Fixed Point Algebras for Easy Quantum Groups

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove tha...

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Дата:	2016
Автори:	Gabriel, O., Weber, M.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2016
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	Fixed Point Algebras for Easy Quantum Groups / O. Gabriel, M. Weber // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ.

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spelling	irk-123456789-1478622019-02-17T01:23:17Z Fixed Point Algebras for Easy Quantum Groups Gabriel, O. Weber, M. Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group Sn⁺, the free orthogonal quantum group On⁺ and the quantum reflection groups Hns⁺. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions. 2016 Article Fixed Point Algebras for Easy Quantum Groups / O. Gabriel, M. Weber // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 46L80; 19K99; 81R50 DOI:10.3842/SIGMA.2016.097 http://dspace.nbuv.gov.ua/handle/123456789/147862 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group Sn⁺, the free orthogonal quantum group On⁺ and the quantum reflection groups Hns⁺. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf-Galois extensions.
format	Article
author	Gabriel, O. Weber, M.
spellingShingle	Gabriel, O. Weber, M. Fixed Point Algebras for Easy Quantum Groups Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Gabriel, O. Weber, M.
author_sort	Gabriel, O.
title	Fixed Point Algebras for Easy Quantum Groups
title_short	Fixed Point Algebras for Easy Quantum Groups
title_full	Fixed Point Algebras for Easy Quantum Groups
title_fullStr	Fixed Point Algebras for Easy Quantum Groups
title_full_unstemmed	Fixed Point Algebras for Easy Quantum Groups
title_sort	fixed point algebras for easy quantum groups
publisher	Інститут математики НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/147862
citation_txt	Fixed Point Algebras for Easy Quantum Groups / O. Gabriel, M. Weber // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 44 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT gabrielo fixedpointalgebrasforeasyquantumgroups AT weberm fixedpointalgebrasforeasyquantumgroups
first_indexed	2025-07-11T02:59:02Z
last_indexed	2025-07-11T02:59:02Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 097, 21 pages Fixed Point Algebras for Easy Quantum Groups Olivier GABRIEL † and Moritz WEBER ‡ † University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark E-mail: olivier.gabriel.geom@gmail.com URL: http://oliviergabriel.eu ‡ Fachbereich Mathematik, Universität des Saarlandes, Postfach 151150, 66041 Saabrücken, Germany E-mail: weber@math.uni-sb.de Received June 13, 2016, in final form September 26, 2016; Published online October 01, 2016 http://dx.doi.org/10.3842/SIGMA.2016.097 Abstract. Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+ n , the free orthogonal quantum group O+ n and the quantum reflection groups Hs+ n . Our fixed point-algebra con- struction provides concrete examples of free actions of free orthogonal easy quantum groups, which are related to Hopf–Galois extensions. Key words: K-theory; Kirchberg algebras; easy quantum groups; noncrossing partitions; fusion rules; free actions; free orthogonal quantum groups; quantum permutation groups; quantum reflection groups 2010 Mathematics Subject Classification: 46L80; 19K99; 81R50 In memory of the late Professor John E. Roberts. 1 Introduction This article was initiated from the meeting of the two authors and their respective interests in easy quantum groups and the fixed point algebra construction. Let us start by reminders on the setting of the present article. Compact quantum groups (CQGs) were defined by Woronowicz and further studied in a series of papers [41, 42, 44]. Following the paradigm of noncommutative geometry, the general idea is to describe all properties of a compact group G in terms of its algebra C(G) of (continuous) functions, using in particular a comultiplication ∆: C(G)→ C(G×G) ' C(G)⊗C(G) to realise the group law µ : G×G→ G. If we then consider (possibly noncommutative) C∗-algebras with such a comultiplication, we get CQGs as an extension of compact groups. Of course, additional assumptions are needed to make the above rigorous (see Section 2.1 below). To be more precise, we will mainly deal with compact matrix quantum groups (CMQGs). Among CMQGs, there is a class of particular examples, called easy quantum groups. Cate- gories of partitions and easy quantum groups were first defined by Banica and Speicher in [5] in the orthogonal case. Tarrago and the second author extended their approach to the unitary setting, see [37]. To each easy quantum group is associated a category of partitions, which pro- vides a way to “visualise” it. The basic idea of easy quantum groups is that they should form a tractable sub-class of CMQGs, since they can be described and manipulated via their category of partitions, by a Tannaka–Krein type argument [43]. This line of argument is illustrated by mailto:olivier.gabriel.geom@gmail.com http://oliviergabriel.eu mailto:weber@math.uni-sb.de http://dx.doi.org/10.3842/SIGMA.2016.097 2 O. Gabriel and M. Weber the article [18], where Freslon and the second author provided a description of fusion rules for easy quantum groups based on their categories of partitions. On another note, the Cuntz algebra On has been defined as a universal C∗-algebra by Cuntz in his paper [11] and has evolved over time into one of the most important examples of C∗-algebras, with applications to classification theory and physics. An example of an application is provided by Doplicher and Roberts’s abstract, Tannaka–Krein like duality results in a series of articles (see, e.g., [15, 16]) for actions of (ordinary) compact groups on C∗-algebras. This discovery motivated a considerable interest (see for instance [8, 33, 34]). A basic step of Doplicher–Roberts’s duality theory is to consider so-called “canonical actions” of compact groups on Cuntz algebras. A source of inspiration for further research in this direction is the article [32], where Pinzari introduces a fixed point algebra Oλ(G) from the regular representation λ acting on OL2(G) and proves that given two compact groups G,G′, the fixed point algebras Oλ(G) and Oλ(G′) are isomorphic as Z-algebras if and only if C∗(G) ' C∗(G′). Motivated by the desire of generalising Doplicher–Roberts theory and following the articles [10, 27, 31], the first author considered an action of a CQG G on a Cuntz algebra and de- scribed its fixed point algebra. More precisely, two conditions (C1) and (C2) were introduced, which ensure that the fixed point algebra is actually a Kirchberg algebra in the UCT class N . Kirchberg–Phillips’s classification theory (see [23, 24]) then proves that up to ∗-isomorphism, the fixed point algebra is characterised by its K-theory. In [19], examples of computations of the K-theory of the fixed point algebra are given – they only depend on the fusion rules of G. In the present article, we combine these two directions of research to describe the fixed point algebras of actions of the free orthogonal quantum group O+ n , the quantum permutation group S+ n and the quantum reflection group Hs+ n . An interesting feature of the present fixed point algebra construction is that it provides a very concrete realisation of the intertwiner spaces defining the easy quantum group (see Proposition 3.2 below). The main results of this paper are the reformulation and characterisation of the hypothe- ses (C1) and (C2) of the fixed point algebra construction theory of [19] in terms of partition categories (see Theorems 3.6 and 3.7 below), together with the identifications of K-theory for the fixed point algebras associated to the natural representations of O+ n , S+ n and Hs+ n (see Theo- rems 4.1 and 5.11 below). The fixed point algebras for O+ n and S+ n are isomorphic while the one for Hs+ n is very different. Thus, in some sense, the actions of S+ n and O+ n are somehow “similar” while Hs+ n acts very differently. Moreover, since these fixed point algebras depend only on the fusion rules of the CQGs at hand, this phenomenon manifests concretely that the fusion rules of Hs+ n are very different from those of O+ n and S+ n . This article is organised as follows: in Section 2, we start by a review of the notions of CQGs and CMQGs, before presenting the notions of categories of partitions and easy quantum groups and discussing Cuntz algebras and actions of CMQGs on these. Section 3 is devoted to the fixed point algebra construction properly speaking, while Sections 4 and 5 are detailed studies of examples, namely the free orthogonal quantum group O+ n and the quantum permutation group S+ n on the one hand, and the quantum reflection group Hs+ n on the other hand. 2 Reminders and review 2.1 Compact matrix quantum groups In this article, we consider only minimal tensor products of C∗-algebras. We will deal with compact quantum groups (CQGs) which we denote by G. They are defined by a separable unital C∗-algebra C(G) together with a unital ∗-algebra homomorphism ∆: C(G) → C(G) ⊗ C(G) which satisfies coassociativity and cancellation properties – for more details on these objects and their representations, see [41, 44]. Fixed Point Algebras for Easy Quantum Groups 3 These compact quantum groups admit (unitary) representations or, equivalently actions on Hilbert spaces and C∗-algebras. A unitary representation u of G on a Hilbert space H of finite dimension d is a C(G)-valued d×d matrix U = (uij) ∈Md(C(G)) which is unitary and satisfies the coassociativity property ∆(uij) = ∑ k uik ⊗ ukj . In particular, for any G, we have a trivial representation denoted by ε, defined by the (1× 1)- matrix 1 ∈ C(G). For two fixed representations u ∈ Md1(C(G)) and v ∈ Md2(C(G)) acting on the Hilbert spaces H1 and H2, respectively, a linear map T : H1 →H2 is an intertwiner (see [41]) if Tu = vT. We denote by Hom(u, v) the set of interwiners between u and v. If Hom(u, v) includes an invertible map, then we say that u and v are equivalent. Just like for ordinary compact groups, any representation of G is equivalent to a unitary one, therefore we will only consider unitary representations and we will refer to these as “representations”. A representation u is called irreducible if Hom(u, u) ' C. Representations of CQGs admit notions of direct sum – with the above notations, u ⊕ v is a representation on H1 ⊕H2 – and tensor product – u ⊗ v is then a representation on H1 ⊗H2. It is a well-known property of CQGs (see, e.g., [44, Theorem 3.4]) that every unitary repre- sentation of a CQG is unitarily equivalent to a direct sum of irreducible unitary representations and any irreducible representation is finite-dimensional. Given two representations u and v, we use the notation u 6 v to express that the representation u is included in v (i.e., there is an isometry in Hom(u, v)). Of particular interest for our investigations are compact matrix quantum groups (CMQGs), a particular class of CQGs. A CMQG is given by a so-called fundamental representation u, whose coefficients generate a dense subalgebra of C(G). One can define CMQGs in the following way: Definition 2.1. A compact matrix quantum group is defined by a unital C∗-algebra A generated by elements uij , 1 ≤ i, j ≤ n such that u = (uij) and ut = (uji) are invertible and the map ∆: A→ A⊗A, uij 7→ ∑ k uik ⊗ ukj is a ∗-homomorphism. A CMQG (A, u) is a quantum subgroup of (B, v), if there is a surjective ∗-homomorphism from B to A mapping vij to uij . Of course, in the definition of a CMQG (A, u), we should think of A as the functions on the “quantum space” GA (and (B, v) as associated to GB). Gelfand duality, which is contravariant, therefore explains why the “inclusion GA ↪→ GB” is represented by a surjective morphism B → A. In this paper, we use the notations N0 = {0, 1, 2, . . .} and N = {1, 2, . . .} (as opposed to [19], where N = {0, 1, . . .}). 2.2 Categories of partitions and easy quantum groups Colored partitions are key tools for the introduction of unitary easy quantum groups, as done in [37]. They generalize the orthogonal easy quantum groups of Banica and Speicher [5]. Let k, l ∈ N0 and consider a finite ordered set with k+ l elements each being colored either in white or in black. A partition is a decomposition of this set into disjoint subsets, the blocks. Let 4 O. Gabriel and M. Weber P ◦•(k, l) denote the set of all such partitions and put P ◦• := ⋃ k,l∈N0 P ◦•(k, l). We usually use a pictorial representation of a partition involving lines representing the block structure, and we assume k of these points to be placed on an upper row and l on a lower row, see [36]. If these lines may be drawn in a way such that they do not cross, we call the partition noncrossing. Let NC◦• be the set of all noncrossing partitions. Some examples of partitions are the singleton partitions ↑◦ ∈ P ◦•(0, 1) consisting of a single white lower point or ↑• ∈ P ◦•(0, 1), the pair partitions •◦ and ◦• in P (0, 2) consisting of two lower points of different colors which are in the same block, the identity partitions ◦ ◦ , • • ∈ P (1, 1) consisting of one upper and one lower point both of the same color and both in the same block, or the partitions bs ∈ P ◦•(0, s) consisting of s lower white points in a single block. We have the following operations on the set P ◦• of partitions. The tensor product of p ∈ P ◦•(k, l) and q ∈ P ◦•(k′, l′) is p ⊗ q ∈ P ◦•(k + k′, l + l′) obtained by placing p and q side by side. The composition of p ∈ P ◦•(k, l) and q ∈ P ◦•(l,m) is qp ∈ P ◦•(k,m) obtained by placing p above q. We may only perform it when the color pattern of the lower points of p matches the upper color pattern of q. The involution of p ∈ P ◦•(k, l) is p∗ ∈ P ◦•(l, k) obtained by reflecting p at the horizontal axis. The rotation of the left upper point of p ∈ P ◦•(k, l) to the lower row is a partition in P ◦•(k − 1, l+ 1). When rotating a point, its color is inverted but its membership to a block remains untouched. Likewise we have a rotation of the left lower or right upper/lower points. These operations (tensor product, composition, involution and rotation) are called the category operations. A collection C of subsets C(k, l) ⊆ P ◦•(k, l) (for every k, l ∈ N0) is a category of partitions if it is invariant under the category operations and if the identity partitions ◦ ◦ , • • ∈ P ◦•(1, 1) and the pair partitions •◦ , ◦• ∈ P ◦•(0, 2) are in C. Note that rotation may be deduced from the other category operations. We write C = 〈p1, . . . , pm〉, if C is the smallest category of partitions containing the partitions p1, . . . , pm. We say that C is generated by p1, . . . , pm. Let n ∈ N. Given p ∈ P ◦•(k, l) and two multi-indices (i(1), . . . , i(k)), (j(1), . . . , j(l)) with entries in {1, . . . , n}, we can label the diagram of p with these numbers (the upper and the lower row both are labelled from left to right, respectively) and we put δp(i, j) = { 1, if each block of p connects only equal indices, 0, otherwise. We fix a basis e1, . . . , en of Cn and define a map Tp : (Cn)⊗k → (Cn)⊗l associated to p by Tp(ei(1) ⊗ · · · ⊗ ei(k)) = ∑ 1≤j(1),...,j(l)≤n δp(i, j) · ej(1) ⊗ · · · ⊗ ej(l). We use the convention that (Cn)⊗0 = C, i.e., for a partition p ∈ P ◦•(0, l) with no upper points, Tp(1) is actually a vector in (Cn)⊗l. Note that the colors of the points of p do not play a role in the definition of Tp. The operations on the partitions match nicely with canonical operations of the linear maps Tp, namely we have Tp⊗q = Tp ⊗ Tq, Tqp = n−b(p,q)TqTp and Tp∗ = (Tp) ∗. Here, b(p, q) denotes the number of removed blocks when composing p and q. The maps Tp can be normalized in such a way that they become partial isometries, see [18]. From a category C and the realisation p 7→ Tp, we may construct a concrete monoidal W ∗- category with a distinguished object u, and we thus may assign a CMQG (A, u) to it. We call it the easy quantum group associated to C. If C ⊂ NC◦•, then it is called a free easy quantum group. Given a color string r ∈ {◦, •}k, we may define ur as the tensor product of copies u and ū. Here ◦ corresponds to u while • corresponds to ū. We then may say: a CMQG Sn ⊂ G ⊂ U+ n is Fixed Point Algebras for Easy Quantum Groups 5 easy, if there is a category of partitions C such that the intertwiner spaces of G are of the form {T \|Tur = usT} = span{Tp \| p ∈ C(k, l) with color strings r (upper) and s (lower)}. Easy quantum groups are a class of CMQGs with a quite intrinsic combinatorial structure. They are completely determined by their associated category of partitions and in many cases certain quantum algebraic properties of an easy quantum group may be traced back to certain combinatorial properties of its category of partitions. A simple criterion for verifying that a given CMQG is easy is contained in the following lemma. Let n ∈ N and let A be a C∗-algebra generated by n2 elements uij , 1 ≤ i, j ≤ n. Let p ∈ P ◦•(k, l) be a partition with upper color string r ∈ {◦, •}k and lower color string s ∈ {◦, •}l. We say that the generators uij fulfill the relations R(p), if for all β(1), . . . , β(l) ∈ {1, . . . , n} and for all i(1), . . . , i(k) ∈ {1, . . . , n}, we have n∑ α(1),...,α(k)=1 δp(α, β)ur1α(1)i(1) · · ·u rk α(k)i(k) = n∑ γ(1),...,γ(l)=1 δp(i, γ)us1β(1)γ(1) · · ·u sl β(l)γ(l). The left-hand side of the equation is δp(∅, β) if k = 0 and analogously δp(i,∅) for the right-hand side if l = 0. Furthermore, u◦ij := uij and u•ij := u∗ij . Lemma 2.2 ([37, Corollary 3.12]). Let p1, . . . , pm ∈ P ◦• be partitions and let A be the universal C∗-algebra generated by elements uij, 1 ≤ i, j ≤ n such that u = (uij) and ū = (u∗ij) are unitary (i.e., ∑ k u ∗ ikujk = ∑ k u ∗ kiukj = ∑ k uiku ∗ jk = ∑ k ukiu ∗ kj = δij) and uij satisfy relations R(pl) for l = 1, . . . ,m. Then A is an easy quantum group with associated category C = 〈p1, . . . , pm〉. Proof. The proof relies on the fact that the relations R(p) are fulfilled if and only if Tp inter- twines ur and us. See [37, Lemma 3.9, Corollary 3.12] for details. � Note that for p = • ◦ the relation R(p) effects that all uij are selfadjoint. On the combinatorial level this amounts to having partitions whose points have no colors. In this case, we recover the orthogonal easy quantum groups of Banica and Speicher [5]. Example 2.3. (a) The free orthogonal quantum group O+ n is given by the universal unital C∗-algebra C(O+ n ) generated by selfadjoint elements uij , 1 ≤ i, j ≤ n such that u = (uij) is an orthogonal matrix. It is easy with category C = 〈 • ◦ 〉 = NC2, the set of all (noncolored) noncrossing pair partitions (each block consists of exactly two elements). We omit to write down the generating partitions •◦ , ◦• and ◦ ◦ , • • since they are contained in every category, by definition. (b) The free unitary quantum group U+ n is given by the universal unital C∗-algebra C(U+ n ) generated by elements uij , 1 ≤ i, j ≤ n such that u = (uij) and ū = (u∗ij) are unitary matrices. It is an easy quantum group with C = 〈∅〉. (c) The quantum permutation group S+ n is given by the universal unital C∗-algebra C(S+ n ) generated by projections uij , 1 ≤ i, j ≤ n such that ∑ k uik = ∑ k ukj = 1. It is easy with category C = 〈 • ◦ , ↑◦ , ◦◦ ◦◦ 〉. The quantum groups in (a), (b) and (c) were introduced by Wang [39, 40]. 6 O. Gabriel and M. Weber (d) For s ∈ N, the quantum reflection group Hs+ n is given by the universal unital C∗-algeb- ra C(Hs+ n ) generated by elements uij , 1 ≤ i, j ≤ n such that u = (uij) and ū = (u∗ij) are unitaries, all uij are partial isometries and we have usij = uiju ∗ ij = u∗ijuij . We have H1+ n = S+ n and H2+ n = H+ n , the latter one being Banica, Bichon and Collins’s hyperocta- hedral quantum group [4]. The quantum reflection groups Hs+ n were studied by Banica, Belinschi, Capitaine and Collins in [3]. They are easy with C = 〈bs, ◦◦•• 〉, see [37]. Both H+ n and Hs+ n can be traced back to Bichon’s work on free wreath product [9]. 2.3 Cuntz algebra The Cuntz algebra On is the universal unital C∗-algebra generated by isometries S1, . . . , Sn such that ∑ i SiS ∗ i = 1. It has been defined and studied by Cuntz in [11]. He first proved that On is a crossed product by an endomorphism and then introduced a criterion for (what we call today) purely infinite and simple algebras. He used it to show that all On are simple. By an action of a CQG on a C∗-algebra A we mean a faithful unital ∗-homomorphism α : A → A ⊗ C(G) such that (α ⊗ Id) ◦ α = (Id⊗∆) ◦ α holds and (1 ⊗ C(G))α(A) is linearly dense in A⊗C(G). In our case, we have the following action on the Cuntz algebra, as observed by Cuntz [12] (see also [25]). Proposition 2.4. Let G be a CMQG such that G ⊂ U+ n (i.e., the matrices u and ū are unitaries). It acts on On by α(Si) = n∑ j=1 Sj ⊗ uji. Proof. For the convenience of the reader, we give the proof since it is a short argument. By the universal property of On, the unital ∗-homomorphism α exists, and it is faithful since On is simple. It is straightforward to check that (α ⊗ Id) ◦ α = (Id⊗∆) ◦ α is satisfied, and the computation (using orthogonality of u)∑ k (1⊗ u∗ik) (∑ l Sl ⊗ ulk ) = Si ⊗ 1 shows that the linear span of (1⊗ C(G))α(A) equals A⊗ C(G). � It is a well-known fact that we may find copies of matrix algebras inside the Cuntz algebra. Indeed, the monomials Sj(1) · · ·Sj(k)S ∗ i(k) · · ·S ∗ i(1) satisfy the relations of matrix units, thus we have span{Sj(1) · · ·Sj(k)S ∗ i(k) · · ·S ∗ i(1) \| 1 ≤ i(t), j(t) ≤ d} ∼= B (( C n )⊗k) . Thus, Sj(1) · · ·Sj(k)S ∗ i(k) · · ·S ∗ i(1) corresponds exactly to the rank one operator mapping ei(1) ⊗ · · · ⊗ ei(k) to ej(1) ⊗ · · · ⊗ ej(k). More generally, we may identify linear maps T : (Cn)⊗k → (Cn)⊗l, ei(1) ⊗ · · · ⊗ ei(k) 7→ ∑ j(1),...,j(l) a(i(1), . . . , i(k), j(1), . . . , j(l))ej(1) ⊗ · · · ⊗ ej(l) (where a(i(1), . . . , i(k), j(1), . . . , j(l)) ∈ C) with elements∑ i(1),...,i(k),j(1),...,j(l)=1 a(i(1), . . . , i(k), j(1), . . . , j(l))Sj(1) · · ·Sj(l)S∗i(k) · · ·S ∗ i(1) Fixed Point Algebras for Easy Quantum Groups 7 in On. Hence, we may view the maps Tp : (Cn)⊗k → (Cn)⊗l indexed by partitions p ∈ P (k, l) as elements in On via Tp ↔ n∑ i(1),...,i(k),j(1),...,j(l)=1 δp(i, j)Sj(1) · · ·Sj(l)S∗i(k) · · ·S ∗ i(1). 3 Actions of easy quantum groups on the Cuntz algebra 3.1 The fixed point algebra As explained in Section 2.3, any given CQMG acts naturally on a Cuntz algebra On for a suitable choice of n. Our aim is to understand the fixed point algebra in the case of easy quantum groups. Definition 3.1. Let G be a CMQG with fundamental representation u = (uij) and let α be the action as in Proposition 2.4. The fixed point algebra Oα is defined as Oα := {x ∈ On \|α(x) = x⊗ 1}. In the case of easy quantum groups, we may read the fixed point algebras directly from the categories of partitions. The following statement appeared in [27, Proposition 3.4], see also [19, Lemma 2.5]. Proposition 3.2. Given C, let G be the associated easy QG. The intersections of the fixed point algebra Oα with the copies of B((Cn)⊗k, (Cn)⊗l) in On (as described in Section 2.3) are given by Oα ∩B ( (Cn)⊗k, (Cn)⊗l ) = span{Tp ∈ On \| p ∈ C(k, l) all points are white}. Proof. The general proof may be found in [27, Proposition 3.4], but we give a direct argument here. Let p ∈ C(k, l) be a partition with only white points. Hence, in C(G) the relations R(p) hold. Moreover, we use the fact that u is unitary. Now α(Tp) = ∑ i,i′,j,j′ δp(i, j)Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ uj′(1)j(1) · · ·uj′(l)j(l)u∗i′(k)i(k) · · ·u ∗ i′(1)i(1) = ∑ i′,j′ Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ (∑ i (∑ j δp(i, j)uj′(1)j(1) · · ·uj′(l)j(l) ) u∗i′(k)i(k) . . . u ∗ i′(1)i(1) ) = ∑ i′,j′ Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ (∑ i (∑ s δp(s, j ′)us(1)i(1) · · ·us(k)i(k) ) u∗i′(k)i(k) · · ·u ∗ i′(1)i(1) ) = ∑ i′,j′ Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ (∑ s δp(s, j ′) (∑ i us(1)i(1) · · ·us(k)i(k)u ∗ i′(k)i(k) · · ·u ∗ i′(1)i(1) )) = ∑ i′,j′ Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ (∑ s δp(s, j ′)δsi′ ) = ∑ j′ δp(i ′, j′)Sj′(1) · · ·Sj′(l)S∗i′(k) · · ·S ∗ i′(1) ⊗ 1 = Tp ⊗ 1. 8 O. Gabriel and M. Weber Conversely, let T ∈ Oα∩B((Cn)⊗k, (Cn)⊗l). Reversing the above computation, we see that T intertwines u⊗k and u⊗l. But Hom(u⊗k, u⊗l) consists exactly of the linear span of all Tp ∈ On such that p ∈ C(k, l) has only white points. � Proposition 3.3. The algebraic result of Proposition 3.2 extends into a topological one, namely, Oα = span{Tp ∈ On \| p ∈ C(k, l) all points are white, ∀ k, l ∈ N0}. Proof. Lemma 2.7 of [19, p. 1017] (see also [25, Lemma 6]) ensures that “algebraic elements” (i.e., those obtained by (finite) polynomial combination of Sj) of Oα are dense in Oα. Propo- sition 3.2 above ensures that any such algebraic element is in the linear span of Tp, so the conclusion follows. � 3.2 Obtaining Kirchberg algebras In [19], the first author isolated two conditions which turn Oα into a Kirchberg algebra. This class of algebras plays a central role in Kirchberg–Phillips classification theory, which proves that Kirchberg algebras are completely classified by their K-groups – see [23, 24] for the original papers, [35] for an overview of classification for nuclear simple C∗-algebras and [38] for the latest developments in this area. We denote by Tu the set of (classes of) irreducible representations appearing in the iterated tensor products u⊗l for l ∈ N0. (C1) For any v ∈ Tu, we can find v′ ∈ Tu such that the representation v⊗v′ possesses a nonzero invariant vector. (C2) There are integers N, k0 ∈ N0 such that u⊗N is contained in u⊗(N+k0) and for all inte- gers t, l with 0 < t < k0, Hom(u⊗l, u⊗(l+t)) = 0. Condition (C1) is actually satisfied for all finite-dimensional semisimple Hopf algebras overC, see [19, Remark 7.5] and [22, Theorem of Section 4.2]. The following result has been proven by the first author, see [19, Lemma 2.10, Corollary 4.7, Lemma 6.4]. It holds true in a much more general setting but we restrict it to CMQG. Proposition 3.4. Let G be a CMQG and let α be its action on On as described in Propo- sition 2.4. If the conditions (C1) and (C2) are satisfied, then the fixed point algebra Oα is a Kirchberg algebra, i.e., it is purely infinite, simple, separable, unital and nuclear (satisfying the UCT). We are now going to study which easy quantum groups satisfy conditions (C1) and (C2). In order to do so, let us rephrase these conditions in the language of partitions. The representation theory of easy quantum groups – i.e., the set Tu – is completely understood and can be given in terms of partitions, see [18]. We review some facts from [18]. We say that a partition p ∈ P ◦•(k, k) is projective, if p = p∗ = p2. It follows that Tp is a projection up to normalization. Moreover, the upper points of p are colored exactly like the lower points of p. If q ∈ P ◦•(k, k) is another projective partition, we write q ≺ p if pq = qp = q and p 6= q. In this case, Tq is a subprojection of Tp. Given a category C of partitions, we write ProjC(k) for the set of all projective partitions in C(k, k). For p ∈ ProjC(k) with upper (and equivalently lower) color string s, we put Rp := ∨ q∈ProjC(k), q≺p Tq, and Pp := Tp −Rp ∈ Hom(us, us). We denote by up the subrepresentation (Id⊗Pp)(us) of us. Two such subrepresentations up and uq are unitarily equivalent if and only if there is a partition r ∈ C such that p = r∗r and q = rr∗. Fixed Point Algebras for Easy Quantum Groups 9 Proposition 3.5 ([18, Theorem 5.5]). If C ⊂ NC◦•, then up is irreducible. Furthermore, any irreducible representation of the associated G is unitarily equivalent to some up for some p ∈ ProjC, thereby inducing a one-to-one correspondence. Since only tensor powers of u play a role for conditions (C1) and (C2), let us denote by Projwhite C and Cwhite the restrictions of ProjC and C to those partitions consisting only of white points. Theorem 3.6. Let G be a free easy quantum group, namely the associated category of parti- tions C contains only noncrossing partitions. The following two conditions imply the condi- tions (C1) and (C2). (CP 1) For any projective partition p ∈ Projwhite C (a), we can find a projective partition q ∈ Projwhite C (b) and a partition r∈Cwhite(0, a+b) with no upper points such that (Pp⊗Pq)Tr 6=0. (CP 2) There are integers k0, N ∈ N0 and a partition r ∈ Cwhite(N+k0, N) such that rr∗ = ◦ ◦⊗N . Moreover, for all t ∈ N0 with 0 < t < k0 and for all l ∈ N0 we have Cwhite(l, l + t) = ∅. Proof. Due to Proposition 3.5, the set Tu is in bijection with equivalence classes of projective partitions in C. Let us first prove that (CP 1) implies (C1). Let p ∈ ProjC be a projective partition with only white points and let up ∈ Tu be the associated irreducible representation. Let q ∈ ProjC and r ∈ C(0, a + b) be partitions according to (CP 1). Then Tr(1) is a vector in (Cn)⊗a+b as described in Section 2.2 and hence (Pp ⊗ Pq)Tr(1) is a non-zero vector. Now, since Pp, Pq and Tr are intertwiners for tensor powers of u, we have: (up ⊗ uq)(Pp ⊗ Pq)Tr = (Id⊗(Pp ⊗ Pq)) ( u⊗a+b ) (Pp ⊗ Pq)Tr = (Id⊗(Pp ⊗ Pq)) ( u⊗a+b ) Tr = (Pp ⊗ Pq)Tr. This proves (C1). As for deducing (C2) from (CP 2), observe that u⊗NTr = Tru ⊗N+k0 implies u⊗NTrT ∗ r = Tru ⊗N+k0T ∗r . Since TrT ∗ r = 1 up to normalization, this proves that u⊗N is a subrep- resentation of u⊗N+k0 . Moreover, Hom(u⊗l, u⊗(l+t)) is spanned by all Tp with p ∈ Cwhite(l, l+ t). We have proved (C2). � The advantage of dealing with free easy quantum groups is that they are all known: the class of categories C ⊂ NC◦• – and hence the class of free easy quantum groups – is completely classified. The complete list may be found in [36, Section 7]. Throughout the classification process, a parameter k(C) ∈ N0 is assigned to any category of partitions C, see [36, Definition 2.5]. It is given by the following. For a partition p ∈ P ◦• denote by c◦(p) the sum of the number of white points on the lower line of p and the number of black points on its upper line. Likewise put c•(p) to be the number of lower black points plus the number of upper white points. Let c(p) := c◦(p) − c•(p). The number c(p) is designed in such a way that it yields the difference of the number of white and black points, if p has no upper points; moreover c(p) is invariant under rotation. Now, let k(C) be the minimum of all numbers c(p) > 0 for p ∈ C if such a number exists, and k(C) := 0 otherwise. One can show that for every partition p ∈ C the number c(p) is a multiple (from the integers) of k(C). One of the main results in [37] is then to detect the cyclic group of order k(C) as a building block in the easy quantum group associated to C. The parameter k(C) also appears in connection with conditions (CP 1) and (CP 2). Theorem 3.7. Let G be an easy quantum group with C ⊂ NC◦•. (a) If k(C) = 0, then neither condition (CP 1) nor (CP 2) are satisfied. 10 O. Gabriel and M. Weber (b) If k(C) 6= 0 and ◦◦ ⊗ •• ∈ C, then condition (CP 1) holds. (c) If k(C) 6= 0 and ◦◦•• ∈ C, then condition (CP 1) holds. (d) If k(C) 6= 0, then condition (CP 2) holds. Proof. (a) Observe that if a partition p is in Cwhite(l, l+ t), then c(p) = t. Thus, if k(C) = 0, all sets Cwhite(l, l + t) are empty for all l ∈ N0 and all t 6= 0, using [36, Proposition 2.7]. Therefore neither condition (CP 1) nor condition (CP 2) hold. (b) Let p ∈ Projwhite C (a) be a projective partition. We basically want to show that the contragredient representation of up does the job for choosing q, but the colorization of the points turns this into a nontrivial problem (see also the next Examples 3.8 and the remarks in [19, p. 1019] on condition (C1)). Let q1 be the partition obtained by reflecting p about the vertical axis (without inverting the colors). It is in C, since the verticolor reflected partition p̃ is in C [36, Lemma 1.1(a)], and p ⊗ p ⊗ p̃ is in C; using color permutation [36, Lemmas 1.3(a) and 1.1(b)], we infer q1 ∈ C. Now, let b ∈ N0 be such that 2(a + b) is a multiple of k(C). Let q0 = ◦◦ nest(b) ⊗ ◦◦ nest(b) be the partition obtained by nesting b copies of the pair partition ◦◦ into each other, both on the upper and the lower line respectively, see [37, Lemma 2.4]. It is in C since we may apply [36, Lemmas 1.3(a) and 1.1(b)] on the following partition which is in C: •◦ nest(b) ⊗ •◦ nest(b) ⊗ ( ◦◦ ⊗ ◦◦ )⊗b. We then put q := q0⊗q1 ∈ Projwhite C (a+2b) and r := ◦◦ nest(a+b) . We have r ∈ Cwhite(0, 2(a+b)), see for instance Step 3 in the proof of [37, Theorem 4.13]. If now s ∈ Projwhite C (a) is a projective partition with s ≺ p, then (s⊗ q)r 6= (p⊗ q)r by [18, Lemma 2.23]. Hence T(s⊗q)r is linearly independent from T(p⊗q)r by [18, Lemma 4.16]. Likewise T(p⊗t)r is linearly independent from T(p⊗q)r for t ≺ q, t ∈ Projwhite C (a + 2b). This proves that (Tp ⊗ Tq)Tr is linearly independent from (Rp ⊗ Tq)Tr, (Tp ⊗ Rq)Tr and (Rp ⊗ Rq)Tr. Thus, (Pp ⊗ Pq)Tr 6= 0. (c) Let p ∈ Projwhite C (a) be a projective partition. Using the through-block decomposition of p, we may bring it into the following form (see Proposition 2.9 and the remarks after Propo- sition 2.12 in [18]): p = p∗upu, pu = s0 ⊗ t1 ⊗ s1 ⊗ · · · ⊗ tl ⊗ sl. Here, si are partitions with no upper points while each ti has exactly one upper point. Let αi be the sum of the number of points of si and ti, with α1 being the sum of the number of points of s0, t1 and s1. Let βi be numbers such that αi + βi is a multiple of k(C), for all i. Let q be the partition q := q1 ⊗ · · · ⊗ ql, where each qi is the partition consisting of a single block on βi upper white points and βi lower white points. Since ◦◦•• ∈ C, we have q ∈ C, applying [36, Lemma 1.3(c)] on ◦ ◦⊗βi . Let r ∈ Cwhite(0, ∑ i(αi+βi)) be the partition obtained from nesting l blocks bi into each other, each of size αi + βi, such that block bi+1 has αi legs of bi to its left and βi legs of bi to its right. We then conclude (Pp ⊗ Pq)Tr 6= 0 similarly to (b). (d) Put N := 1 and k0 := k(C) > 0. We find a partition p ∈ C such that c(p) = k0, by definition of k(C). Using rotation and [36, Lemma 2.6(d)], we may assume that p consists only of lower points, hence p ∈ P ◦•(0,m) for some m > k0. Then m = c◦(p) + c•(p) and k0 = c(p) = c◦(p)− c•(p), thus p has m+k0 2 white points and m−k0 2 black points. Fixed Point Algebras for Easy Quantum Groups 11 Using [36, Lemma 1.1(b)], we may erase m−k0 2 pairs of a white and a black point and we obtain a partition p0 ∈ Cwhite(0, k0) on k0 white points. Put r := ◦ ◦ ⊗ p∗0 ∈ Cwhite(1 + k0, 1). It satisfies rr∗ = ◦ ◦⊗N for N = 1. Moreover, Cwhite(l, l + t) = ∅ for 0 < t < k(C), since if we had p ∈ Cwhite(l, l + t), then c(p) = t, but c(p) is an integer multiple of k(C) by [36, Proposition 2.7]. � Example 3.8. (a) If G is free orthogonal easy (u = ū) with category C ⊂ NC◦•, then k(C) = 1 if ↑◦ ∈ C and k(C) = 2 otherwise, see [36, Section 7]. Moreover, as • ◦ is in C, we also have ◦◦ ⊗ •• ∈ C. Hence, conditions (CP 1) and (CP 2) hold for S+ n , O+ n and the other five free orthogonal easy quantum groups. (b) For U+ n , we have k(C) = 0, thus conditions (CP 1) and (CP 2) are violated. (c) The quantum reflection groups Hs+ n of Example 2.3 have the parameter k(C) = s and ◦◦•• ∈ C, thus conditions (CP 1) and (CP 2) hold. Theorem 3.7 is about as far as we can go for free easy quantum groups in general: indeed, if Theorem 3.7 together with Proposition 3.4 provide a way to prove that some fixed point algebras are Kirchberg algebras, the exact identification of the fixed point algebra requires a computation of K-theory that can only be performed for definite fusion rules. For this reason, we focus on examples in the rest of the paper. 3.3 Free actions In this subsection, we investigate the relation between our fixed point construction and free actions. Free actions appear in articles such as [7, 13, 14]. However, there are not that many concrete examples available, and for this reason we prove that our construction generates new examples. We remind the reader of the following Definition 2.4 of [17] (see also [14]): Definition 3.9. Given a CQG G, an action δ : A→ A⊗C(G) on a C∗-algebra A is called free if (A⊗ 1)δ(A) is dense in A⊗ C(G). In order to discuss more easily the above definition, we follow the terminology and notations of [7, 13, 21] and introduce the canonical map can: A⊗A→ A⊗C(G) given by can(a⊗ a′) := (a⊗ 1)δ(a′). The condition above is therefore that the map can has a dense image. Remark 3.10. The (C∗-)freeness property defined above has strong ties with the notion of Hopf–Galois extension (first defined in [26] – see for instance [28]). Indeed, Woronowicz proved [44] that any CQG contains a canonical dense Hopf ∗-algebra O(G). Using the action of G on A, we can in turn define the Peter–Weyl subalgebra PG(A) of A (see, e.g., [7]) – which is not a C∗-algebra but just a ∗-algebra. The preprint [7, p. 3] proves in its Theorem 0.4 that the action δ is (C∗-)free if and only if PG(A) is a Hopf–Galois extension over its fixed point al- gebra – compare the Peter–Weyl–Galois condition of [7, Definition 0.2, p. 3] with Definition 2.2 of Hopf–Galois extensions in [28, p. 372]. Going back to our initial motivations concerning free actions, we prove: Proposition 3.11. Let G be a compact matrix quantum group and u its fundamental represen- tation. If G satisfies condition (C1) then the action δ induced from Proposition 2.4 is free. 12 O. Gabriel and M. Weber Remark 3.12. At this point, it may seem plausible that condition (C1) is equivalent to freeness of the action of G on On. However, as we have seen in [19, Notation 3.1], (C1) is not satisfied for the natural representation of G = U(1) (or multiple thereof). Using the same kind of argument as in the proof below, it appears that the action of U(1) on O2 is free. Of course, this action is not strictly induced from U(1) by Proposition 2.4. But we can build on this example by considering SU(2) × U(1). This is a CMQG which does not satisfy condition (C1), but which still acts freely on O2. Proof of Proposition 3.11. For our proof, we will consider the set S := { s ∈ C(G) \| ∃ km ∈ N0, vm, wm ∈H ⊗km , can ( finite∑ m v∗m ⊗ wm ) = 1⊗ s } . It is clear from the definition of δ that for any i, j, can(S∗j ⊗ Si) = (S∗j ⊗ 1) ( n∑ k=1 Sk ⊗ uki ) = 1⊗ uji. This means that S contains all uji. Now, if vi,m, wi,m are elements in H ⊗ki for i = 1, 2 and such that can (∑ m v ∗ i,m ⊗ wi,m ) = 1⊗ si, then can (∑ m,l v∗2,mv ∗ 1,l ⊗ w1,lw2,m ) = (∑ m,l v∗2,mv ∗ 1,l ⊗ 1 ) δ(w1,lw2,m) = (∑ m v∗2,m ⊗ 1 )(∑ l v∗1,l ⊗ 1 ) δ(w1,l)δ(w2,m) = (∑ m v∗2,m ⊗ 1 ) (1⊗ s1)δ(w2,m) = (1⊗ s1) (∑ m v∗2,m ⊗ 1 ) δ(w2,m) = (1⊗ s1)(1⊗ s2) = 1⊗ s1s2. This in turn proves that S is stable under multiplication. It follows that S is an algebra, which contains all polynomials in uij . If the generators of G are selfadjoint, the polynomials in uij coincide with the ∗-polynomials in uij and they are all contained in S, thus S is dense in C(G). In general, we can always decompose u into irreducible representations, which all correspond to unitary matrices with coefficients in C(G). Up to a conjugation by a scalar-valued unitary matrix, the algebra generated by the entries of u and their adjoints is the same as the algebra generated by the irreducible components and their adjoints. So without loss of generality, we assume that u is an irreducible representation. It is well known (see, e.g., [30, Definition 1.3.8, p. 11]) that the contragredient of (uij) is (u∗ij) in a suitable basis. u is irreducible, so condition (C1) implies that up to equivalence, we can recover the coefficients (u∗ij) through a subrepre- sentation of a high enough power H ⊗k. It follows that S contains the ∗-polynomials in uij and thus is dense in C(G). Given any T ∈ On, for any can (∑ v∗j ⊗ wj ) = 1⊗ s we have can (∑ Tv∗j ⊗ wj ) = (T ⊗ 1) can (∑ v∗j ⊗ wj ) = T ⊗ s. It follows that the image of can is dense in On ⊗ C(G) and thus the action δ is free. � Fixed Point Algebras for Easy Quantum Groups 13 Remark 3.13. On the one hand, it follows from the proof above that assumption (C1) is not needed for Proposition 3.11 in the orthogonal case (i.e., if all the entries of the fundamental representation are selfadjoint). On the other hand, the argument above also shows that in this orthogonal case condition (C1) is automatically satisfied. Theorem 3.14. The actions of S+ n , O+ n and Hs+ n are free. 4 Examples – O+ n and S+ n To illustrate our previous results, we consider three cases, namely those of O+ n , S+ n and of the quantum reflection groups Hs+ n – which we are going to consider separately in the next section. We start with the case of O+ n . This free easy quantum group was introduced by S. Wang in [40] and its fusion rules were studied by Banica in [1]. It appears that it shares the same fusion rules as SU(2), i.e., its irreducible representations are denoted by uk for k ∈ N0 – u0 being the trivial representation – and the tensor products decompose into uk ⊗ ul = u\|k−l\| ⊕ u\|k−l\|+2 ⊕ · · · ⊕ uk+l. (4.1) It is clear from the statement of conditions (C1) and (C2) that they only depend on the fusion rules of the quantum group and not on the quantum group itself. This provides an elementary way to recover the result of Example 3.8(a) in this case. In [19], the case of SUq(2) was studied in detail (see Section 7.1, p. 17, therein). The computation of the K-theory of the fixed point algebra also depends only on the fusion rules. Since the fundamental representation of O+ n corresponds to the natural representation of SU(2), the proof of Proposition 7.10 [19, p. 1031] applies verbatim and we get Theorem 4.1 below for G = O+ n . The case of S+ n is very similar: this CMQG was introduced by S. Wang in [39] and its fusion rules were computed by Banica in [2]. The fusion rules are the same as those of SO(3), i.e., we take the fusion rules (4.1) of SU(2) but consider only even representations u2k, k ∈ N0. By Example 3.8(a), conditions (C1) and (C2) are satisfied. The fundamental representation u of S+ n decomposes into u = u0 ⊕ u2 = u1 ⊗ u1 and this shows that the proof of Proposition 7.10 of [19] applies again: Theorem 4.1. For G = O+ n and G = S+ n , the fixed point algebra Oα obtained from Proposi- tion 2.4 via the fundamental representation u is a Kirchberg algebra in the UCT class N whose K-theory is K0(Oα) = Z, K1(Oα) = 0. Moreover, [1Oα ]0 = 1 and therefore Oα is C∗-isomorphic to the infinite Cuntz algebra O∞. 5 Examples – quantum reflection groups As mentioned in Example 2.3(d), the quantum reflection groups Hs+ n were studied by Bani- ca, Belinschi, Capitaine and Collins in [3]. Their fusion rules were computed by Banica and Vergnioux in their article [6] – see in particular Theorem 7.3, p. 348, therein. We follow the notations used in their article up to a point: for a reason that will be clear later on, we write rs for the representation denoted by r0 in the original article. For the reader’s convenience, we reproduce here the fusion rules of these quantum groups, as described in [6, Theorem 7.3]. The monoid F = 〈Z/sZ〉 of words over Z/sZ is equipped with an involution and a fusion operation: (1) Involution: (i1 . . . ik )̄ = (−ik) · · · (−i1). (2) Fusion: (i1 . . . ik) · (j1 . . . jl) = i1 . . . ik−1(ik + j1)j2 . . . jl. 14 O. Gabriel and M. Weber Using these relations, the fusion rules can be written rx ⊗ ry = ∑ x=vz, y=z̄w rvw + rv·w, (5.1) where v ·w is not defined when v or w is the empty word. In the case z = ∅ (“leading terms of the fusion rule”), we distinguish between the term rxy – that we call the concatenation term – and the term rx·y that we call the product term. We are going to compute the K-theory of the fixed point algebra generated from the natural representation (indexed by r1). In the lemma below, we gather useful results: Lemma 5.1. For any quantum reflection group G = Hs+ n , • for all 0 < ` 6 s, the representation r` appears in (r1)`, i.e., r` 6 (r1)`; • moreover, rs−1 and r1 are contragredient to one another, i.e., 1 6 (r1)s; • all irreducible representations of G appear as irreducible components of some r`1 for ` large enough. Actually, rσ1...σk 6 (r1)σ1+···+σk . (5.2) Remark 5.2. It follows immediately from this lemma together with Woronowicz’s abstract existence of a contragredient for any irreducible representation that condition (C1) is satisfied for Hs+ n and its representation r1. Proof of Lemma 5.1. We prove the first point by induction on `: the result is true and obvious for ` = 1. For ` = 2, • if s = 2 then r1 · r1 = r11 + r2 + 1 and the result is true; • if s > 2, then r1 · r1 = r11 + r2, i.e., r2 6 (r1)2 and thus the property is true for ` = 2. Let us now assume the property for ` > 1, then (provided `+1 < s), r` ·r1 = r`1 +r`+1 6 (r1)`+1, which shows the result for `+ 1. If `+ 1 = s, then the product becomes r` · r1 = r`1 + r`+1 + 1. The first two terms above correspond to z = ∅, while the third one corresponds to z = `, i.e., z̄ = −` = 1 (equality in Z/sZ). Let rσ1...σk be any irreducible representation in Hs+ n (where all σj are taken between 1 and s), then it appears as an irreducible component of rσ1+···+σk 1 : our first point proved that rσ appears in rσ1 . Therefore, in the decomposition into irreducible components of rσ1+···+σk 1 = (r1)σ1 · · · (r1)σk , there appears a product of rσ1 · rσ2 · · · rσk , which in turn produces a copy of rσ1...σk – by using iteratively only the first (concatenation) term of the fusion rules, for z = ∅. � This lemma enables us to set: Definition 5.3 (degree function for Hs+ n ). Given an irreducible representation p ofG, we denote by δ(p) the smallest integer ` such p 6 r`1. We can actually give an explicit estimate of δ: Proposition 5.4. The degree of the irreducible representation rσ1...σk , where the σjs are chosen with 0 < σj 6 s, is δ(rσ1...σk) = σ1 + · · ·+ σk. Fixed Point Algebras for Easy Quantum Groups 15 Proof. Given any irreducible representation rx1...xk with for all i, 0 < xi 6 s, we define σ(rx1...xk) = x1 + · · ·+ xk. It follows by direct examination of (5.1) that if γ 6 α · β, then σ(γ) 6 σ(α) + σ(β). (5.3) Iterating the argument and combining it with (5.2), it appears that for all irreducible represen- tation, γ 6 (r1)σ(γ), i.e., δ(γ) 6 σ(γ). Conversely, if γ 6 (r1)`, then iterating the “subadditivity property” (5.3) of σ and using σ(r1) = 1, we get: σ(γ) 6 `. Since this is valid for all `, we get σ(γ) 6 δ(`). This proves that δ(γ) = σ(γ) for all irreducible γ and concludes the proof. � Lemma 5.5. For all irreducible representations α = rx, β = ry and γ with γ 6 α · β, δ(γ) 6 δ(α) + δ(β). (5.4) For G = Hs+ n , the cases of equality in (5.4) can only occur for the terms z = ∅ of (5.1). For those terms, the equality is true unconditionally for rxy, and only if xk + y1 6 s for rx·y. Remark 5.6. The above lemma is the reason why we use the notation rs instead of r0. Proof of Lemma 5.5. The inequality (5.4) follows from γ 6 α · β 6 (r1)δ(α)(r1)δ(β) = (r1)δ(α)+δ(β). This is just a variation on the proof of Lemma 5.1 above. The equality requires to study the behavior of the total degree in the fusion rules, starting from two irreducible representations α = rx = rx1...xk and β = ry = ry1...yl with finite sequen- ces (xi) and (yj) taking their values in {1, . . . , s}. If γ arises from a term in (5.1) with z 6= ∅, then the inequality (5.4) is strict. Indeed, γ could then arise from the z = ∅ term of v and w, where x = vz and y = z̄w. Assuming now that z = ∅, using the estimate of the degree of Proposition 5.4, the term rx1...xky1...yl yields an equality case for (5.4). The same is true of the term rx1...(xk+y1)...yl provided xk + y1 6 s. It remains to treat the case of xk + y1 > s, but such a term corresponds to a strict inequality in (5.4) and this completes the proof. � We will use the following notations extensively: let R` (resp. ∂R`) be the Z-free module constructed on irreducible representations appearing in (r1)` (resp. appearing in (r1)` and not in any (r1)k for 0 6 k < `). It follows immediately from the definition of R` that R` · Rk ⊆ R`+k, where the product R` ·Rk is taking place in the fusion ring of G. Lemma 5.7. The fusion rules (5.1) actually ensure that in Z/sZ: [δ(γ)] = [δ(α)] + [δ(β)]. (5.5) Proof. Direct examination and Proposition 5.4 show that if z = ∅, then rxy = ri1...ikj1...jl has degree δ(rxy) = i1 + · · · + ik + j1 + · · · + jl = δ(rx) + δ(ry). The degree of rx·y is the same in Z/sZ (since a simplification in Z/sZ may happen in the fusion x · y). More generally, if z 6= ∅, then the definition of the involution on the monoid F ensures that taking out both z and z̄ do not change [δ(rvw)] in Z/sZ. However, simplifying by z and z̄ lessen the total degree of the expression, i.e., the degree δ(rvw) has to be strictly less than δ(rx)+δ(ry). The same argument applies to rv+w. � We are now in position to compute the chain group C(G) as introduced in [8, 29]. This object is also known as universal grading group, see, e.g., [20]. 16 O. Gabriel and M. Weber Proposition 5.8. The chain group of G = Hs+ n is C(G) = Z/sZ. Proof. The equation (5.5) shows that if τ1, . . . , τk, p, q are irreducible representations and that they satisfy p, q 6 τ1 · τ2 · · · τk (both p and q appear in the fusion product), then (in the group Z/sZ) [δ(p)] = [δ(τ1)] + · · ·+ [δ(τk)] = [δ(q)]. In other words, if p and q have the same class in the chain group C(G), then [δ(p)] = [δ(q)]. Conversely, take [δ(p)] = [δ(q)] in Z/sZ. If δ(p) = δ(q) = `, then both representations appear in (r1)` and they have the same class in the chain group. Otherwise, without loss of generality, we can assume that δ(p) > δ(q) and thus there is an integer k such that δ(p) = δ(q) + ks. By definition of δ(q), q 6 (r1)δ(q). We can then use Lemma 5.1 (and especially the part 1 6 (r1)s) to show q = q · 1k 6 (r1)δ(q)(r1)s · · · (r1)s = (r1)δ(q)+ks. This in turn proves that p and q share the same class in C(G). The equation (5.5) then ensures that the group law in C(G) and Z/sZ coincide. � Remark 5.9. It follows from this evaluation of the chain group C(G) together with the pro- perty 1 6 (r1)s of Lemma 5.1 that condition (C2) is satisfied for G = Hs+ n equipped with its representation r1, for the integers k0 = s and N = 1. Corollary 5.10. There is a decomposition R` = ⊕ 06k6`,[k]=[`] ∂Rk, (5.6) where the equality [k] = [`] takes place in Z/sZ. Proof. This is a consequence of p 6 (r1)` =⇒ [δ(p)] = [`] in Z/sZ together with 1 6 (r1)s. � Finally, we will need the notion of length λ(rs) of an irreducible representation rs, which is just the length (number of letters) of its indexing sequence s. It is clear from (5.4) that for all γ 6 α · β, λ(γ) 6 λ(α) + λ(β) with equality only for the concatenation term of z = ∅. Theorem 5.11. For G = Hs+ n and its representation α = r1, condition (C2) is satisfied for the integer k0 = s and the computation of K-theory yields K0(Oα) = ⊕ N Z, K1(Oα) = 0. The proof of this theorem is going to require a few intermediate lemmas and a restatement of the problem. Indeed, Theorem 5.11 above is stated in terms of the representation α = r1, but the compu- tations below will be easier if we consider the case of α = rs1 – which yields isomorphic results, according to Proposition 5.8 above and [19, Proposition 7.8, p. 1029]. Consider the maps ϕ : R→ R and ψ : R→ R given on all a ∈ R by ϕ(a) = a(rs1 − 1), ψ(a) = ars1. The previous properties of degree show that these maps induce ϕ` : R` → R`+s and ψ` : R` → R`+s. The K-theory of the fixed point algebra Fα is given by the inductive limit of the system · · · → R` ψ`−→ R`+s ψ`+s−−−→ R`+2s → · · · . Fixed Point Algebras for Easy Quantum Groups 17 The general theory presented in [19] (see in particular Theorem 5.4, p. 1025) shows that K0(Oα) is obtained as the cokernel of the map ϕ : lim → R` → lim → R` defined from the system · · · // R` ψ` // R`+s ψ`+s // R`+2s ψ`+2s // R`+3s ψ`+3s // R`+4s // · · · · · · // R` ψ` // ϕ` << R`+s ψ`+s // ϕ`+s :: R`+2s ψ`+2s // ϕ`+2s :: R`+3s ψ`+3s // ϕ`+3s :: R`+4s // · · · . (5.7) All the squares in the diagram above are commutative – indeed, it amounts to proving that for any a ∈ R, ars1(rs1 − 1) = a(rs1 − 1)rs1. Thus, the map ϕ : lim → R` → lim → R` is well-defined and we can compute its cokernel. To this end, we start by computing the cokernels at each finite level ` and it is a well-known property that we will obtain the overall cokernel as inductive limit of those finite cokernels. The evaluation below is the corner stone of our argument: Lemma 5.12. For any irreducible representation rx with δ(rx) = `, we have ϕ(rx) = rx(rs1 − 1) = rx1 . . . 1︸︷︷︸ s terms + rxs +m, (5.8) where m is a Z-linear combination of irreducible representations in R`+s which do not contain any term rx1...1 and rxs. Proof. It is clear from the fusion rules that by taking only the concatenation term for z = ∅ in the s successive fusion products of rx with r1, we obtain a term rµ = rx1 . . . 1︸︷︷︸ s terms . Moreover, this irreducible irrepresentation has maximal length (namely λ(rµ) = λ(rx) + s) among those appearing in the product rx(rs1−1). Given γ 6 α·β, we know that λ(γ) 6 λ(α)+λ(β) is actually an equality only for the concatenation term of z = ∅. It follows that there is only one way to obtain a representation of such length. Thus, no further term involving rµ appear in ϕ(rx). For rxs, the argument is slightly different: first, we remark that it has maximum degree (δ(rxs) = δ(rx) + s). This implies that it was obtained by taking only terms with z = ∅ in the successive fusion products. We then remark that its length is minimal among those terms obtained by taking only leading terms (z = ∅) in the fusion. This in turn ensures that it is (and can only be) obtained by taking product terms in the s successive fusion products. Thus, no further term involving rxs appear in ϕ(rx). � Lemma 5.13. For all `, there is free Z-module C` ⊆ R` such that R`+s = C`+s ⊕ ϕ`(R`). (5.9) Moreover, this free Z-module decomposes according to the degree into C` = ⊕ 06k6`, [k]=[`] ∂Ck. (5.10) Remark 5.14. The statement above calls for several remarks: • The notation R`+s = C`+s ⊕ ϕ`(R`) indicates that any element of R`+s can be written in a unique way as a sum of an element of C`+s and an element of ϕ`(R`). • An immediate consequence of (5.9) is that we can thus identify C`+s with the cokernel R`+s/ϕ(R`). • In the decomposition (5.10), we use obvious notations similar to those of (5.6). 18 O. Gabriel and M. Weber Proof of Lemma 5.13. We proceed by induction: for a minimal level 0 6 ` < 2s, the de- composition (5.9) shows that we just have to find C` s.t. R` = C` ⊕ ϕ`−s(R`−s). Given any a ∈ R`, we can use relation (5.8) to cancel any term of the form rx1 . . . 1︸︷︷︸ s terms appearing in a. If we then define C` as the free Z-module generated on all irreducible representations appearing in R` which are not of the form rx1 . . . 1︸︷︷︸ s terms , then clearly R` = C` ⊕ ϕ`−s(R`−s). Let us now assume that at level `, we have a decomposition R` = C` ⊕ ϕ(R`), where C` = ⊕ 06k6`,[k]=[`] ∂Ck, we want to prove that R`+s admits a similar decomposition. A consequence of the decomposition (5.6), is that ϕ(R`) = ⊕ 06k6`,[k]=[`] ϕ(∂Rk). Let us now introduce ∂C`+2 as the free Z-module generated by all irreducible representation of degree exactly `+ s which are not of the form rx1 . . . 1︸︷︷︸ s terms , then R`+s = ∂C`+s ⊕ ϕ(∂R`)⊕R` = ∂C`+s ⊕ ⊕ 06k6`,[k]=[`] ∂Ck ⊕ ϕ(R`+s). This completes the proof of the existence of the Z-free module C` = ⊕ 06k6`,[k]=[`] ∂Ck which implements the cokernel in R`. � Let us now study the connecting maps between these cokernels. Remember from the com- mutation relations appearing in (5.7) that all connecting maps ψ` : R` → R`+s induce quotient maps at the level of cokernels, which we denote by ψ̃` : R`/ϕ(R`−s)→ R`+s/ϕ(R`). Lemma 5.15. The connecting maps between the cokernels are the identity: for any a ∈ R`, ψ̃`([a]`) = [a]`+s, where [a]` and [a]`+s are the class of a in R`/ϕ(R`−s) and R`+s/ϕ(R`), re- spectively. Proof. Indeed, take any a ∈ R`, then ψ(a) = ϕ(a) + a. We know that in the cokernel R`+s/ϕ(R`), [ϕ(a)]`+s = 0, thus [ψ(a)]`+s = [a]`+s. � Remark 5.16. A consequence of the above Lemma 5.15 is that the inductive limit lim → C` is simply the increasing union of free Z-modules and it suffices to estimate the number of irreducible representations of Hs+ n of degree ` + s which are not of the form rx1 . . . 1︸︷︷︸ s terms . We do precisely this in the next lemma. Lemma 5.17. Let m`+s be the number of irreducible representations of Hs+ n of degree ` + s which are not of the form rx1 . . . 1︸︷︷︸ s terms , then m`+s →∞. Proof. Let us introduce the number n` of irreducible representations of degree exactly `, then relation (5.8) ensures that n`+s > 2n`. (5.11) Indeed, for each irreducible representation rx of degree `, there are at least two irreducible representations of degree ` + 2, namely rx1 . . . 1︸︷︷︸ s terms and rxs. This also forces m`+s > n`. The rela- tion (5.11), together with the equality n0 = 1, shows that n` (and therefore m`) tends to infinity when ` tends to infinity. � Fixed Point Algebras for Easy Quantum Groups 19 Proof of Theorem 5.11. It follows from Example 3.8(a) that conditions (C1) and (C2) are satisfied for the fusion rules of the quantum reflection group G = Hs+ n . Consequently, Proposi- tion 3.4 applies to the fixed point algebra – which is therefore determined up to ∗-isomorphism by its K-theory. To compute K∗(Oα), we use the inductive system (5.7). We first evaluate K1(Oα): according to [19, Theorem 5.4, p. 1025] this K-group is the kernel of the map ϕ defined by the inductive system (5.7). If c is a nonzero element in R, it can be realised on a finite level `. Let us consider the top length nonvanishing irreducible representations appearing in c ∈ R` and write c =∑ αjrxj1...x j λ +m where αj ∈ Z\{0}, λ is the maximum length of irreducible representations in c and m is a combination of irreducible representations with lower length. Following Lemma 5.12, ϕ(c) = ∑ αjrxj1...x j λ1...1 +m′, where m′ is a linear combination of irreducible representations which do not contain any term r xj1...x j λ1...1 (i.e., maximal length terms). It follows that ϕ(c)− c 6= 0 (since irreducible represen- tations in c have length at most λ and r xj1...x j λ1...1 has length λ + s, no cancellation can occur). Essentially the same argument proves that if c ∈ R` is nonvanishing, then ψ`(c) ∈ R`+s is also nonvanishing. Consequently, ker(1− ϕ∗) = K1(Oα) = {0}. The computation of K0(Oα) is an easy consequence of Lemma 5.13, Remark 5.14 and Lem- mas 5.15 and 5.17. The proof of Theorem 5.11 is thus complete. � As a final comment, this paper shows how techniques from classification theory for C∗- algebras and a thorough understanding of fusion rules can be combined to identify free actions of compact quantum groups on C∗-algebras. The characterisation of the fixed point algebra requires a concrete computation of K-theory, and explains why we restricted ourselves to exam- ples in the second part of the paper. Similar results should however be possible for other classes of CQGs, as soon as we have a fine comprehension of their fusion rules. Acknowledgements The second author was partially funded by the ERC Advanced Grant on Non-Commutative Distributions in Free Probability, held by Roland Speicher, Saarland University. 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Fixed Point Algebras for Easy Quantum Groups

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