e j 1 l ,  j j = δ ( ⊗ , ⊗ )( ⊗···⊗ ⊗  ⊗···⊗  ) p⊗q i i j j e j1 e jl e j e j 1 l j, j

= ( ⊗···⊗ ⊗  ⊗···⊗  ). Tp⊗q ei1 eik ei ei 1 k CMQGs and Banica–Speicher quantum groups 911

Thus we conclude that Tp ⊗ Tq = Tp⊗q . (b) Let p ∈ P(k, l) and q ∈ P(l, m). Then  ( ⊗···⊗ ) = δ ( , ) ( ⊗···⊗ ) Tq Tp ei1 eik p i j Tq e j1 e jl ⎛ j ⎞   = ⎝ δ ( , ) · δ ( , )⎠ ( ⊗···⊗ ). p i j q j h eh1 ehm h j  ⊗···⊗ δ ( , )·δ ( , ) From the above calculation, as the coefficient of eh1 ehm is j p i j q j h , we take a closer look at the terms δp(i, j) · δq ( j, h) for different choices of the labelling set j to identify the origin of the scaling term nα(p,q). We have that δqp(i, h) = 1 if and only if the alignment of p and q to form the composition qp connects equal-valued indices in i and h. Let δqp(i, h) = 1. We try to find all possible values of indices in j for which δp(i, j) · δq ( j, h) = 1. As the removed blocks arise when blocks purely in the lower part of p are aligned to blocks purely in the upper part of q, the value of the indices in j that correspond to the removed block must all be equal. Thus each removed block gives rise to n choices for the common value of the indices of j in that block. The values of the remaining indices of j for which δp(i, j) · δq ( j, h) = 1 are uniquely determined by their connections to the δ ( , ) = upper part in p and to the lower part in q. Hence, whenever qp i h 1, we have that δ ( , ) · δ ( , ) = α(q,p) j p i j q j h n . δ ( , ) = δ ( , ) = δ ( , ) = δ ( , ) = If p i j 1 and q j h 1forsome j, then qp i h 1. Thus if qp i h 0, δ ( , ) = δ ( , ) = δ ( , ) · we have that for any j, either p i j 0or q j h 0. As a result j p i j δq ( j, h) = 0. Summing up the above conclusions, we conclude that  α(q,p) n δqp(i, h) = δp(i, j) · δq ( j, h) j holds for all i, h. α(q,p) Thus Tq Tp = n Tqp. (c) Let p ∈ P(k, l). Note that

δ ∗ ( , ) = ⊗···⊗ , ∗ ( ⊗···⊗ ) , p j i ei1 eik Tp e j1 e jl δ ( , ) = ( ⊗···⊗ ), ⊗···⊗ p i j Tp ei1 eik e j1 e jl = ⊗···⊗ , ∗( ⊗···⊗ ) . ei1 eik Tp e j1 e jl

∗ As the partition p is obtained by turning p upside down, we see that δp∗ ( j, i) = δp(i, j) and thus

∗ ⊗···⊗ , ∗ ( ⊗···⊗ ) = ⊗···⊗ , ( ⊗···⊗ ) ei1 eik Tp e j1 e jl ei1 eik Tp e j1 e jl

∗ , = ∗  for all i j. From this, we conclude that Tp Tp . See [31]. 912 Moritz Weber

We illustrate how α(q, p) arises in part (b) of Proposition 5.5 with an example. Let p = and q = . Then ⎛ ⎞   ( ⊗ ) = ⎝ δ ( , )δ ( , )⎠ . Tq Tp ei1 ei2 p i j q j h eh1 h1 j1,..., j4 Looking at the picture below,

i1 i2

j1 j2 j3 j4

j1 j2 j3 j4

h1  it is clear that the non-zero coefficients arise in the sum δ (i, j)δ ( j, h) when j1,..., j4 p q δp(i, j) = δq ( j, h) = 1, which holds when j2 = i2 = j1 = h1 and j3 = j4. Thus ( ⊗ ) = δ ( , ) (= ) Tq Tp ei1 ei2 n qp i s (as there are n arbitrary choices for j3 j4 ).

DEFINITION 5.6 (Banica–Speicher quantum groups) ⊆ ⊆ + ACMQGG is called a Banica–Speicher QG,ifSn G On and if there is a category of partitions C ⊆ P such that

⊗k ⊗l Mor(u , u ) = span{Tp | p ∈ C(k, l)} for all k, l ∈ N0. It is called free,ifC ⊆ NC. An alternative name (in the past few years, the alternative name came up in order to honor the two pioneers in the theory of easy quantum groups, and also to avoid confusion with the common meaning of the word ‘easy’ in the sense of ‘not difficult’) for Banica–Speicher quantum groups (in fact the original one) is easy quantum groups. See [31].

Remark 5.7 (Tracing back Banica–Speicher QGs to Tannaka–Krein). The definition of Banica–Speicher quantum groups is based on the Tannaka–Krein Theorem 4.18. Let us make this more precise in the sequel. We also refer to [108].

PROPOSITION 5.8 (From categories of partitions to W*-categories) Let C ⊆ P be a category of partitions and let n ∈ N. We put

• R := N0 with the binary operation r · s := r + s. n • Hr := C for r ∈ R, with H0 := C. • Mor(r, s) := span{Tp | p ∈ C(r, s)} This defines a concrete W*-category R generated by 1 = 1. CMQGs and Banica–Speicher quantum groups 913

Proof. It is straightforward to check the axioms of a concrete W*-category. Moreover, n n 1 = 1 since j : C → C defined by j( αi ei ) = αi ei yields t j (1) = ei ⊗ ei = ( ), ∈ ( , ) ( ⊗ ) = δ = ∗ ( ⊗ ), ∗ ∈ ( , ), Tp 1 Tp Mor 0 2 and t j ei1 ei2 i1,i2 Tp ei1 ei2 Tp Mor 2 0 where p is the pair partition . Finally, {1} generates R, since 1 +···+1(k times) = k;for k ∈ N0,takem = 1, b1 = idk ∈ Mor(k, k). 

COROLLARY 5.9 (From Tannaka–Krein to Banica–Speicher QGs) Given a category of partitions C and n ∈ N, we obtain a CMQG G = (A, u) by Proposi- ⊆ ⊆ + tion 5.8 and Theorem 4.18. It is a Banica–Speicher quantum group with Sn G On . More precisely, ( ,( ⊗r ) ) = , (r·s) = (a) A u r∈N0 is a model of the W*-category of Proposition 5.8 with u u u ur ⊗ us and T u⊗r = u⊗s T for all r, s ∈ R and T ∈ Mor(r, s). In particular,

⊗k ⊗l Tpu = u Tp for all p ∈ C(k, l). (b) A is the smallest C∗-algebra containing the matrix elements of u and it is universal, i.e. if (B, u) is another model of the above W*-category, then there is a homomorphism from A to B which carries u to u.

Remark 5.10 (Banica–Speicher QGs defined by relations). From Corollary 5.9, we know ∗ , ≤ , ≤ = ∗ that A is a universal C -algebra generated by elements uij 1 i j n with uij uij ⊗k ⊗l satisfying the relations Tpu = u Tp for partitions p ∈ C(k, l). As  ⊗k( ⊗···⊗ ⊗ ) = ⊗···⊗ ⊗ ··· , u ei1 eik 1 et1 etk ut1i1 utk ik t we infer  ⊗k( ⊗···⊗ ⊗ ) = ( ⊗···⊗ ) ⊗ ··· Tpu ei1 eik 1 Tp et1 etk ut1i1 utk ik ,..., t1tk = ⊗···⊗ es1 esl ,..., s1 sl   ⊗ δ ( , ) ··· p t s ut1i1 utk ik t1,...,tk as well as  ⊗l ( ⊗···⊗ ⊗ ) = ⊗···⊗ u Tp ei1 eik 1 es1 esl ,..., s1 ⎛sl ⎞  ⊗ ⎝ δ ( , ) ··· ⎠ . p i j us1 j1 usl jl j1,..., jl 914 Moritz Weber

⊗k Comparing coefficients, we have the following set of relations in A if and only if Tpu = ⊗l u Tp:   δ ( , ) ··· = δ ( , ) ··· , , . p t s ut1i1 utk ik p i j us1 j1 usl jl for any s i ,..., t1 tk j1,..., jl

Example 5.11 (Relations coming from partitions). Below we consider examples of rela- tions in A we may obtain by analysing various partitions using Remark 5.10. (a) Let . We get   δ = δ ( , ) = δ ( , ) = . s1s2 1A p 0 s1s2 1A p 0 j1 j2 us1 j1 us2 j2 us1 j1 us2 j1 j1, j2 j1 t Thus, we get the relation uu = 1M (A). n  p := ∈ P (0, 1) (b) Let . We note that δp(0, s1)1A = δp(0, j1)us j . The relations  j1 1 1 = ∈{ , ,..., } we get are k uik 1A for all i 1 2 n .

p := ∈ P (2, 2) (c) Let . This partition gives relations of the form uiku jk = ukiukj = 0 if i = j. (d) Let p := ∈ P(2, 2). We conclude that uijukl = ukluij for all i, j, k, l ∈ {1, 2,...,n} which means A is commutative.

+ + Remark 5.12 (Sn and On are Banica–Speicher QGs). From the above examples, we deduce that if G = (A, u) is associated to the category of non-crossing partitions NC ( +) in the sense of Definition 5.6 and Corollary 5.9, there is a homomorphism from C Sn to  ( +) A . One may check that in C Sn all relations corresponding to NC are fulfilled. Thus we +  conclude that Sn is isomorphic to G and hence it is a Banica–Speicher quantum group. Similarly, from the additional commutativity relations obtained from the crossing partition in (d), we see that the group Sn may be associated with the category of all partitions P. Further, P2 may be associated with the orthogonal group On and NC2 may be associated + with On in the sense of Defiinition 5.6 and Corollary 5.9 and they are all Banica–Speicher quantum groups.

Since for all categories of partitions C we have NC2 ⊆ C ⊆ P, we infer that Sn ⊆ G ⊆ + ( )  ( )  ( +) On holds for all Banica–Speicher quantum groups G (i.e. C Sn C G C On mapping generators to generators). Summarizing, we have the following one-to-one correspondence: Combinatorics CMQGs Categories of partitions ←→ Banica–Speicher QGs. See also [31,103,128,130] for more on Banica–Speicher quantum groups.

6. Classification of Banica–Speicher quantum groups Lemma 6.1 (Rotation invariance). Any category C ⊆ P of partitions is closed under rotation, i.e. if p ∈ P(k, l) is in C , then so is the partition obtained by shifting the left- most upper point to the left of all lower points, and similarly by moving the right-most CMQGs and Banica–Speicher quantum groups 915 upper point to the right of all lower points, and also under the reverse operations. We illustrate the rotation operation with the following examples: Let ∈ C . If we perform a left-rotation, the above lemma says that ∈ C . Similarly a further right-rotation gives us that is in C . We may play around with both operations to conclude that is also in C .

Proof. Let p ∈ P(k, l) be a partition in C .AsC is a category, it contains the identity ⊗(k−1) partition |, the pair partition and thus contains p1 :=| ⊗ p and p2 := ⊗| . In order to do the left rotation manoeuvre, we compose p1 ∈ P(k + 1, l + 1) with p2 ∈ P(k − 1, k + 1) to get an element p1 p2 of P(k − 1, l + 1). Pictorially:

···

⊗ p .

Thus the composition yields a left rotation as described in the lemma. Similarly the closure of C under the other rotation operations may be proved by suitable choices of p1, p2 as above. See [31]. 

Lemma 6.2 (Generators of categories). Let p1,...,pn denote the smallest category which contains p1,...,pn ∈ P. We describe generating sets for different categories of partitions:

(a) NC2 = ∅ , (b) P2 = , (c) NC = , , (d) P = , , . Keeping in mind that we always have the identity partition | and the pair partition in a category of partitions C , we omit to write them down in the list of generators.

Proof. Before proceeding forward, we make a few useful observations in order to stream- line the proof. (i) For a partition in a category C ⊆ P with k upper points and l lower points, one may perform k left-rotations to obtain an element of C with no upper points and k + l lower points. Keeping in mind that the process is reversible, all we need to check is that the generating set for C generates all elements of C(0, n) for all n ∈ N. (ii) Let p ∈ NC(0, n). In general, as p is a non-crossing partition with only lower points, it looks like

··· ··· ··· ··· ··· ··· 916 Moritz Weber

Using a minimality argument for block size, we may conclude that p has at least one block bk of the form ··· , where k denotes the number of points in the block. Bear in mind that in order to isolate the block, we do not perform any category operation (like rotation, involution, tensor product, etc). We just consider a block as a subset of the partition. (iii) Using , we get all transpositions in the category and we may permute points in the upper part, or the lower part arbitrarily to directly conclude (b), (d) once we have proved (a), (c).

Proof of (a). Now, clearly ∅ ⊆NC2. As for the opposite inclusion, let p ∈ NC2(0, n).  Removing one of the blocks of two consecutive points, we obtain p ∈ NC2(0, n − 2). By induction hypothesis, p ∈ ∅ . Composing it with for suitable α, β yields p ∈ ∅ . In order to visualize this better, we consider the following example. Let the correspond- ing partitions be

and

.

Choosing α = 1,β = 5 gives back p from p in this case. Proof of (c). Clearly , is in NC as the generating elements are in NC.Weturn our attention to the opposite inclusion. Let bn be the partition in P(0, n) consisting in exactly one block (i.e. all n points are connected).

Since b4 = , composing with ,wehave:

.

, Using the above trick, an inductive argument yields that b2n is in , for all n ∈ N, n ≥ 2. By an involution operation, since , we have that . , Thus we conclude that all partitions bn consisting of exactly one block are in . In order to complete the proof, we may mimic the inductive argument given in (a) by replacing the role of with one of the blocks, using observation (ii). See [128]. 

Example 6.3 (Further categories of partitions). Having extracted generators for natural categories of partitions in the previous lemma, it is now straightforward to define fur- ther categories of partitions. We also list the CMQGs they are corresponding to, due to Corollary 5.9. CMQGs and Banica–Speicher quantum groups 917

(i) corresponds to the free bistochastic quantum group B+ given by n  n n ( +) := ∗ | = ∗ , , = = . C Bn C uij uij uij u orthogonal uik ukj 1 k=1 k=1 + (ii) corresponds to the hyperoctahedral quantum group Hn of Example 2.16.

The category is the same as the category . (iii) corresponds to the hyperoctahedral group Hn := Z2 Sn, see also Example 2.1. See [15,31,128].

Theorem 6.4 (Classification in the free case). There are exactly seven free Banica– Speicher quantum groups i.e. seven categories C in NC. These seven categories of parti- tions are illustrated below with the inclusion relations between them:  ⊗ ⊆⊆ ⊆ ⊆

NC2 = ∅  ,  = NC. ⊆ ⊆   ⊆  ⊗ , 

Proof. Let C be a category of partitions in NC. We consider all possible categories of partitions by a case-by-case study.

(1) Let ∈ C and ∈ C . Then NC = , ⊆C ⊆ NC.

(2) Let ∈ C and ∈/ C . Then ⊆C . On the other hand, if p ∈ C , then all blocks have size one or two. Indeed, assume that p has a block of size at least three. Keeping in mind that p is a non-crossing partition, after suitable rotation, p = ···

p = ··· ··· ··· ··· ··· ··· and composition with

··· ··· ···

= yields that is in C . Thus we conclude that is in C , contradicting our initial assumption. This proves that p has only blocks of size one or two and thus must be in . This implies that C = . (3) Let ∈/ C , ⊗ ∈/ C and ∈/ C . Then all blocks in C have size exactly two and we infer C = NC2. (4) Let ∈/ C , ⊗ ∈/ C and ∈ C . Then all blocks in C have even size and we may prove C = . 918 Moritz Weber

(5)–(7) Continue case study. See [31,128]. 

Remark 6.5 (Dimensions issue). In principle, different categories will yield different CMQGs, in the following sense: Given categories C1 = C2, there will be no *-isomorphism ∗ between their corresponding C -algebras mapping generators uij to generators uij.How- ever, for smaller values of n, some of the categories may yield the same quantum group.

Theorem 6.6 (Classification in the group case). There are exactly six Banica–Speicher groups, i.e. six categories with ∈ C .

(1) (On). (2) , (Bn) (3) , ⊗ (Bn × Z2). (4) , (Hn) (5) , , (Sn). (6) , ⊗ , (Sn × Z2) (Note that .)

Proof. As C0 := C ∩ NC ⊆ NC is a category, the previous theorem gives us all possibil- ities for C0. Clearly C0, is contained in C . For the reverse inclusion, we note that the crossing partition can be used to transform a partition with crossings to a non-crossing partition. Thus C = C0, . See [31,128]. 

Theorem 6.7 (Classification in the half-liberated case). We may list all half-liberated Banica–Speicher quantum groups, i.e. quantum groups whose category of partitions C contains but not . ∗ (1) corresponds to On given by ( ∗) := ∗( | = ∗ , , C On C uij uij uij u orthogonal

abc = bca where a = uij, b = ukl, c = u pq).

⊆ ∗ ⊆ + We have On On On . (2) , ⊗ . (3) , . (4) , ,hs , s ≥ 3, where hs is the two block partition in P(0, 2s) such that all odd points form one block while all even points form a second block. See [128].

DEFINITION 6.8 (Hyperoctahedral case)

A category C ⊆ P is said to be hyperoctahedral if is in C , and ⊗ is not in C .

Theorem 6.9 (Classification in the non-hyperoctahedral case). There are exactly 13 non-hyperoctahedral categories. CMQGs and Banica–Speicher quantum groups 919

Proof. (1) Let C ⊂ NC. By Theorem 6.4, we know that there are six non-hyperoctahedral categories (only is hyperoctahedral). (2) Let C ⊂ NC and ∈ C . By Theorem 6.6, we know that there are five non- hyperoctahedral categories (only , is hyperoctahedral). (3) Let C ⊂ NC and ∈/ C . We may prove that we are in the situation ∈/ C and ∈ C , thus by Theorem 6.7, there are two more non-hyperoctahedral categories. See [128]. 

Theorem 6.10 (Classification in the hyperoctahedral case).IfC is a category of par- titions in the hyperoctahedral case, then

(a) Either we are in the first case: C = πk for some k ∈ N or C = πl , l ∈ N , where πk ∈ P(0, 4k) is the partition given by k blocks each on four points arranged as

πk = a1a2 ...akak ...a2a1a1a2 ...akak ...a2a1, (b) or we are in the second case: For n ∈ N, the CMQG G associated to C is given by the semi direct product

∼ ∗ C(G) = C () ⊗ C(Sn), ↔ v , uij ugi ij

 Z∗n ,..., v where is a quotient group of 2 with generators g1 gn and ij are the gener- ators of C(Sn).

Proof. Note that we may view partitions in P(0, n) as words of the length n, where blocks are represented by equal letters, thus corresponds to the word aaaa for instance, whereas ⊗ is written as the word ab.IfC is a category in the hyperoctahedral case containing the partition

aabaab ∈ P(0, 6), we are in Case (b); otherwise we are in Case (a). Case (a). If aabaab is not in C , we may infer that all partitions in C must obey some ‘pair nesting rule’; a typical partition looks like

abccbddaaeebba.

We may show that all such partitions may be constructed using the partitions πk and the category operations. See [103]. Case (b). If aabaab is in C , we may restrict our attention to words (aka partitions) such that neighbouring letters are different – put differently, if a word contains two neighbouring equal letters, we may remove them (obtaining a word in C from which we may reconstruct Z∗∞ our original word). This resembles words in 2 . (C ) ⊂ Z∗∞ We thus define a subset F 2 consisting of all possible ways to represent a C Z∗∞ (C ) partition in as a word in the generators of 2 . Then, F is 920 Moritz Weber

• Z∗∞ a subgroup of 2 (using the tensor product of partitions for the product of group elements and the involution of partitions for the inversion of group elements), • normal (using the pair partition and the partition aabaab together with the composi- tion), • Z∗∞ invariant under certain endomorphisms of 2 . Z∗∞ Conversely, to every such normal subgroup of 2 , we obtain a hyperoctahedral category Z∗n (C ) of partitions in Case (b). We then take the quotient of 2 by the restriction F to n letters in order to obtain . See [101,102]. 

COROLLARY 6.11 (The classification of Banica–Speicher quantum groups is complete)

The classification of Banica–Speicher quantum groups is complete. Moreover, Theo- rem 6.10(b) shows that the world of Banica–Speicher quantum groups is very rich: To  Z∗∞ every variety of a group, we may associate a quotient group of 2 in order to obtain a Banica–Speicher quantum group in the sense of Theorem 6.10(b). Since there are uncount- ably many varieties of groups, the same holds true for the class of Banica–Speicher quan- tum groups.

7. Applications and use of Banica–Speicher quantum groups, state-of-the-art, open problems 7.1 Use of Banica–Speicher quantum groups for CQGs Banica–Speicher quantum groups give rise to many new examples of CMQGs in the spirit of ‘liberation’ (i.e. omitting fg = gf for the function algebras, see Remark 2.18). + They provide a whole class of quantum subgroups of On and are thus a big step in the direction of understanding all kinds of quantum rotations. Note that the machinery of Banica–Speicher quantum groups has recently been extended to unitary Banica–Speicher + quantum subgroups, i.e. we may now cover even quantum subgroups of Un ,see[110]. The class of Banica–Speicher quantum groups may be understood by combinatorial means. In fact, the general philosophy is: ‘Everything about Banica–Speicher quantum groups should be visible in terms of partitions.’ This has been proven successful, for instance, regarding the Haar measure (‘Weingarten calculus’) [18] or the representation theory (‘fusion rules’) [73]. Thus, understanding Banica–Speicher quantum groups via their combinatorics might help to get a better understanding for general CMQGs. The result of Theorem 6.10(b) is a first example of a structure theorem which was discovered by studying the combinatorics of Banica–Speicher quantum groups first [101], but could then be extended also to non-Banica–Speicher quantum groups in a very general way [102]. One of the strategies to produce new results about CMQGs using Banica–Speicher quantum groups is the following:

• Take any result that holds for Sn, On or Un. • +, + + Make it ‘quantum’ by proving an analog for Sn On or Un . • Generalize it to Banica–Speicher quantum groups, ideally by finding a uniform proof (possibly using partitions). • Generalize it to CMQGs. CMQGs and Banica–Speicher quantum groups 921

7.2 Use of Banica–Speicher quantum groups for free probability Free probability [99,118] profits from Banica–Speicher quantum groups in many ways. Firstly, we can do free probability on Banica–Speicher quantum groups. Indeed, any CMQG comes as a C∗-algebra A with a (Haar) state h – thus, we have a noncommutative probability space and we may ask for the noncommutative distributions of the elements uij, for instance, the ‘laws of characters’. This is doable for Banica–Speicher quantum groups, as the Haar state is well-understood here [18]. It turns out, that the law of i uii is + a semicircle in the case of On (which is in a way the ‘Gauß distribution in free probability’) + and a free Poisson distribution in the case of Sn . Secondly, Banica–Speicher quantum groups provide distributional symmetries for free probability in a de Finetti sense. Recall the classical de Finetti theorem: If (xn)n∈N is a sequence of classical, real random variables, then (xn)n∈N is independent (over the tail σ-algebra) and identically distributed if and only if the distribution of (xn)n∈N is invariant under the actions of Sn for all n ∈ N. A free version of this theorem holds true [85]: If (xn)n∈N is a sequence of noncommutative, self-adjoint random variables, then (xn)n∈N is free (over the tail von Neumann algebra) and identically distributed if and only if the ( ) + ∈ N distribution of xn n∈N is invariant under the actions of Sn for all n . Refinements of this de Finetti theorem to other Banica–Speicher quantum groups have been proven [24],seealso§7.4.

7.3 Use of Banica–Speicher quantum groups for von Neumann algebras As any CQG comes with a Haar state, we may perform the GNS construction and we may thus associate a von Neumann algebra to a CQG. These ‘quantum group von Neumann alge- bras’ are an own class of examples of von Neumann algebras, besides the well-known group + von Neumann algebras. Those associated to On share many properties with the famous free group factors LFm; however, they are not isomorphic to the free group factors [49]. + + Amongst the properties for the von Neumann algebraic side of On and Un , we know • their reduced C∗-algebras are non-nuclear, exact, simple and have the metric approx- imation property. They are K -amenable and their K -theory is ( ( +)) = Z, ( ( +)) = Z, ( ( +)) = Z, ( ( +)) = Z2. K0 C On K1 C On K0 C Un K1 C Un • the von Neumann algebras associated to them are strongly solid, non-injective, full, prime II1 factors having the Haagerup approximation property and no Cartan subal- gebra. • + + moreover, On and Un are weakly amenable and have the Akemann-Ostrand property as well as the property of rapid decay. See [130] for an overview on the operator algebraic properties of Banica-Speicher quantum groups, see also Section 7.4.

7.4 Overview on the literature on Banica–Speicher quantum groups and related work The literature on Banica–Speicher quantum groups (also called easy quantum groups) is constantly growing. The official starting point was the initial article by Banica and Speicher in 2009 [31]. However, there are many articles predating and preparing the theory 922 Moritz Weber

+, + of Banica–Speicher quantum groups. They mainly deal with the quantum groups On Un + and Sn and even nowadays when being interested in Banica–Speicher quantum groups one should certainly consider the literature on these three special quantum groups as an ‘extended literature on Banica–Speicher quantum groups’. In the sequel, we try to give a commented and complete list of the literature on Banica– Speicher quantum groups and a commented but certainly incomplete list of the literature +, + + on On Un and Sn . Note that our comments on the articles only highlight some aspects and never present a full summary of the contents repsectively.

7.4.1 The roots within Woronowicz’s framework. • [132,135]: Introduction of CMQGs. • [134]: Tannaka–Krein theorem for CMQGs, enabling a category approach to CMQGs and thus building the basis of the Banica–Speicher quantum group machine. See also [96] for a recent approach to Tannaka–Krein for CMQGs.

7.4.2 Predating work leading to Banica–Speicher quantum groups. • +, + [122]: Introduction of On Un . See also [115] for their deformed versions. • + + [2,3]: Representation theory of On and Un . • + [124]: Introduction of Sn . • + [4]: Representation theory of Sn . • + [37]: Definition of Hn . • + [1,15,32]: Representation theory of Hn and related quantum groups. • +, + + [18,19]: Computation of the Haar state of On Un and Sn using partitions (Weingarten calculus). • + + [85]: De Finetti theorems for Sn . See also [58] for de Finetti theorems for On and + Un and [57,59,61] for more on distributional quantum symmetries and further links to free probability.

7.4.3 Predating work on a diagrammatic calculus. The diagrammatic calculus has diverse roots and appears in many disguises at many places in mathematics. We can only approx- imate the question of how partitions and CMQGs got combined at some point. However, it was mainly Banica who – over the time and in several articles, partially with coauthors – in the end manifested partitions and a diagrammatic calculus as an indispensable tool in the theory of CMQGs.

• [50]: Using pair partitions (allowing crossings) in order to study the representation theory of On. • [131]: A similar calculus is used when computing integrals on groups with respect to the Haar measure, today known as the Weingarten calculus. See, for instance, the work of Collins [56] for modern approaches. • [83]: Diagrammatic approach to the Jones polynomial; pushed further by Bisch and Jones [42] in 1997 in order to understand Fuss–Catalan algebras and Temperly–Lieb algebras. • [134]: A diagrammatic calculus resembling the one in categories of partitions shows up throughout the investigation of quantum SU(n). CMQGs and Banica–Speicher quantum groups 923

• [105]: Combinatorial approach to free probability theory using partitions; linking it to CMQGs in joint work with Köstler in 2009 [85]; see also [60,61] for further links to free probability. • [3,5,15,18,32]: Using partitions at several places when dealing with CMQGs, finally leading to the theory of Banica-Speicher quantum groups.

7.4.4 Literature on Banica–Speicher quantum groups. • [31]: Introduction of Banica–Speicher quantum groups (under the name ‘easy quantum groups’); partial classification in the free case. (s) [s] • [22]: Definition of the series Hn and Hn as Banica–Speicher quantum groups; partial classification of non-hyperoctahedral Banica–Speicher quantum groups; computation of several laws of characters. • [23]: Stochastic aspects of Banica–Speicher quantum groups. • [24]: Uniform de Finetti theorems for some Banica–Speicher quantum groups such as +, +, + + Sn On Hn and Bn . • + =∼ + [100]: Isomorphism Bn On−1 and others; fusion rules. • [30]: Quantum homogeneous spaces in certain Banica–Speicher quantum groups aris- ing as rectangular parts of the fundamental unitary (for instance as the first k lines). • [128]: Complete classification in the free case, the non-hyperoctahedral case and the half-liberated case. • [73]: Uniform fusion rules theorem for Banica–Speicher quantum groups; later extended by Freslon in [68]. • [101–103]: Complete classification of all Banica–Speicher quantum groups (in partic- ular, in the hyperoctahedral case). • [130]: Chapter on Banica–Speicher quantum groups in lecture notes on free probability theory [118]. • [109,110]: Introduction of unitary Banica–Speicher quantum groups; complete clas- sification in the free case. • [75]: Fixed point algebras for Banica–Speicher quantum groups. • [97]: Computation of the Martin boundary of the semi-direct product Banica–Speicher quantum groups (hyperoctahedral case).

7.4.5 Generalizations of Banica–Speicher quantum groups. • +, + [29]: Two-parameter versions of On Sn and other CMQGs; leading to Banica’s [11] definition of super-easy quantum groups in 2017. • [69]: Group elements decorating points of partitions; see also [70]. • [51]: Banica–Speicher quantum groups associated to three-dimensional partitions.

+, + + ( ) 7.4.6 Further related work on On Un and Sn incomplete list . Again, let us empha- size that the following list is composed and ordered from a subjective point of view (in fact, the order is vaguely regarding subject matter, but it is not optimized). Many results +, + + about On Un and Sn are hidden in further articles, possibly just as side remarks. More- + over, we decided to refrain from including all results about the deformations of On and + Un (as defined in [115]), because sometimes the undeformed cases behave special and are excluded though, and also, as Woronowicz’s SUq (2) can be seen as a deformation of 924 Moritz Weber

+ O2 (see Example 2.15), this would then exceed our capacities for assembling a list of literature. So, we more or less restricted our attention to the exposition of work directly +, + + related to On Un and Sn , and we sincerely apologize for any gap in our list – and there are certainly many! – as well as for the quite restricted (and again, incomplete) expo- sition of the contents of the respective articles. In particular, many articles by Banica and coauthors are missing, one could almost add his whole list of publications to the list below.

• + [16]: Survey on Sn . • ∗ [17]: There are no intermediate quantum groups between On and On . • ∗ [33,39]: On On and other half-liberated quantum groups. • ∈ ( +) [21]: Spectral measure of uij C On . • + [26]: On as the quantum isometry group (in Goswami’s sense) of the free sphere, within the framework of Connes’s noncommutative geometry; see also [8,9,76,106] for related work. • + [112]: The factor associated to On is solid, prime and has the Akemann–Ostrand ∗ + property; the reduced C -algebra associated to On is exact. • +, + + [43–46]: Properties of von Neumann algebras associated to On Un and Sn and oth- ers, such as the Haagerup approximation property (HAP), the metric approximation property, Haagerup inequalities and others. • + + + [62,67]: CCAP for On , weak amenability for On and Un . • + [79,80]: The von Neumann algebra associated to On has no Cartan subalgebra. • + [91]: HAP for Hn and others. • + + [34]: Kirchberg’s factorization property for On and Un . • + [71]: MASA on On . • + + [48]: Connes Embedding Property (CEP) for On and Un . • + [49]: The factors associated to On are not isomorphic to the free group factors. • + [47]: On and entanglement in quantum information theory. • + + + [119,120]: K -theory of On and Sn , see also [117] for more on this and the case Un . • + [72,78,93]: Other de Finetti theorems related to Sn (Boolean, monotone and bi-free independence). • 2 + + [41,89,116]: Computation of -Betti numbers of On and Un . • + [38]: Hochschild cohomology on Sn ; see also [40]. • + + [125]: Sn and On are simple; see also [53,126,127]. • + [54]: Un is residually finite dimensional. • + + [81]: Hopf images in S4 and automorphisms of S4 . • + ( ( )) [20]: Models for Sn , i.e. matrices in Mk C X for some compact space X providing ( +) representations of C Sn ; see also [10,13,25,27] for more on this subject. • [37]: Introduction of free wreath products of CMQGs; see also [64,70,90,92,107,121] for more on this subject. • [6,36]: Quantum automorphisms of graphs; see also [12,14,37,52,74,104,106]for more on this subject. • + [28]: Links between Sn and the Hadamard conjecture; see also [7]. • + [55]: Stochastic processes on On ; see also [65,66] for more on this subject. CMQGs and Banica–Speicher quantum groups 925

7.5 Open problems Although the theory of Banica–Speicher quantum groups is already quite well studied, there are still many open problems and questions. We only list a few of them. Others – and more technical ones – may be found in various articles of §7.4.4.

7.5.1 Intermediate and non-Banica–Speicher quantum groups. From [17], we know   ∗ that there is no quantum group G such that On G On . How about those with ∗   + On G On ? In general, we know, that the only Banica–Speicher quantum group sitting strictly + ∗ between On and On is the half-liberated quantum group On , but we do not know whether + there is any other (non-Banica–Speicher) quantum group G in between On and On . More- + over, it is unkown whether there is a quantum group G sitting in between Sn and Sn ,ifso   + is Sn G Sn ? It is easy to show, that there cannot be a Banica–Speicher QG, but we do not know whether there is any non–Banica–Speicher quantum group in that strip. More generally, we ⊂ ⊂ + can ask for examples of non-Banica–Speicher quantum groups G such that Sn G Un . Almost nothing is known about this question, see [102] for some indirect examples.

+ 7.5.2 Deformations of Banica–Speicher quantum groups. The quantum groups On + +( ) +( ) and Un admit deformations to non-Kac quantum groups O Q and U Q , see Exam- ples 2.13, and 2.14, and [115]. It is not so difficult to see that the intertwiner spaces of + O (Q) may again be described by noncrossing pair partitions, just like in the case Q = 1n, + which is On . However, it is absolutely unclear how to define Q-deformations of Banica– Speicher quantum groups, i.e. how to define analogs G(Q) ⊂ U +(Q) of Banica–Speicher ⊂ + quantum groups G Un such that their intertwiner spaces are again given by the same category of partitions. In other words, given a category of partitions C , the assignment ∗ p → Tp (see Definition 5.3) allows us to associate a concrete W -category to C ,see Proposition 5.8. The crucial fact is that the operations on the partitions, like tensor prod- uct, composition and involution, behave nicely with respect to the assignment p → Tp,see Proposition 5.5. We may therefore ask: Is there a way to deform the phase functor p → Tp such that suitable (maybe slightly weaker) analogs of Propositions 5.5 and 5.8 hold? Can we then associate ‘deformed Banica-Speicher’ quantum groups G(Q) ⊂ U +(Q) to cate- gories of partitions?

7.5.3 Classification in the unitary case. The classification of all orthogonal Banica– Speicher quantum groups is settled, see Corollary 6.11 and [103]. In the unitary case however, the situation is quite the converse: Almost nothing is known about unitary Banica– Speicher quantum groups. The task is to classify all categories of colored partitions, see [109]; they correspond to Banica–Speicher quantum groups G such that ⊂ ⊂ +. Sn G Un In [109], we classified all categories of noncrossing colored partitions, i.e. the case + ⊂ ⊂ +. Sn G Un Moreover, we found all classical Banica–Speicher groups

Sn ⊂ G ⊂ Un. 926 Moritz Weber

It is natural to ask for the case ⊂ ⊂ +. Un G Un Further natural questions about partial classifications are (derived from the observations in the orthogonal case): Which are all half-liberated unitary Banica–Speicher quantum groups? Which are all categories/quantum groups in the non-hyperoctahedral case? Do we have a unitary semi-direct product construction as in [102]? See also [110]formore questions. Note that in the situation of generalizations of Banica–Speicher quantum groups (see §7.4.5) even further classification problems await their completion. The flavour of these problems is by large means of combinatorial nature.

7.5.4 Laws of characters. One important characteristics of a CMQG is its law of characters. More precisely, let (A, u) be a CMQG and let h be its Haar state. We put χ := ∈ ( , ) i uii A.Thelawof A u is by definition the collection of all moments: h(χ k), k ∈ N. One can show that h(χ k) coincides with ⊗ ⊗ dim Mor(u k, u k). The law of a CMQG thus contains important information about the intertwiner spaces as well as about the Haar state. If (A, u) is a Banica–Speicher quantum group and if the maps Tp for p ∈ C are linearly independent (for instance, when C ⊂ NC), the computation of the law of characters amounts to a combinatorial counting problem: |C (k, k)|=? In order to express the law of a quantum group, the language of free probability theory is appropriate (see also §7.2) – we need to find a noncommutative random variable x in a noncommutative probability space (B,ϕ)such that ϕ(xk) = h(χ k), for all k ∈ N holds, i.e. the noncommutative distributions of x and χ coincide. We then say that the law of the CMQG (A, u) is the law of x ∈ (B,ϕ). Examples of such laws are the semicircle + law (somehow the ‘Gaussian law of free probability’) for On or the free Poisson law (also + called Marchenko–Pastur law) for Sn , in analogy with the classicial situation: Gaussian for On and Poisson for Sn. See [128] for more on laws of characters for CMQG and Banica– Speicher quantum groups and [129] for more on noncommutative probability spaces and noncommutative distributions. There are many Banica–Speicher quantum groups for which we do not know the law of characters including those from the semi-direct product construction [102]aswellas those arising from the categories πk in [103].

7.5.5 Uniform proofs for Banica–Speicher quantum groups. One of the striking features of Banica–Speicher quantum groups is that they are in a way a class of ‘combinatorial quantum groups’ in the sense that most of the properties of a Banica–Speicher quantum group should be hidden in its combinatorial data – the philosophy is that we should be able to read the properties of a Banica–Speicher quantum group from its category of partitions. As a consequence, for any property that holds for all Banica–Speicher quantum groups (or maybe for a large subclass) we should find a uniform proof in terms of partitions. CMQGs and Banica–Speicher quantum groups 927

Examples for such a strategy are the uniform de Finetti proof [24], the Weingarten calculus for Banica–Speicher quantum groups [18], the fusion rules [73] or the semi-direct product quantum groups [101]. A generic open problem is now to find uniform proofs for properties of Banica–Speicher quantum groups. Ideally, this would lead to some general insight for CMQGs in the sense that certain properties of CMQGs may be derived directly from their intertwiner spaces, see for instance [102]. See also §7.1. +, + + Now, many of the properties listed in §7.4.6 are known only for On Un and/or Sn . For each of these properties, we may now ask: +, + + (1) Is it true for all three quantum groups On Un and Sn ? Is it true for all free Banica– + ⊂ ⊂ + Speicher quantum groups Sn G Un ? Is it true for all Banica–Speicher quantum groups? (2) Is there a uniform proof of this phenomenon? Can we express the property by means of partitions, i.e. do we have a theorem/proof like: If C is a category of partitions with property X, then the associated Banica– Speicher quantum group has the property Y . (3) Can we generalize the uniform proof in a way such that it holds for a large class of CMQGs, even non-Banica–Speicher ones?

One property for which a uniform proof seems possible is the Haagerup property (HAP).

7.5.6 A diagrammatical calculus for partially commutative quantum groups. It is a natural desire to extend the powerful machine of indexing intertwiners by partitions to other CMQGs. One class for which first ideas are around are the partially commutative quantum ,   ∈ ({ , }) groups On Sn , etc. as introduced in [106]. Here, Mn 0 1 is a symmetric matrix  with ii = 0 and the quantum groups G arise from imposing some partial commutation  relations on the generators uij ∈ C(G ). In the extreme case ij = 1 for all i = j,we  =  =  = , obtain On On and Sn Sn, whereas the other extreme case ij 0 for all i j yields  = +  = +  On On and Sn Sn . Moreover, may be seen as the adjacency matrix of a finite,   undirected graph  without loops, and Sn is exactly the quantum automorphism group of  as defined by Bichon [36]. Hence, understanding the intertwiner spaces of these -quantum groups would help to understand the intertwiner spaces of quantum automorphism groups of graphs, amongst others – besides understanding this new class of CMQGs itself, formed by the -quantum groups. Since the extreme cases include Banica–Speicher quantum groups, it seems plau- sible that a diagrammatic calculus based on partitions (with some extensions and possibly restricting to matrices  with certain technical conditions) should work for the -intertwiner spaces as well, see [106] for first attempts in this direction. However, for the moment, basi- cally nothing is known about the intertwiner spaces of these quantum groups.

7.5.7 Further open problems. Further open problems may be found in the various arti- cles mentioned in §7.4.4 as well as in §7.4.6. In the following list, we will only very briefly link to some of them. • + [7]: Many open problems, questions and conjectures regarding the links between Sn and the Hadamard conjecture. • [24]: Find de Finetti theorems for all Banica–Speicher quantum groups (linked to the question of finding the laws of characters). 928 Moritz Weber

• [30]: How to fill up a rectangular matrix (uij)1≤i≤n,1≤ j≤k to a square matrix (uij)1≤i, j≤n of a Banica–Speicher quantum group (A, u)? (In the language of the paper: When is the map in [30, Prop. 5.4] proper for general k, n?) • [126]: Wang’s program of classifying simple compact quantum groups; see in partic- ular, [126, Sect. 6]. • [76]: How can Banica–Speicher quantum groups be seen as the quantum symmetries of some quantum spaces, in particular: as quantum isometry groups in Goswami’s framework? See, in particular, [76, Sect. 10].

Acknowledgements The author would like to thank V.S. Sunder for his invitation to IMSc, Chennai, for his great hospitality and his vision and support to get these lecture notes produced. He would like to wish him all the best for his retirement, in deepest admiration for all of his mathematical achievements. Furthermore, he thanks Soumya for all his great work regarding the contents and the design of these notes and for providing all the figures. The author thanks Pierre Fima and Issan Patri for their various and valuable critical remarks throughout the lectures at IMSc. He also thanks Felix Leid and Simon Schmidt for improvements regarding the exposition in the chapter on Tannaka–Krein theory. The author was partially supported by the ERC Advanced Grant NCDFP, held by Roland Speicher.

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