Gluing Compact Matrix Quantum Groups

We study glued tensor and free products of compact matrix quantum groups with cyclic groups – so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.


Introduction
The subject of this article are compact matrix quantum groups as defined by Woronowicz in [28]. A lot of attention has recently been devoted to quantum groups possessing a combinatorial description by categories of partitions. Those were originally defined in [7]. Since then, their full classification was obtained [20] and many generalizations were introduced [4,10,11,15,22].
Studying and classifying categories of partitions or generalizations thereof is useful for the theory of compact quantum groups for several reasons. The primary motivation is finding new examples of quantum groups since every category of partitions induces a compact matrix quantum group. Those are then called easy quantum groups. In addition, since the categories of partitions are supposed to model the representation categories of quantum groups, we immediately have a lot of information about the representation theory of such quantum groups (see also [12]).
Categories of partitions provide a particularly nice way to describe the representation categories of quantum groups. Understanding the structure of partition categories and obtaining some classification results, we may obtain analogous statements also for the associated quantum groups. Such results can then be generalized and go beyond categories of partitions and easy quantum groups. Let us illustrate this on a few examples.
The classification of ordinary non-coloured categories of partitions involves a special class of so-called group-theoretical categories. Those induce group-theoretical quantum groups described by some normal subgroups A Z N 2 . However, the latter definition turns out to be more general -not every group-theoretical quantum group can be described by a category of partitions [19]. Another example are glued products, which were defined in [22] in order to interpret some classification result on two-coloured categories of partitions. The definition of glued products was inspired by partitions, but it is independent of the partition description. Last example comes from [15], where a certain classification result for so-called categories of partitions with extra singletons was obtained. Some of the new categories were interpreted by some new Z 2 -extensions of quantum groups. In addition, this extension procedure was generalized to a new product construction interpolating the free and the tensor product of quantum groups.
The goal of the current paper is to study some additional results obtained in previous works on coloured partition categories [13,15,22,23]. We reformulate those results purely in terms of quantum groups and their representation categories without referring to partitions. Below, we give a detailed overview of the results of this paper. Let us start by recalling the above mentioned glued product construction.
For both quantum groups, we can consider the representation vz = v⊗z. This representation may not be faithful, so it defines a quotient quantum group called the glued product (2) The representation category of G×Ẑ k looks as follows otherwise.
(3) The quantum group G×Ẑ k is topologically generated by G andẐ k .
We also generalize the concept of global colourization introduced in [23] for categories of partitions. It turns out that this concept characterizes the tensor k-complexification of orthogonal quantum groups for k = 0. The case k ∈ N remains open. The above two theorems can be understood as a generalization of the work [13] on globally-colourized categories of partitions to arbitrary quantum groups. In addition, we provide the irreducible representations of tensor complexifications in Proposition 4.23.
We continue by studying the free complexification. In this case, we do not have many results even for easy quantum groups. In the recent work [15,Section 4.3], partitions with alternating colouring were introduced and linked to free k-complexifications for k = 2. In this article, we show that the free k-complexification actually often do not depend on the number k. The following two results form an analogy to Theorems A and B.  Finally, the glued tensor product and the glued free product, which are used to define the quantum group complexifications, can be also understood as a special case of some gluing procedure. In Section 5, we ask whether we can perform this procedure in the converse direction and unglue some cyclic groupẐ k from a unitary quantum group. A particularly nice result can be obtained if we are ungluingẐ 2 . It is a generalization of [15,Theorem 4.10].

Theorem E (Theorem 5.13) There is a one-to-one correspondence between
(1) quantum groups G ⊂ O + (F ) * Ẑ 2 with degree of reflection two and (2) quantum groupsG ⊂ U + (F ) that are invariant with respect to the colour inversion. This correspondence is provided by gluing and canonical Z 2 -ungluing.
In addition, we characterize coamenability and provide irreducible representations of the canonical Z 2 -ungluings. These results are applied to the new Z 2 -extensions introduced recently in [15] as those are special examples of canonical Z 2 -ungluings.

Graded Algebras
Through the whole by Z k the cyclic group of order k ∈ N 0 putting Z k := Z for k = 0.
A Z k -grading of a * -algebra A is a decomposition of the algebra into a vector space direct sum A = i∈Z k A i such that the multiplication and involution of the algebra respect the operation on Z k , that is, The elements of the i-th part a ∈ A i are called Z k -homogeneous of degree i.
By definition, every element f ∈ A uniquely decomposes as f = i∈Z k f i with f i ∈ A i . We call the elements f i the homogeneous components of f .
An ideal I ⊂ A is called Z k -homogeneous if it contains with every element f all its homogeneous components f i . A quotient of the algebra with respect to a homogeneous ideal inherits the grading.
The definition of a Z k -grading for C*-algebras is quite simple for k ∈ N. In the case of the group Z or other groups, it gets a bit complicated and we will not mention it here. Let A be a C*-algebra. A Z k -grading on A is defined by a grading automorphism, that is, an automorphism α : A → A satisfying α k . Its spectrum consists of k-th roots of unity and the corresponding eigenspaces can be identified with the homogeneous parts of A satisfying the properties of the algebraic definition above.
If A is a Z k -graded * -algebra by the algebraic definition, we can define the grading automorphism by setting α(x) = e 2π ij/k x for x ∈ A k . The grading automorphism can be then extended to the C*-envelope C * (A) by the universal property.

Compact Matrix Quantum Groups
We provide here only a brief overview of the notions concerning compact matrix quantum groups. For more information, see for example [18,21].
Let A be a C*-algebra, u ∈ M N (A), N ∈ N. The pair (A, u) is called a compact matrix quantum group if (1) the elements u ij i, j = 1, . . . , N generate A, (2) the matrices u and u t = (u ji ) are invertible, (3) the map : A → A ⊗ min A defined as (u ij ) := N k=1 u ik ⊗ u kj extends to a * -homomorphism.
The * -subalgebra O(G) generated by the elements u ij is dense in A and generalizes the coordinate ring of G. It is actually a Hopf * -algebra, that is, it is closed under the above defined comultiplication , it is further equipped with a counit (a * -homomorphism : O(G) → C mapping u ij → δ ij ), and an antipode (an antihomomorphism mapping Two compact matrix quantum groups G = (A, u) and G = (A , u ) are considered to be identical if there is a * -isomorphism O(G) → O(G ) mapping u ij → u ij . Note that the C*-algebras A and A might not be isomorphic -those might be two different completions of O(G). To overcome this ambiguity, we work with the universal C*-algebra C u (G) := C * (O(G)). Given a compact matrix quantum group G = (A, u), its maximal version G = (C u (G), u) is again a compact matrix quantum group. From now on, we will assume that every quantum group appearing in the paper is in its maximal version and denote C(G) := C u (G).
The above notion of identical compact matrix quantum groups indeed generalizes the notion of matrix groups being the same (i.e. not only isomorphic, but also represented by the same matrices). Similarly, we can define H = (C(H ), v) to be a quantum subgroup ) mapping u ij → v ij assuming both matrices u and v have the same size.
On the other hand, two compact (matrix) quantum groups G and H are said to be iso- A coideal that is also a * -ideal is called a * -biideal. A Hopf * -ideal is a * -biideal that is invariant under the antipode, that is, S(I ) ⊂ I . Hopf * -ideals are in a one-to-one correspondence with quantum subgroups. That is, given H ⊂ G, the kernel of the surjective

Representations of CMQGs
For a compact matrix quantum group G = (C(G), u), we say that v ∈ M n (C(G)) is a representation of G if (v ij ) = k v ik ⊗v kj , where is the comultiplication defined in the previous subsection. In particular, the matrix u is a representation called the fundamental representation.
For every compact quantum group it holds that every non-degenerate representation is equivalent to a unitary one. Hence, given a compact matrix quantum group G = (C(G), u), we may assume that u is unitary. At the same timeū = (u * ij ) is a non-degenerate representation, so there exists an invertible matrix F such that FūF −1 is unitary. Consequently, any compact matrix quantum group G is, up to similarity, a quantum subgroup of the universal unitary quantum group U + (F ) [24] for some F ∈ M N (C), N ∈ N, whose C*-algebra is defined by C(U + (F )) = C * (u ij ; i, j = 1, . . . , N | u and FūF −1 are unitary).
Assuming FF = c 1 N for some c ∈ R, we also define the universal orthogonal quantum group O + (F ) [1] by For a pair of representations u ∈ M n (C(G)), v ∈ M m (C(G)), we denote by It holds that any representation is a direct sum of irreducible ones.
For a given quantum group G, we denote by Irr G the set of equivalence classes of G. For α ∈ Irr G, we denote by u α some unitary representative of the class α. It holds that the matrix elements of all the u α , α ∈ Irr G form a linear basis of O(G).
Remark 2.1 Let G and H be compact (matrix) quantum groups such that C(H ) ⊂ C(G) and H = G | C(H ) . We may say that H is a quotient of G. Then Indeed, any representation of H is by definition a representation of G. The notion of irreducibility does not change if we enlarge the algebra.

Grading on Quantum Group Function Spaces
Given a compact matrix quantum group G = (C(G), u), we denote by ). There is a natural structure of a Z k -grading on the algebra C x ij , x * ij given by associating degree one to the variables x ij , and associating degree minus one to the variables x * ij . In this article, by a Z k -grading we will always mean this particular grading.
If the ideal I G is homogeneous, then the * -algebra O(G) inherits this grading. Moreover, this grading passes also to the fusion semiring of irreducible representations in the following sense. For any α ∈ Irr G, there is d α ∈ Z k such that all the matrix entries of u α are Z k homogeneous of degree d α . We will call d α the degree of the irreducible u α .

Representation Categories
The main point of this section is to formulate the Tannaka-Krein duality for unitary compact matrix quantum groups. The Tannaka-Krein duality for quantum groups was first formulated by Woronowicz [29]. Essentially it says that any compact quantum group can be recovered from its representation category. Since then, many formulations of this statement appeared -from very categorical ones such as in [18] to very concrete ones such as [17]. We will stick here to the latter approach reformulating it a bit in the spirit of [11] to fit into our setting.
All the concepts presented in this section are well known to the experts. Therefore, we try to keep the section very brief. On the other hand, since the topic is quite new and rapidly developing, it is hard to give some general reference here. Some of the concrete notation and formulations of definitions and propositions are actually author's original. We refer to the author's PhD thesis [14] for a more detailed discussion of the concepts. See also [5].

Two-coloured Categories
Consider a compact matrix quantum group G = (C(G), u) ⊂ U + (F ), F ∈ GL(N, C). Denote u • := u, u • := FūF −1 . Denote by W the free monoid over the alphabet {••}. For any word w ∈ W , we denote u ⊗w the corresponding tensor product of the representations u • and u • .
For a pair of words w 1 , w 2 ∈ W , denote Such a collection of vector spaces forms a rigid monoidal * -category in the following sense For the last point, we can write explicit formulae Definition 3.1 Consider a natural number N ∈ N. Let C (w 1 , w 2 ) be a collection of vector spaces of linear maps (C N ) ⊗|w 1 | → (C N ) ⊗|w 2 | satisfying the conditions (1)-(5) above. Then we call C a two-coloured representation category. For any collection of sets C(w 1 , w 2 ) of linear maps (C N ) ⊗|w 1 | → (C N ) ⊗|w 2 | satisfying (5), we denote by C the smallest category containing C. We say that C generates this category.
The justification for this name is given by the following formulation of the Tannaka-Krein duality, which comes from [11]. Theorem 3.2 (Woronowicz-Tannaka-Krein duality for CMQG) Let C be a two-coloured representation category. Then there exists a unique compact matrix quantum group G such that C = C G . This quantum group is determined by the ideal We give only a sketch of proof here. For a detailed explanation, see [13,Theorem 3.4.6]. The proof closely follows the proof formulated for orthogonal quantum groups in [17].
Sketch of proof In order to give a sense to the formula for I G , we need to specify the matrix F , so that the matrix u • is well defined. We fix the duality morphisms satisfying (3.3) and define F according to Eq. 3.2. Note that the duality morphisms are not defined uniquely by Eq. 3.3. Part of the "uniqueness" statement is that the resulting quantum group does not depend on the particular choice of F .
It is straightforward to check that I G is a biideal. We obviously have I G ⊃ I U + (F ) . Then we can check that I G /I U + (F ) ⊂ O(U + (F )) is a Hopf * -ideal. Hence I G defines a compact quantum group. (See [14,17] for details.) Now, letξ ,ξ be alternative solutions of Eq. 3.3 and letF be the alternative matrix andG the alternative resulting quantum group. Then we have This means that u • =F F −1 u • FF −1 =FūF −1 and hence G ⊂G. From symmetry, we have G =G.
It remains to prove that we indeed have C G = C (from construction, we can actually easily see that C G ⊃ C ) and that G is a unique quantum group with this property, that is, if CG = C for some quantum groupG ⊂ U + (F ), then surely G =G (again, from construction, we obviously have G ⊃G). This can be proven using the double commutant theorem, see [14,17].

Proposition 3.3
Let G ⊂ U + (F ) be a compact matrix quantum group. Suppose that the associated category C G is generated by some collection C(w 1 , w 2 ). Then I G ⊂ C x ij , x * ij as an ideal is generated by Proof Denote by I the ideal generated by C as formulated above. Obviously, we have I ⊂ I G . To prove the opposite inclusion, it is enough to prove that form a category. Then, since obviously C ⊂ C and hence C G ⊂ C , we must have I G ⊂ I . So, denote A := C x ij , x * ij /I and by v ij denote the images of x ij by the natural homomorphism. Taking T 1 ∈ C (w 1 , w 2 ), T 2 ∈ C (w 2 , w 3 ). Then For tensor product and involution, the proof is similar.

Remark 3.4
Recall the universal orthogonal quantum group O + (F ) ⊂ U + (F ), which is defined by the relation u = FūF −1 . The relation can also be written as u • = u • . Consequently, we have u ⊗w = u ⊗|w| for any w ∈ W , so only the length of the word w matters. For any G ⊂ O + (F ), we define where w 1 , w 2 are any words with |w 1 | = k and |w 2 | = l.

Representation Categories of Quantum Subgroups, Intersections, and Topological Generation
In this section, we would like to briefly explain the concepts of the quantum group intersection and topological generation. Together with quantum subgroups, we relate those notions with the associated ideals of algebraic relations I G and the representation categories C G . See also [14, ] for more detailed discussion. (1) H ⊂ G, Proof The equivalence (1) ⇔ (2) follows directly from the definition of quantum subgroup. The equivalence (2) ⇔ (3) follows from Tannaka-Krein duality (Theorem. 3.2).
Consider H 1 , H 2 ⊂ U + (F ). Their intersection H 1 ∩ H 2 is the largest compact matrix quantum group contained in both H 1 and H 2 . This notion was recently heavily used especially in the work of Teodor Banica. See [5].
Proof The equivalence follows from Proposition 3.5. G being the largest quantum group contained in H 1 and H 2 is equivalent to I G being the smallest ideal containing I H 1 and I H 2 , which is equivalent to C G being the smallest category containing C H 1 and C H 2 .
Consider H 1 , H 2 ⊂ U + (F ). The smallest quantum group G containing both H 1 and H 2 is said to be topologically generated by H 1 and H 2 . We denote it by G = H 1 , H 2 . This notion goes back to [6,9].
Proof Again, the equivalence follows from Proposition 3.5. G being the smallest quantum group containing H 1 and H 2 is equivalent to I G being the largest ideal contained in I H 1 and I H 2 , which is equivalent to C G being the largest category contained in C H 1 and C H 2 .

Frobenius Reciprocity
We define an involution on the set of two colours {••} mapping • → •, • → •. We extend this operation as a homomorphism on the monoid W , denote it by bar w →w, and call it the colour inversion. We denote by w * the colour inversion composed with reflection on w (that is, reading the word backwards).
Consider a two-coloured representation category C and fix the duality morphisms ξ , ξ . We define a map Rrot: We call this map the right rotation.
This map has an inverse Rrot −1 : Similarly, we can define the left rotation Lrot : C (aw 1 , w 2 ) → C (w 1 ,āw 2 ). As a consequence, we have the following. Proposition 3.8 Let C be a two-coloured representation category. Then C is generated by the collection C (∅, w) with w running through W .

Corollary 3.9
Let G be a compact quantum group. Then the ideal I G is generated by relations of the form u ⊗w ξ = ξ .
Not only that the collection C (∅, w) already determines the whole category C . We can even characterize representation categories in terms of those fixed point spaces by introducing some alternative operations. The following essentially reformulates the operations defined in [13]. Definition 3.10 Consider a two-coloured representation category C . Denote for simplicity ξ • := ξ and ξ • := ξ . We define the following operations on the sets C (∅, w).
• If a i and a i+1 have opposite colours, we define the contraction: On elementary tensors, it acts as On elementary tensors, it acts as

Proposition 3.11 For any two-coloured representation category C , the collection of sets C (∅, w), w ∈ W is closed under tensor products, contractions, rotations, inverse rotations, and reflections. Conversely, for any collection of vector spaces C (w) ⊂ (C N ) ⊗|w| that is closed under tensor products, contractions, rotations, inverse rotations, and reflections and satisfies axiom (5) of two-coloured representation categories, the sets
Proof The first part of the proposition follows from the fact that all the new operations are defined using the category operations of tensor product, composition, and involution.
The converse statement is proved by expressing the category operations in terms of the new operations:

Free and Tensor Product
The following two constructions were defined by Wang.

Proposition 3.12 ([25]) Let
is a compact matrix quantum group. For the co-multiplication we have that The tensor product of quantum groups is a generalization of the group direct product. The free product should also be seen as some free version of the direct product. Note in particular that the free product of compact quantum groups does not generalize the group free product since the freeness occurs in the C*-algebra multiplication not in the quantum group comultiplication. For this reasons, some authors call it rather dual free product.
We will focus here mainly on products of quantum groups H = (C(H ), v) with cyclic group dualsẐ k . Denote by z the generator of the C*-algebra C * (Z k ), so the fundamental representation of H * Ẑ k or H ×Ẑ k is of the form u := v ⊕ z. Now, we would like to describe the corresponding representation category. For our purposes, it will be convenient not to restrict only to the intertwiner spaces between tensor products of u • and u • , but to keep track of the blocks v and z.
Consider G ⊂ U + (F ) * Z k for some k ∈ N 0 with fundamental representation of the form u = v ⊕ z, where z is one-dimensional. Denote by N the size of v. We define a monoid W k with generators , , , and relations = = ∅, k = ∅, where ∅ is the monoid identity. We denote u := v • , u := v • , u := z, u = z * , so u ⊗w is the tensor product of the corresponding representations. We denote by [w] the number of white and black squares in a given word w (which is well defined in contrast with the overall length of w). Then we can associate to G the following category The axiomatization of a two-coloured representation category extends to this case as follows.
In other words, Z k -extended categories are rigid monoidal * -categories with W k being the monoid of objects and morphisms realized by linear maps. Note that it is not necessary to assume the existence of duality morphisms for the triangles since we automatically have C (∅, ) = C (∅, ) = C (∅, ∅) 1 being the duality morphism.
We can reformulate many results mentioned above to this Z k -extended case. In particular, the Tannaka-Krein duality associates a compact matrix quantum group G ⊂ U + (F ) * Ẑ k to any Z k -extended representation category.
Also the Frobenius reciprocity holds for Z k -extended representation categories and the operations on the fixed point spaces can be defined in a similar way. In particular, the contraction C (∅, is defined the same way as in Section 3.3. On the other hand the contraction C (∅, w 1 w 2 ) → C (∅, w 1 w 2 ) is simply the identity since already on the level of objects we have w 1 w 2 = w 1 w 2 . Similarly, the rotation of squares is defined the same way as the rotation of circles in Section 3.3 and will be denoted by R. The rotation of triangles is then simply the identity.
A way how to model Z 2 -extended representation categories using partitions is described in [15]. Many new examples of quantum groups were obtained using this approach. In particular, several new product constructions interpolating the free and tensor product.

Glued Products
In [13,15,22], the representation categories of glued products of orthogonal easy quantum groups with cyclic groups were studied. Here, we revisit the theory dropping the easiness assumption.

Gluing Procedure
Definition 4.1 Let G be a compact matrix quantum group with fundamental representation of the form u = v 1 ⊕ v 2 . Denote byÃ the C*-subalgebra of C(G) generated by elements of the form is a compact matrix quantum group called the glued version of G.
A particular example of the gluing procedure is the glued free product H 1 * H 2 defined as the glued version of the free product H 1 * H 2 and the glued tensor product H 1× H 2 defined as the glued version of the tensor product H 1 × H 2 . Those two definitions were first formulated in [22].
Again, we will be interested in particular in the case when u = v ⊕ z, where z is onedimensional. The glued free product H * Ẑ k is also called the free k-complexification of H and the glued tensor product H×Ẑ k is called the tensor k-complexification. The free complexification was studied already by Banica in [2,3].

Remark 4.2
The glued versionG of a quantum group G is by definition a quotient of G. It may happen that the elements In this case, we have thatG is isomorphic to G. However, theG and G are still distinct as compact matrix quantum groups since their fundamental representations are different.
The same holds in particular for the glued products and for the complexifications. Con- Now, we are going to characterize the representation categories of the glued versions.

Definition 4.3
Let us fix k ∈ N 0 . Then for any word w ∈ W we associate its glued versioñ w ∈ W k mapping • → , • → .

Proposition 4.4 Consider
That is CG is a full subcategory of C G given by considering the glued words only. The ideal associated toG can be described as Proof Denote byṽ = vz the fundamental representation ofG. Consider a word w and its glued versionw. Directly from the definitions ofṽ andw, we haveṽ ⊗w = u ⊗w . So, For the ideal, we have

Glued Version and Projective Version
As a side remark, let us relate the new notion of glued version with already existing notion of projective version.
Proof By definition, we have that P G is determined by a C*-subalgebra C(P G ) ⊂ C(G) generated by matrix elements of the fundamental representation of P G , which is of the We need to show that C(G) and C(P G ) coincide as subalgebras of C(G). (2) and ξ = ξ (1) ⊕ ξ (2) are the duality morphisms corresponding to v 1 ⊕ v 2 .

Remark 4.7
Often it happens that G and G are identical as compact matrix quantum groups. For instance, in the case of tensor product or free product withẐ k . So, we can write

Degree of Reflection
We characterize the notion of the degree of reflection, which was introduced in [23] in the categorical language and in [15] for general compact matrix quantum groups. Recall that given a quantum group G = (C(G), u), we can construct a quantum subgroup of G -so called diagonal subgroup -imposing the relation u ij = 0 for all i = j . If we, in addition, impose the relation u ii = u jj for all i and j , we get a quantum group corresponding to a C*-algebra generated by a single unitary. Therefore, it must be a dual of some cyclic group.  (1) The mapping u ij → δ ij z extends to a * -homomorphism φ : If k 0 is the degree of reflection of G, then directly by the definition there is a * -homomorphism φ : C(G) → C * (Ẑ k 0 ). Such an homomorphism obviously exists also if k is a divisor of k 0 . By definition,Ẑ k 0 is the largest group with this property, so k must be a divisor of k 0 .
(2) ⇒ (3): By Corollary 3.9, I G is generated by the relations u ⊗w ξ = ξ , ξ ∈ Mor(1, u ⊗w ). Since the entries of u ⊗w are monomials of degree c(w), the relations u ⊗w ξ = ξ are Z c(w) -homogeneous (of degree zero). Consequently, they are also Z k -homogeneous and hence generate a Z k -homogeneous ideal.
(3) ⇒ (4): We need to show that u ij → u ij z extends to a * -isomorphism C(G) → C(G×Ẑ k ). To prove that it extends to a homomorphism, take any f ∈ I G . Suppose f is Z khomogeneous of degree l. Then, since u ij and z commute, we have f (u ij z) = f (u ij )z l = 0. It is surjective directly from definition. For injectivity, note that the projection to the first tensor component C(G) ⊗ max C * (Z k ) → C(G) restricts to the inverse of α.
(4) ⇒ (1): We define φ := ( ⊗ id)α, where is the counit of G and α is the As a consequence, we have the following four equivalent characterizations of the degree of reflection.

Proposition 4.11
Let G be a compact matrix quantum group, k ∈ N 0 . The following are equivalent.
(1) The number k is the degree of reflection of G.
In items (3) and (4), we consider zero to be larger than every natural number (equivalently, consider the order defined by "is a multiple of").
Proof We just take the maximal k (in the above mentioned sense) satisfying the equivalent conditions in  (1) The degree of reflection of G is one and C G (0, k) = {0} for some odd k ∈ N 0 .
(2) The degree of reflection of G is two and C G (k, l) = {0} for every k + l odd.
Proof Recall from Remark 3.4 that the intertwiner spaces C G (w 1 , w 2 ) depend only on the length of the words w 1 and w 2 for G ⊂ O + (F ). This allowed us to introduce the notation C G (k, l). Now the proposition follows from Proposition 4.11. First of all, the degree of reflection must be a divisor of two (that is, either one or two) since we have C G (∅, Then G has degree of reflection one if and only if C G (∅, w) = C G (0, |w|) = {0} for some word w with c(w) = 1. Such a word with c(w) = 1 must be of odd length.

Global Colourization
The notion of globally-colourized categories was introduced in [23] and studied in more detail in [13]. Here, we reformulate the results in the non-easy case. Assuming G ⊂ U + (F ), this can be equivalently expressed using the entries of the unitary representations u • = u and u • = FūF −1 as Proposition 4.14 A compact matrix quantum group G = (C(G), u) is globally colourized if and only if for every w 1 , w 2 , w 1 , w 2 ∈ W satisfying |w 1 | = |w 1 |, Proof The equality (4.2) can be also expressed as u • ⊗ u • = u • ⊗ u • , so it is equivalent to saying that the identity is an intertwiner between u • ⊗ u • and u • ⊗ u • . From this, the right-left implication follows directly.
For the left-right implication, from Frobenius reciprocity, it is enough to show the equality for w 1 = w 1 = ∅. It is easy to infer that if the identity is in Mor(u • ⊗ u • , u • ⊗ u • ), we must also have the identity in Mor(u • ⊗ u • , u • ⊗ u • ) and hence also in Mor(u ⊗w 2 , u ⊗w 2 ), and Mor(u ⊗w 2 , u ⊗w 2 ), which implies the desired equality.
It is easy to check that the tensor complexification H×Ẑ k is a globally colourized quantum group with degree of reflection k for every k ∈ N 0 . In the following theorem, we prove the converse for k = 0. Proof We denote by u, v, z the fundamental representations of G, H , andẐ k , respectively. The quantum group H is the quantum subgroup of G defined by the relation v • = v • . As mentioned above, the right-left implication is clear since v ij commute with z, so kl . Now, let us prove the left-right implication. First, we show that there is a surjective * -homomorphism To show this, take any element f ∈ I G . Since I G is Z-homogeneous, we can assume that f is also Z-homogeneous of some degree l. Then This proves the existence of such a homomorphism. Its surjectivity is obvious.
Now it remains to prove that α is injective and hence is a * -isomorphism. Denote by To prove the existence of such a homomorphism, we need the following.
Using the fact that 1 N ⊗ξ * ∈ Mor(u • ⊗u • ⊗u • , u • ) = Mor(u • ⊗u • ⊗u • , u • ), we derive and similarly yu • = u • . This allows us to see that v ij z = z v ij = u ij 1 2 . Now, it only remains to show that all relations of the generators v ij are satisfied by v ij . For this, note that I H is generated by the relations v • = v • and the ideal I G . For the first part, we use the assumption FF = c 1 N to derive For the second part, take any f ∈ I G . Assume it is Z-homogeneous of degree i. Then we have This concludes the proof of existence of β. Now, noticing that β • α is the embedding of C(G) into diagonal matrices over C(G), we see that α must be injective.
Remark 4. 16 We leave the situation for general degree of reflection k ∈ N open. Modifying the proof, it is actually easy to show that, for any globally colourized G with degree of reflection k, we have H×Ẑ k ⊂ G ⊂ H×Ẑ.
However, we were unable to prove the inclusion G ⊂ H×Ẑ k . This problem is actually equivalent to proving a stronger version of Proposition 4.14: Consider G globally colourized with degree of reflection k. Taking w 1 , w 2 ∈ W with |w 1 | = |w 2 |, does it hold that Mor(1, u ⊗w 1 ) = Mor(1, u ⊗w 2 ) whenever c(w 1 ) ≡ c(w 2 ) modulo k?

Tensor Complexification
In this section, we study the tensor complexification with respect to representation categories and algebraic relations. (1) The ideal I G×Ẑ k is the Z k -homogeneous part of I G . That is, (2) The representation category of G×Ẑ k looks as follows (3) G×Ẑ k is topologically generated by G andẐ k . More precisely, G×Ẑ k = G, E×Ẑ k , where E denotes the trivial group of the appropriate size, so E×Ẑ k is the quantum groupẐ k with the representation z ⊕ · · · ⊕ z = z 1 N .
Proof For (1), we can express where f = l f l is the decomposition of into the homogeneous components f l of degree l. If all f l ∈ I G , so f l (v ij , v * ij ) = 0, we have f (v ij z, z * v * ij ) = 0, so f ∈ I G×Ẑ k . Conversely, if there is some l ∈ Z k such that f l ∈ I G , then f (v ij z, z * v * ij ) = 0 and hence f ∈ I G×Ẑ k . For (2), first we prove that Mor (1, u To prove (3), note that the category corresponding to G×Ẑ k (given by (2)) is indeed the intersection of the category C G and the category C E×Ẑ k , whose morphism spaces are given by

Remark 4.18
An alternative proof of the proposition above could go as follows. One can easily see that the Z k -extended category associated to G ×Ẑ k looks as follows where w 1 , w 2 ∈ W are created from w 1 , w 2 ∈ W k mapping → •, → •, → ∅, → ∅ and by t (w) we mean the number of white triangles minus the number of black triangles in w (which is a well-defined element of Z k ). The item (2) of the proposition then follows from Proposition 4.4. Then z k ∈ C(G×Ẑ l ) for every l. Consequently, z nk 0 ∈ C(G×Ẑ l ) for every n ∈ N 0 , where k 0 := gcd(k, l).
Proof From Proposition 4.11, we can find a vector ξ ∈ Mor(1, v ⊗w ) with c(w) = k and ξ = 1. Recall that C(G×Ẑ l ) is generated by the elements v ij z and that v ij commute with z, so Consequently, z nk ∈ C(G×Ẑ k ) for every n and obviously {z nk } n∈N 0 = {z nk 0 } n∈N 0 .

Proposition 4.21 Let G ⊂ U + (F ) be a quantum group with degree of reflection k.
Consider a number l ∈ N 0 . Then G× Z l G × Z l if and only if k is coprime with l.
Proof Assume we have G×Ẑ l G ×Ẑ l . Suppose d is a divisor of both k and l. Then we must have also G×Ẑ d G ×Ẑ d . But from Lemma 4.10, we have that G×Ẑ d = G, which is a contradiction unless d = 1.
For the converse, denote by v the fundamental representation of G and by z the generator of C * (Z l ). It is enough to show that we have z ∈ C(G×Ẑ l ) ⊂ C(G)⊗ max C * (Z l ) since this already implies the equality of the C*-algebras. This follows directly from Lemma 4.20.

Remark 4.22
If l is not coprime with k, but l 0 := l/ gcd(k, l) is coprime with k, we can use Remark 4.19, Lemma 4.10, and Proposition 4.21 to obtain Finally, we are going to characterize irreducible representations of the tensor complexification. Note that the irreducibles of the standard tensor product G ×Ẑ l (or G × H in general) was obtained already by Wang in [26]. where l 0 = l/ gcd(k, l) and z is the generator of C * (Z k ).
Proof Since G has degree of reflection k, the ideal I G is Z k -homogeneous by Proposition 4.11. This means that the algebra O(G) is Z k -graded assigning degree one to v ij and degree minus one to v * ij , where v is the fundamental representation of G. Consequently, the entries of any irreducible representation u α , α ∈ Irr G are Z k -homogeneous of some degree d α (recall Section 2.4).
By Theorem 4.17, I G×Ẑ l is the Z l -homogeneous part of I G . Consequently, I G×Ẑ l is Z lcm(k,l) -homogeneous and O(G×Ẑ l ) is Z lcm(k,l) -graded. However, this time the degree is computed with respect to the variables u ij := v ij z. The irreducible representations of the standard tensor product G ×Ẑ l were described by Wang in [26]. Namely, those are exactly all u α z n with α ∈ Irr G, n = 0, . . . , l − 1. We just have to choose those whose matrix entries are elements of C(G×Ẑ l ) ⊂ C(G ×Ẑ l ). That is, we need to determine all the pairs (α, n) such that u α z n is a matrix with entries in C(G×Ẑ l ) ⊂ C(G ×Ẑ l ).
We first prove that every irreducible of G×Ẑ l is equivalent to one from Eq. 4.3. As we just mentioned, it must be of the form u α z n for some α, n. Since it is a representation of G×Ẑ l , it must be a subrepresentation of u ⊗w = v ⊗w z c(w) for some w ∈ W . Consequently, u α is a subrepresentation of v ⊗w , so d α ≡ c(w) modulo k. In addition, we must also have n ≡ c(w) modulo l. As a consequence, n ≡ d α modulo k 0 := gcd(k, l). Thus, we must have n = k 0 i + d α for some i ∈ Z. Obviously, For the converse inclusion, we need to show that the entries of u α z ki+d α are elements of C(G×Ẑ l ) for every α, i. Since u α is an irreducible representation of G, it must be a subrepresentation of v ⊗w for some w ∈ W . Consequently, u α z c(w) is a subrepresentation of u ⊗w = v ⊗w z c(w) . Hence, it is a representation of G×Ẑ l . From Lemma 4.20, it follows that also u α z ki+c(w) is a representation of G×Ẑ l . Since d α ≡ c(w) modulo k, this is equivalent to considering representations u α u ki+d α .

Free Complexification
The goal of this section is to characterize the representation categories of the free complexifications, that is, the quantum groups H * Ẑ l . For the free complexification, we do not have many results yet even in the easy case. In [22], the two-coloured categories corresponding to free complexifications of free orthogonal easy quantum groups are provided. For us, the motivating result is [15,Proposition 4.21] linking the free complexification by Z 2 with the category Alt C generated by alternating coloured partitions. This proposition was proven with the help of categories of partitions with extra singletons describing the free product with Z 2 and the functor F describing the gluing procedure. Also here, we will make use of the Z l -extended representation categories describing the free product H * Ẑ l and then we will glue the factors and apply Proposition 4.4 to find the corresponding representation category. An interesting result is that the free complexification H * Ẑ l actually does not depend on the number l unless the degree of reflection of H equals to one.

Definition 4.24 A monomial of even length of the form
where the variables with and without star alternate, is called alternating. A linear combination of alternating monomials, where either all start with non-star variable or all start with star variable, is called an alternating polynomial. A quantum group G is called alternating if I G is generated by alternating polynomials.
Considering a compact matrix quantum group G ⊂ U + (F ) with unitary fundamental representation u, recall the notation u • := u, u • := FūF −1 . So, the relations of G can be alternatively expressed by polynomials in variables u • ij , u • ij instead of u ij and u * ij . Since the transformation between those two sets of variables is linear, the definition of an alternating quantum group can be stated in unchanged form also using the alternative ideal.

Lemma 4.25
Let H be a compact matrix quantum group, k, l ∈ N 0 such that gcd(k, l) = 1 (using the convention gcd(0, k) = k). Then we have Proof We denote by v, z, r, s the fundamental representations of H ,Ẑ,Ẑ k , andẐ l , respectively. We need to find a * -isomorphism C(H * Ẑ) → C((H×Ẑ k ) * Ẑ l ) mapping v ij z → v ij sr.
First, we see that there exists a * -homomorphism This * -homomorphism then restricts to a surjective * -homomorphism of the form we are looking for. It remains to prove that it is injective. To prove this, we construct a * -homomorphism where d = 1 is some common divisor of k and l.
We can check that the images satisfy all the defining relations between the generators, so such a homomorphism indeed exists. Now, we can see that β • α is injective, so α must be injective. Thus, the restriction of α we are interested in is also injective.

Proposition 4.26
Let H be a compact matrix quantum group and l ∈ N. Then the Z lextended category C H * Ẑ l is generated by the collection C(ι(w 1 ), ι(w 2 )) := C H (w 1 , w 2 ), where ι : W → W l is the injective homomorphism mapping • → , • → . Moreover, we have the following inductive description. If w ∈ W k contains no triangles, i.e. w = ι(w ) for some w ∈ W , then C H * Ẑ l (∅, w) = C H (∅, w ). Otherwise, Proof Let C be the Z l -extended category generated by C. Then the associated quantum group G = (C(G), v ⊕ z) is a quantum subgroup of U + (F ) * Ẑ l defined by the relations of H for v and no relations for z (except for zz * = z * z = 1 = z l ). But this is exactly the free product H * Ẑ l . Now, it remains to prove that C is given by the above described recursion. The inclusion ⊃ follows from the fact that C has to be closed under the category operations. To check the inclusion ⊂, it is enough to check that the right-hand side defines a category. That is, we need to check that it is closed under tensor products, contractions, rotations, inverse rotations, and reflections as defined in Sections 3.3 and 3.4. Checking this is straightforward using induction. Nevertheless it may become a bit lengthy to check all the details. We will do it here for the rotation and tensor product.
So, denote the sets given by the inductive description byC . If we take words w without triangles, that is, w = ι(w ), then the setsC (∅, w) = C H (∅, w ) are closed under all the operations since C H is a category. To show closedness under rotations in general, we do an induction on the length of the word w. So, consider an element ξ ∈C (∅, w) with w = w 0 w 1 · · · w l , so it is of the form ξ = R [w 0 ] (ξ 1 ⊗ · · · ⊗ ξ l ). First, suppose that w l is not empty and denote by x its last letter. Then we directly have Rξ = R [w 0 ]+1 (ξ 1 ⊗ · · · ⊗ ξ l ) ∈ C (∅, Rw). For the case w l = ∅, the last letter of w is a triangle. So, we need to check that C (∅, w 0 w 1 · · · w l−1 ) ξ = (R [w 0 ] ξ l ) ⊗ ξ 1 ⊗ · · · ⊗ ξ l−1 . This is true thanks to the fact that R [w 0 ] ξ l ∈C (∅, w 0 ) by induction. For the inverse rotations, the proof goes exactly the same way. Now, we can also prove closedness under the tensor product. Take ξ ∈C (∅, w), η ∈ C (∅, w ). We will do the induction on the length of w. Actually, we can assume that |w| ≥ |w | since we can swap the factors by rotation: by assumption,

Lemma 4.27
We can arrange the recursion of the above proposition in such a way that the words w 1 , . . . , w l−1 contain no triangles, so we have Proof We prove this by induction. Take an arbitrary word w ∈ W l and suppose that the above description works for any shorter word. Now consider an element ξ ∈ C H * Ẑ l (∅, w), so it is of the form ξ = R [w 0 ] (ξ 1 ⊗ · · · ⊗ ξ l ) corresponding to the decomposition w = w 0 w 1 · · · w l . Suppose now that w i contains some triangles for some i ∈ {1, . . . , l − 1}. By induction hypothesis, we can write ξ i = R [a 0 ] (η 1 ⊗ · · · ⊗ η l ) corresponding to w i = a 0 a 1 · · · a l , where a 1 , . . . , a l−1 contain no triangles, so η i ∈ C H (∅, a i ). But this means that we can write also ∈ C H * Ẑ l (∅, a l w i+1 · · · w l w 0 w 1 · · · w i−1 a 0 ).
In the following theorem, we describe the representation category of the free complexification. In the formulation, we use the following notation. Given an element w ∈ W or w ∈ W l , we use negative powers to indicate the colour inversion, that is, w −j =w j . For example, So, let C be the two-coloured representation category generated by C. We need to prove that C H * Ẑ l (∅, w) = C (∅, w) for every w ∈ W . In order to do that, we will use Proposition 4.4, whose statement can be, in this case, formulated as wherew ∈ W l is the glued version of w ∈ W . Let us start with the easier inclusion ⊃. Since C H * Ẑ l is a category, it is enough to show that C H * Ẑ l (∅, w) ⊃ C(∅, w) for every w = (••) j , j ∈ Z. Note that the glued version of w is in this casew = ( ) j = ( ) j . Combining Proposition 4.26 and Eq. 4.4, we have We will prove the opposite inclusion ⊂ by induction on the length of w. Take some Suppose ξ = 0. According to Lemma 4.27, we can assume thatw = w 0 w 1 · · · w l , where w 1 , . . . , w l−1 contain no triangles, and then ξ = R w 0 (ξ 1 ⊗ · · · ⊗ ξ l ) with ξ i ∈ C H (∅, w i ) and ξ l ∈ C H * Ẑ l (∅, w l w 0 ). Sincew is the glued version of w, this means that in all the words w 1 , . . . , w l−1 the colours alternate (two consecutive white squares would necessarily have a white triangle between them, two consecutive black squares would have = l−1 between them). Moreover, since we assume ξ i = 0, we must have c(w i ) ∈ kZ and, since k = 1, this means that the w i 's are of even length. So, Finally note that if we delete •• or •• from some word w, its glued version will be given by deleting resp. . In particular, denote byŵ the element w after deleting all the subwords w 1 , . . . , w l−1 . Its glued version is then w 0 l w l = w 0 w l . Using the induction hypothesis, this finishes the proof as we have We may ask what happens if we iterate those free complexifications. The following statement was again already formulated in [22]; however, without a proof. (Note that it generalizes Lemma 4.25 dropping the assumption gcd(k, l) = 1.)

Proposition 4.29
Let H be a compact matrix quantum group, k, l ∈ N 0 \ {1}. Then Proof The second equality follows directly from Theorem 4.28 -we see that iterating the operation on the categories for the second time cannot change it since C H (∅, (••) j ) = C H * Ẑ k (∅, (••) j ). For the first equality, we use, in addition, Theorem 4.17. Since Again, we can ask in what situations does it happen that the glued free product H * Ẑ l is isomorphic to the standard one. Obviously, the necessary condition is that H has degree of reflection one since H * Ẑ l H * Ẑ l implies H×Ẑ l H ×Ẑ l and here we can use Proposition 4.21. We can formulate the converse in the case of globally-colourized quantum groups H (in particular, if H ⊂ O + (F )).

Proposition 4.30
Let H be a globally colourized compact matrix quantum group with degree of reflection one. Then H * Ẑ k H * Ẑ k for every k ∈ N 0 .
Proof Denote by v the fundamental representation of H and by z the generator of C * (Z k ). Again, it is enough to show that we have z ∈ C(H×Ẑ k ) ⊂ C(H ) ⊗ max C * (Z k ) since this already implies the equality of the C*-algebras. From Proposition 4.11, we can find a vector ξ ∈ Mor(1, v ⊗w ) with c(w) = 1 and ξ = 1. Since H is globally colourized, we have

Free Complexification of Orthogonal Quantum Groups
In this section, we will study more in detail the free complexification H * Ẑ k with H ⊂ O + (F ). Recall that we define O + (F ) ⊂ U + (F ) only for F satisfying FF = c 1 N for some c ∈ R. We will use this assumption in the whole section. Let us explain a bit this definition. First of all, note that the required * -homomorphism maps Indeed, the first assignment is exactly the definition. For the second one, we have thanks to the assumption FF = c 1 N . In the Kac case F = 1 N , the homomorphism maps u ij → u * ij . But let us stress that for general elements of C(G) the homomorphism does not coincide with the * -operation (simply because the * is not a homomorphism).
Secondly, we have the following alternative formulations. ( More precisely, we mean one of the following equivalent conditions. Proof The equivalence (1) ⇔ (2a) follows from the universal property of C(G).
For (3) ⇒ (2b), we use the Tannaka-Krein, namely the fact that I G is spanned by the relations of the form T x ⊗w 1 = x ⊗w 2 T . Those relations are invariant with respect to the homomorphism x • → x • , x • → x • since this homomorphism maps x w → xw. Consequently, the whole ideal I G must be invariant with respect to this homomorphism.
The implication (2b) ⇒ (1) again follows from the universal property of C(G). We Using the assumption FF = c 1 N , we can show this actually must be a * -homomorphism.
As an example, note that all the universal unitary quantum groups U + (F ) with FF = c 1 N have this property. In addition, any quantum group G ⊂ O + (F ) has this property. Proof The right-left implication follows from Theorem 4.28: The fact that H * Ẑ is alternating is precisely the statement of Theorem 4.28. As we mentioned above, H ⊂ O + (F ) is surely invariant with respect to the colour inversion. According to Proposition 4.32, this is equivalent to saying that the associated category C H is invariant with respect to the colour inversion. In particular, we must have C H (∅, (••) j ) = C H (∅, (••) j ), which are the generators of C H * Ẑ according to Theorem 4.28. Consequently, also H * Ẑ must be invariant with respect to the colour inversion.
In order to prove the left-right implication, we construct a surjective * -homomorphism To prove that such a homomorphism exists, take any alternating element f ∈ I G . Since H ⊂ G, we have f (v ij ) = 0. We need to prove that f (u ij ) = 0.
If all terms of f start with a non-star variable, then f ( It remains to prove that α is injective. To do that, we define a * -homomorphism We immediately see that indeed z z * = z * z = 1. In exactly the same way as in the proof of Theorem 4.15, we also prove that v • : Finally, take f ∈ I G and, for convenience, use the representation in variables u by invariance under the colour inversion.
Since obviously β • α is injective, we have proven that α is a * -isomorphism.
Considering a quantum group H = (C(H ), v) ⊂ O + (F ), we have H×Ẑ k ⊂ H×Ẑ ⊂ H * Ẑ. We express those subgroups in terms of relations in the variables u ij = v ij z. Of course, those subgroups are given by the relations v ij z = zv ij and z k = 1, but those may not be well-defined in C(H * Ẑ) as we may not have z ∈ C(H * Ẑ).

Proposition 4.34 Consider H ⊂ O + (F ). Then H×Ẑ is a quantum subgroup of H * Ẑ given by the relation
For k ∈ N, H×Ẑ 2k is a quantum subgroup of H×Ẑ with respect to the relation Before proving the statement, note that Relations (4.5) and (4.6) correspond to the twocoloured partitions ⊗ and ⊗k , respectively. Hence, those are exactly the same relations that were used to construct the quantum groups H×Ẑ and H×Ẑ 2k in [13,Theorem 5.1].
Proof Relation (4.5) is the relation of the global colourization (see Definition 4.13) and it is obviously satisfied in H×Ẑ. We just need to show that imposing this relation is enough. By Corollary 3.9 and Theorem 4.17, the ideal I H×Ẑ is generated by relations of the form u ⊗w ξ = ξ , where ξ ∈ C H×Ẑ (∅, w) = C H (∅, 2l), c(w) = 0, l := |w|/2. In H * Ẑ, we have a relation of the form u ⊗(••) l ξ = ξ . The former relation can surely be derived from the latter one and Relation (4.5) since it is obtained just by permuting the white and black circles.
The second statement is proven in a similar way. If we denote u ij = v ij z, then Relations (4.6) say v i 1 j 1 · · · v i k j k z k = v i 1 j 1 · · · v i k j k z −k , which is surely satisfied in H×Ẑ 2k . For the converse, the ideal I H×Ẑ k is generated by relations of the form u ⊗w ξ = ξ , where ξ ∈ C H×Ẑ (∅, w) = C H (0, 2l), where c(w) is a multiple of 2k and l := |w|/2. Again, this relation can be derived from u ⊗(••) l ξ = ξ using Relations (4.5) to permute colours and Relations (4.6) to swap colours of k consecutive white points to black or vice versa.

Free Complexification and Partition Categories
In Sections 4.4 and 4.5, we generalized results that were formulated in the language of categories of partitions in [13]. In contrast, Sections 4.6 and 4.7 were rather new. Hence, it is interesting to look on the special case of easy and non-easy categories of partitions.
We will not recall the theory of partition categories here. Regarding the original definition of categories of partitions as defined in [7], we refer to the survey [27]. See also [23] for the definition of two-coloured categories of partitions and [16] for the definition of linear categories of partitions. Also see [15] for the definition of the two-coloured category Alt C associated to a given non-coloured category C . Alternatively, everything is summarized in the thesis [14].  Proof The base statement that H * Ẑ corresponds to Alt K follows directly from Theorem 4.28. By Proposition 4.12, the distinction of the cases correspond to the situation that either (1) H has degree of reflection one or (2) H has degree of reflection two. So, item (2) is also contained in Theorem 4.28.
For item (1), denote by w := • • · · · • the word of length l with alternating colours. Since H has degree of reflection one, we have H * Ẑ m H * Ẑ m for every m ∈ N 0 by Proposition 4.30. We can actually prove this directly repeating the proof of Proposition 4.30: Denote by v the fundamental representation of H and by z the fundamental representation ofẐ m . Then since we have v ⊗l ξ p = ξ p , we must have Now, H * Ẑ k is just a quantum subgroup of H * Ẑ with respect to the relation z k = 1. Note that So, the relation z k = 1 can be written as u ⊗w k ξ ⊗k p = ξ ⊗k p , which is exactly the relation corresponding top.

Proposition 4.36 Let C ⊂ P be a category of partitions and denote by G ⊂ O +
N the corresponding easy quantum group. Then G * Ẑ is a unitary easy quantum group corresponding to the category Alt C .
(1) If ∈ C , then G * Ẑ k = G * Ẑ for all k ∈ N and it corresponds to the category Alt C .
(2) If ∈ C , then G * Ẑ k corresponds to the category Alt C , ⊗k .
Proof This is just a reformulation of Proposition 4.35 to the easy case. Note that C contains some partition of odd length if and only if it contains the singleton .

Ungluing
The purpose of this section is to reverse the gluing procedure from Definition 4.1. The motivating result is the one-to-one correspondence formulated in terms of partition categories in [15,Theorem 4.10]. According to [15,Proposition 4.15], the functor F providing this correspondence translates to the quantum group language exactly in terms of gluing.
Recall from Definition 4.1 that given a quantum group G ⊂ U + (F ) * Ẑ k with fundamental representation u = v ⊕ z, we define its glued version to be the quantum groupG ⊂ U + (F ) with fundamental representationũ := vz and underlying C*-algebra C(G) ⊂ C(G) generated by the elements v ij z.
In Section 5.1, we are going to study the ungluings in general and show that they always exist. Unsurprisingly, an ungluing of a quantum group is not defined uniquely. The ungluings introduced in Section 5.1 are universal, but not particularly interesting. In Section 5.2, we are going to study more interesting ungluings of the form G ⊂ O + (F ) * Ẑ 2 , which allow us to generalize the above mentioned one-to-one correspondence. We formulate the result as Theorem 5.13, which constitutes the main result of this section.

General Ungluings
Proposition 5.2 There exists a Z k -ungluing for every quantum groupG and for every k ∈ N 0 . Namely, we have the trivial Z k -ungluingG × E, where E ⊂Ẑ k is the trivial group. Moreover,G ×Ẑ k is an ungluing ofG whenever k divides the degree of reflection of G.
Proof The first statement is obvious as we haveG× E =G. The second follows from Lemma 4.10 as we haveG×Ẑ k =G.
Let us denote by ι : C x ij ,x * ij → C x ij , x * ij , z, z * the embeddingx ij → x ij z. Consider G ⊂ U + (F ) and G its ungluing. The fact thatG is a glued version of G is, according to Proposition 4.4, characterized by the equalityĨG := ι(IG) = I G ∩ ι(C x ij ,x * ij ). Consequently, we have I G ⊃ĨG for every ungluing G of a quantum groupG.

Proposition 5.4
The maximal Z k -ungluing always exists. That is, considering the notation of Definition 5.3, G is indeed a compact matrix quantum group andG is indeed its glued version.
Proof First of all, note thatĨG contains the relations vv To prove that G is well defined, we need to check that I G /I U + (F ) is a Hopf * -ideal. Since IG/I U + (F ) is a Hopf * -ideal, we have thatĨG/I U + (F ) is a coideal invariant under the antipode. Consequently, the ideal generated byĨG/I U + (F ) must be a Hopf * -ideal.
From the construction, it is clear that, ifG has some Z k -ungluing, then G must be the maximal one (since we take the smallest possible ideal I G ). But every quantum group has the trivial ungluing as mentioned in Proposition 5.2.
Remark 5. 5 We do not have to know explicitly the whole ideal IG to compute the maximal ungluing. ConsiderG ⊂ U + (F ) and suppose that it is determined by a set of relationsR. That is, IG is generated by the coidealR. Then the maximal Z k -ungluing G ofG is defined by the relations R := ι(R). That is, taking the generating relations forG and exchanging v ij for v ij z andṽ * ij for z * v * ij .
Alternatively, we can describe the maximal ungluing by its representation category. Recall the definition of the gluing homomorphism mapping • → , • → . Given a word w ∈ W , the imagew under this homomorphism is called the glued version of w by Definition 4.3. If G is a quantum group andG its glued version, then CG is a full subcategory of C G according to Proposition 4.4. The full embedding is given exactly by the above mentioned gluing homomorphism. Consequently, the maximal ungluing G of somẽ G should be a quantum group with the minimal representation category containing CG as a full subcategory.
Proof By Tannaka-Krein duality, IG is linearly spanned by relations of the form [Tṽ ⊗w 1 − v ⊗w 2 T ] ji . By definition of the maximal ungluing, the ideal I G is generated by elements of IG, which are exactly the relations [T u ⊗w 1 − u ⊗w 2 T ] ji corresponding to the set C.
For the second statement, notice that ifC 0 generates CG, then C 0 must generate C. This follows simply from the fact that gluing of words is a monoid homomorphism (see also Proposition 4.4). Consequently, by what was already proven, C 0 generates C G .
. Iff starts with r followed by x ij , sof (x ij , r) = rx ijf0 (x ij , r) for some f 0 ∈Ã, then Remark 5.9Ã is the * -subalgebra of A generated by the elements x ij r. Consequently, for any G ⊂ O + (F ) * Ẑ 2 , we can express the coordinate algebra associated to its glued versioñ G as In addition, we can rephrase Recall the definition of quantum groupsG ⊂ U + (F ) invariant with respect to the colour inversion from Definition 4.31.

Proposition 5.10 Consider a compact matrix quantum groupG
is also a Z 2 -ungluing.

Definition 5.11
The quantum group G from Proposition 5.10 will be called the canonical Z 2 -ungluing ofG.
Before proving the proposition, recall that I G ⊂ C x ij , x * ij , r, r * is defined as the smallest ideal containing ι(IG). Consequently, I G = I G /(x • = x • , r 2 = 1) is the smallest ideal of the algebra A = C x ij , x * ij , r, r * /(x • = x • , r 2 = 1) containingĨG = ι(IG)/(x • = x • , r 2 = 1). In other words, the canonical Z 2 -ungluing is determined by relations of the form f (x ij r, rx ij ) with f ∈ IG ⊂ C x ij , x * ij .
Proof Adopting the notation introduced above, we need to prove that I G ∩Ã =ĨG. We prove that I G =ĨG +ĨGr = span{f, f r | f ∈ĨG}.
To prove the equality, it is enough to show that the right-hand side is an ideal since then it must be the smallest one containingĨG, which is exactly I G . So, denote the right-hand side by I . By Proposition 4.32, G being invariant with respect to the colour inversion means that I G is invariant with respect to interchanging x • ij ↔ x • ij . Applying ι A , we get thatĨG is invariant with respect to x ij r ↔ rx ij , soĨG is invariant with respect to conjugation by r, that is, x → rxr. We use that to prove that I is an ideal. The subspace I is obviously invariant under right multiplication by r. For the left multiplication, we can write rx = (rxr)r. For multiplication by x ij , we can write xx ij = (xx ij r)r and x ij x = r((x ij r) • x).

Proposition 5.12
Consider G ⊂ O + (F ) * Ẑ 2 and denote by k its degree of reflection. Then exactly one of the following situations occurs.
(2) If k = 2, then C(G) is Z 2 -graded and C(G) is its even part.
Proof Since G is orthogonal, its degree of reflection must be either one or two by Proposition 4.12. First, let us assume that k = 1. To show that C(G) = C(G), it is enough to prove that r ∈ C(G). The assumption k = 1 means that there is a vector ξ ∈ Mor(1, u ⊗k ), ξ = 1 for some k odd, so we have ξ * u ⊗k ξ = 1 in C(G). Consequently, r = ξ * u ⊗k ξr holds in C(G), and, since it is an even polynomial, it must be an element of C(G) (see Remark 5.9).
If the degree of reflection equals two, then by Proposition 4.11 the Z 2 -grading of A passes to O(G) and hence also C(G). As mentioned in Remark 5.9, O(G) consists of even polynomials in the generators v ij and r and hence is the even part of O(G).
The following theorem provides a non-easy counterpart of Theorem [15,Theorem 4.10].

Theorem 5.13
There is a one-to-one correspondence between (1) quantum groups G ⊂ O + (F ) * Ẑ 2 with degree of reflection two and (2) quantum groupsG ⊂ U + (F ) that are invariant with respect to the colour inversion. This correspondence is provided by gluing and canonical Z 2 -ungluing.
Proof Almost everything follows from Proposition 5.10. The only remaining thing to prove is that, given G ⊂ O + (F ) * Ẑ 2 andG its glued version, thenG is invariant with respect to the colour inversion and G is its canonical Z 2 -ungluing. The first assertion follows from the fact that I G and hence alsoĨ := I G ∩Ã =ĨG is invariant with respect to conjugation by r. For the second assertion, we need to prove that I G =Ĩ +Ĩ r (see the proof of Proposition 5.10). Since G has degree of reflection two, we have that I G is Z 2 -graded and hence it decomposes into an even and odd part, which is preciselyĨ andĨ r.
In [15], we introduced the Z 2 -extensions H × 2kẐ2 exactly to be the canonical Z 2ungluings of H×Ẑ 2k . In the following, we recall the definition of those products and provide a direct proof for the fact that H × 2kẐ2 are canonical Z 2 -ungluings of H×Ẑ 2k for arbitrary H ⊂ O + (F ). Definition 5.14 ( [15]) Let G and H be compact matrix quantum groups and denote by u and v their respective fundamental representations. We define the following quantum subgroups of G * H . The product G × × H is defined by taking the quotient of C(G * H ) by the relations ab * x = xab * , a * bx = xa * b (5.1) the product G × × H is defined by the relations ax * y = x * ya, axy * = xy * a (5.2) the product G × 0 H by the combination of the both pairs of relations and, finally, given k ∈ N, the product G × 2k H is defined by the relations a 1 x 1 · · · a k x k = x 1 a 1 · · · x k a k , which is obviously equivalent to v ij v kl r = rv ij v kl -the defining relation for H × ×Ẑ 2 = H × 0Ẑ2 . For H×Ẑ 2k , k > 0, we get v i 1 j 1 zv i 2 j 2 z · · · v i k j k z = zv i 1 j 1 zv i 2 j 2 · · · zv i k j k , which is exactly the defining relation for H × 2kẐ2 .

Irreducibles and Coamenability for Ungluings and Z 2 -extensions
In this section, we will study properties of the gluing procedure and canonical Z 2 -ungluing. In many cases, we can view the Z 2 -extensions H × 2kẐ2 as a motivating example. It remains an open question, whether one can generalize the statements for arbitrary products H 1 × 2k H 2 .
For this section, we will assume that G ⊂ O + N * Ẑ 2 has degree of reflection two. For quantum groups G and H , the property O(H ) ⊂ O(G) can be understood either as H being a quotient of G or the discrete dualĤ being a quantum subgroup ofĜ. In that case, we can study the homogeneous spaceĜ/Ĥ by defining Proof Consider x ∈ l ∞ (Ĝ). By Proposition 5.12, O(G) is Z 2 -graded. Putting b := 1 in the equality x(ab) = x(b), we get that x is constant on the even part. Putting b := r, we get that x is constant on the odd part.
Recall that a compact quantum group G is called coamenable if the so-called Haar state is faithful on the universal C*-algebra C(G).

Proposition 5.17
Consider G ⊂ O + N * Ẑ 2 andG ⊂ U + N its glued version. Then G is coamenable if and only ifG is coamenable.
Proof It is easy to see that the Haar state onG is given just by restriction of the Haar state h on G. We also have that all positive elements of C(G) are contained in C(G). Hence, the faithfulness of h on C(G) is equivalent to the faithfulness on C(G).
Note that it is well known that the coamenability is preserved under tensor product of quantum groups, but it is not preserved under the (dual) free product. Hence, it is interesting to ask, whether it is preserved under the new interpolating products from Definition 5.14.
Let us also recall another well known fact that can be seen directly from the definition of coamenability: Let G be a coamenable compact quantum group. Then every its quotient, that is, every quantum group H with C(H ) ⊂ C(G) is coamenable.