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’.