aa r X i v : . [ m a t h . QA ] J un GLUING COMPACT MATRIX QUANTUM GROUPS
DANIEL GROMADA
Abstract.
We study glued tensor and free products of compact matrix quan-tum groups with cyclic groups – so-called tensor and free complexifications.We characterize them by studying their representation categories and alge-braic relations. In addition, we generalize the concepts of global colourizationand alternating colourings from easy quantum groups to arbitrary compactmatrix quantum groups. Those concepts are closely related to tensor and freecomplexification procedures. Finally, we also study a more general procedureof gluing and ungluing.
Introduction
The subject of this article are compact matrix quantum groups as defined byWoronowicz in [Wor87]. A lot of attention has recently been devoted to quantumgroups possessing a combinatorial description by categories of partitions . Thosewere originally defined in [BS09]. Since then, their full classification was ob-tained [RW16] and many generalizations were introduced [CW16, Fre17, TW17,Ban18, GW19].Studying and classifying categories of partitions or generalizations thereof isuseful for the theory of compact quantum groups for several reasons. The primarymotivation is finding new examples of quantum groups since every category ofpartitions induces a compact matrix quantum group. Those are then called easy quantum groups. In addition, since the categories of partitions are supposed tomodel the representation categories of quantum groups, we immediately have a lotof information about the representation theory of such quantum groups (see also[FW16]).Categories of partitions provide a particularly nice way to describe the repre-sentation categories of quantum groups. Understanding the structure of partitioncategories and obtaining some classification results, we may obtain analogous state-ments also for the associated quantum groups. Such results can then be generalizedand go beyond categories of partitions and easy quantum groups. Let us illustratethis on a few examples.The classification of ordinary non-coloured categories of partitions involves a spe-cial class of so-called group-theoretical categories. Those induce group-theoretical
Date : June 25, 2020.2010
Mathematics Subject Classification.
Key words and phrases. compact matrix quantum group, representation category, complexifi-cation, gluing.The author was supported by the collaborative research centre SFB-TRR 195 “Symbolic Toolsin Mathematics and their Application”. The article is a part of the authors PhD thesis.The author would like to thank to his supervisor Moritz Weber for numerous comments andsuggestions regarding the text. We also thank to Adam Skalski and Pierre Tarrago for inspiringdiscussions on the topic. quantum groups described by some normal subgroups A E Z N . However, the latterdefinition turns out to be more general – not every group-theoretical quantum groupcan be described by a category of partitions [RW15]. Another example are gluedproducts, which were defined in [TW17] in order to interpret some classificationresult on two-coloured categories of partitions. The definition of glued productswas inspired by partitions, but it is independent of the partition description. Lastexample comes from [GW19], where a certain classification result for so-called cate-gories of partitions with extra singletons was obtained. Some of the new categorieswere interpreted by some new Z -extensions of quantum groups. In addition, thisextension procedure was generalized to a new product construction interpolatingthe free and the tensor product of quantum groups.The goal of the current paper is to study some additional results obtained inprevious works on coloured partition categories [TW17, TW18, Gro18, GW19]. Wereformulate those results purely in terms of quantum groups and their representa-tion categories without referring to partitions. Below, we give a detailed overviewof the results of this paper. Let us start by recalling the above mentioned gluedproduct construction.Consider a compact matrix quantum group G = ( C ( G ) , v ) and a cyclic groupdual ˆ Z k = ( z, C ∗ ( Z k )) (for k = 0, we take ˆ Z k := ˆ Z = T ). We can construct thetensor product or the free product as G × ˆ Z k = ( v ⊕ z, C ( G ) ⊗ max C ∗ ( Z k )) , G ∗ ˆ Z k = ( v ⊕ z, C ( G ) ∗ C C ∗ ( Z k )) . For both quantum groups, we can consider the representation vz = v ⊗ z . Thisrepresentation may not be faithful, so it defines a quotient quantum group calledthe glued product G ˜ × ˆ Z k = ( vz, C ( G ˜ × ˆ Z k )) , G ˜ ∗ ˆ Z k = ( vz, C ( G ˜ ∗ ˆ Z k )) . The glued tensor product with ˆ Z k is also called the tensor k -complexification . Like-wise the glued free product is called the free k -complexifiation . As we alreadymentioned, this definition comes from [TW17].Another important concept appearing throughout the whole article is the degreeof reflection – defined in [TW17] for categories of partitions, generalized in [GW19]for arbitrary quantum groups, and further characterized in Sect. 3.3 of this article.The main topic of this article is the characterization of the tensor and freecomplexifications. Let us start with the tensor case. The following theorem char-acterizes the tensor complexification in terms of algebraic relations, the associatedrepresentation category, and topological generation. Theorem A (Theorem 3.17) . Consider a compact matrix quantum group G =( C ( G ) , v ) , k ∈ N . Denote by z the generator of C ∗ ( Z k ) and by u := vz thefundamental representation of G ˜ × ˆ Z k . We have the following characterizations of G ˜ × ˆ Z k . (1) The ideal I G ˜ × ˆ Z k of algebraic relations in C ( G ˜ × ˆ Z k ) is the Z k -homogeneouspart of the ideal I G corresponding to G . (2) The representation category of G ˜ × ˆ Z k looks as follows Mor( u ⊗ w , u ⊗ w ) = ( Mor( v ⊗ w , v ⊗ w ) if c ( w ) − c ( w ) is a multiple of k , { } otherwise. (3) The quantum group G ˜ × ˆ Z k is topologically generated by G and ˆ Z k . LUING COMPACT MATRIX QUANTUM GROUPS 3
We also generalize the concept of global colourization introduced in [TW18] forcategories of partitions. It turns out that this concept characterizes the tensor k -complexification of orthogonal quantum groups for k = 0. The case k ∈ N remainsopen. Theorem B (Theorem 3.15) . Consider G ⊆ U + ( F ) with F ¯ F = cI , c ∈ R . Then G is globally colourized with zero degree of reflection if and only if G = H ˜ × ˆ Z ,where H = G ∩ O + ( F ) . The above two theorems can be understood as a generalization of the work[Gro18] on globally-colourized categories of partitions to arbitrary quantum groups.In addition, we provide the irreducible representations of tensor complexificationsin Proposition 3.23.We continue by studying the free complexification. In this case, we do nothave many results even for easy quantum groups. In the recent work [GW19,Sec. 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 -complexificationactually often do not depend on the number k . The following two results form ananalogy to Theorems A and B. Theorem C (Theorem 3.28) . Let H be a compact matrix quantum group withdegree of reflection k = 1 . Then all H ˜ ∗ ˆ Z l coincide for all l ∈ N \ { } . (1) The ideal I H ˜ ∗ ˆ Z l of algebraic relations in H ˜ ∗ ˆ Z l is generated by the alternatingpolynomials in I H . (2) The representation category C H ˜ ∗ ˆ Z l corresponding to H ˜ ∗ ˆ Z l is a (wide) sub-category of the representation category C H generated by the sets C ( ∅ , ( ) j ) := C H ( ∅ , ( ) j ) , j ∈ Z .The above characterization holds also for k = 1 and l = 0 . Theorem D (Theorem 3.33) . Consider G ⊆ U + ( F ) with F ¯ F = c N . Then G isalternating and invariant with respect to the colour inversion if and only if it is ofthe form G = H ˜ ∗ ˆ Z , where H = G ∩ O + ( F ) . Finally, the glued tensor product and the glued free product, which are used todefine the quantum group complexifications, can be also understood as a specialcase of some gluing procedure . In Section 4, we ask whether we can perform thisprocedure in the converse direction and unglue some cyclic group ˆ Z k from a unitaryquantum group. A particularly nice result can be obtained if we are ungluing ˆ Z .It is a generalization of [GW19, Theorem 4.10]. Theorem E (Theorem 4.13) . There is a one-to-one correspondence between (1) quantum groups G ⊆ O + ( F ) ∗ ˆ Z with degree of reflection two and (2) quantum groups ˜ G ⊆ U + ( F ) that are invariant with respect to the colourinversion.This correspondence is provided by gluing and canonical Z -ungluing. In addition, we characterize coamenability and provide irreducible representa-tions of the canonical Z -ungluings. These results are applied to the new Z -extensions introduced recently in [GW19] as those are special examples of canonical Z -ungluings. DANIEL GROMADA Preliminaries
Graded algebras.
Through the whole paper, we denote by Z k the cyclicgroup of order k ∈ N putting Z k := Z for k = 0.A Z k -grading of a ∗ -algebra A is a decomposition of the algebra into a vectorspace direct sum A = M i ∈ Z k A i such that the multiplication and involution of the algebra respect the operationon Z k , that is, A i A j ⊆ A ij , A ∗ i ⊆ A − i . 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 = P 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 toa homogeneous ideal inherits the grading.The definition of a Z k -grading for C*-algebras is quite simple for k ∈ N . Inthe case of the group Z or other groups, it gets a bit complicated and we will notmention it here. Let A be a C*-algebra. A Z k -grading on A is defined by a gradingautomorphism , that is, an automorphism α : A → A satisfying α k . Its spectrumconsists of k -th roots of unity and the corresponding eigenspaces can be identifiedwith the homogeneous parts of A satisfying the properties of the algebraic definitionabove.If A is a Z k -graded ∗ -algebra by the algebraic definition, we can define the gradingautomorphism by setting α ( x ) = e π ij/k x for x ∈ A k . The grading automorphismcan be then extended to the C*-envelope C ∗ ( A ) by the universal property.1.2. Compact matrix quantum groups.
We provide here only a brief overviewof the notions concerning compact matrix quantum groups. For more information,see for example [NT13, Tim08].Let A be a C*-algebra, u ∈ M N ( A ), N ∈ N . The pair ( A, u ) is called a compactmatrix 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 ) := P Nk =1 u ik ⊗ u kj extends toa ∗ -homomorphism.The ∗ -subalgebra O ( G ) generated by the elements u ij is dense in A and gen-eralizes the coordinate ring of G . It is actually a Hopf ∗ -algebra, that is, it isclosed under the above defined comultiplication ∆, it is further equipped with a counit (a ∗ -homomorphism ε : O ( G ) → C mapping u ij δ ij ), and an antipode (anantihomomorphism mapping u ij [ u − ] ij ).Two compact matrix quantum groups G = ( A, u ) and G ′ = ( A ′ , u ′ ) are consid-ered 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 betwo different completions of O ( G ). To overcome this ambiguity, we work with theuniversal C*-algebra C u ( G ) := C ∗ ( O ( G )). Given a compact matrix quantum group LUING COMPACT MATRIX QUANTUM GROUPS 5 G = ( A, u ), its maximal version G = ( C u ( G ) , u ) is again a compact matrix quan-tum group. From now on, we will assume that every quantum group appearing inthe paper is in its maximal version and denote C ( G ) := C u ( G ).The above notion of identical compact matrix quantum groups indeed generalizesthe notion of matrix groups being the same (i.e. not only isomorphic, but alsorepresented by the same matrices). Similarly, we can define H = ( C ( H ) , v ) tobe a quantum subgroup of G = ( C ( G ) , u ) if there is surjective ∗ -homomorphism O ( G ) → O ( H ) (or C ( G ) → C ( H )) 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 tobe isomorphic , denoted G ≃ H , if there exists any ∗ -isomorphism ϕ : C ( G ) → C ( H )such that ∆ H ◦ ϕ = ( ϕ ⊗ ϕ ) ◦ ∆ G .An important question is also how to construct quantum subgroups. A set I ⊆ O ( G ) is called a coideal if∆( I ) ⊆ I ⊗ O ( G ) + O ( G ) ⊗ I and ε ( I ) = 0 . A coideal that is also a ∗ -ideal is called a ∗ -biideal . A Hopf ∗ -ideal is a ∗ -biideal thatis invariant under the antipode, that is, S ( I ) ⊆ I . Hopf ∗ -ideals are in a one-to-onecorrespondence with quantum subgroups. That is, given H ⊆ G , the kernel of thesurjective ∗ -homomorphism O ( G ) → O ( H ) is a Hopf ∗ -ideal. Conversely, given anyHopf ∗ -ideal I ⊆ O ( G ), then the quotient O ( H ) := O ( G ) /I is a Hopf algebra thatdefines a quantum subgroup H ⊆ G .1.3. 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 ) = P k v ik ⊗ v kj , where ∆ is the comultiplication defined in the previous subsection. In partic-ular, the matrix u is a representation called the fundamental representation .A representation v is called non-degenerate if it is invertible as a matrix, it iscalled unitary if it is unitary as a matrix, i.e. P k v ik v ∗ jk = P k v ∗ ki v kj = δ ij . Tworepresentations v and w are called equivalent if there is an invertible matrix T suchthat vT = T w .For every compact quantum group it holds that every non-degenerate represen-tation is equivalent to a unitary one. Hence, given a compact matrix quantumgroup G = ( C ( G ) , u ), we may assume that u is unitary. At the same time ¯ u = ( u ∗ ij )is a non-degenerate representation, so there exists an invertible matrix F such that F ¯ uF − is unitary. Consequently, any compact matrix quantum group G is, upto similarity, a quantum subgroup of the universal unitary quantum group U + ( F )[VDW96] 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 F ¯ uF − are unitary) . Assuming F ¯ F = c N for some c ∈ R , we also define the universal orthogonalquantum group O + ( F ) [Ban96] by C ( O + ( F )) = C ∗ ( u ij ; i, j = 1 , . . . , N | u = F ¯ uF − is unitary) . For a pair of representations u ∈ M n ( C ( G )), v ∈ M m ( C ( G )), we denote byMor( u, v ) := { T : C n → C m | T u = vT } the space of intertwiners between u and v . A representation u is called irreducible if Mor( u, u ) = C ·
1. It holds that any representation is a direct sum of irreducibleones.
DANIEL GROMADA
For a given quantum group G , we denote by Irr G the set of equivalence classesof G . For α ∈ Irr G , we denote by u α some unitary representative of the class α . Itholds that the matrix elements of all the u α , α ∈ Irr G form a linear basis of O ( G ). Remark 1.1.
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 Irr H = { α ∈ Irr G | [ u α ] ij ∈ C ( H ) ∀ i, j } ⊆ Irr G. Indeed, any representation of H is by definition a representation of G . The notionof irreducibility does not change if we enlarge the algebra.1.4. Grading on quantum group function spaces.
Given a compact matrixquantum group G = ( C ( G ) , u ), we denote by I G := { f ∈ C h x ij , x ∗ ij i | f ( u ij , u ∗ ij ) = 0 } the ideal determining the algebras O ( G ) = C h x ij , x ∗ ij i /I G and C ( G ) = C ∗ ( O ( G )).There is a natural structure of a Z k -grading on the algebra C h x ij , x ∗ ij i given byassociating degree one to the variables x ij , and associating degree minus one to thevariables x ∗ ij . In this article, by a Z k -grading we will always mean this particulargrading.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 representa-tions in the following sense. For any α ∈ Irr G , there is d α ∈ Z k such that all thematrix entries of u α are Z k homogeneous of degree d α . We will call d α the degree of the irreducible u α . 2. Representation categories
The main point of this section is to formulate the Tannaka–Krein duality forunitary compact matrix quantum groups. The Tannaka–Krein duality for quantumgroups was first formulated by Woronowicz [Wor88]. Essentially it says that anycompact quantum group can be recovered from its representation category. Sincethen, many formulations of this statement appeared – from very categorical onessuch as in [NT13] to very concrete ones such as [Mal18]. We will stick here to thelatter approach reformulating it a bit in the spirit of [Fre17] to fit into our setting.All the concepts presented in this section are well known to the experts. There-fore, we try to keep the section very brief. On the other hand, since the topic isquite new and rapidly developing, it is hard to give some general reference here.Some of the concrete notation and formulations of definitions and propositions areactually author’s original. We refer to the author’s PhD thesis [Gro20] for a moredetailed discussion of the concepts. See also [Ban19].2.1.
Two-coloured categories.
Consider a compact matrix quantum group G =( C ( G ) , u ) ⊆ U + ( F ), F ∈ GL( N, C ). Denote u := u , u := F ¯ uF − . Denote by W the free monoid over the alphabet { , } . For any word w ∈ W , we denote u ⊗ w thecorresponding tensor product of the representations u and u .For a pair of words w , w ∈ W , denote C G ( w , w ) : = Mor( u ⊗ w , u ⊗ w )= { T : ( C N ) ⊗| w | → ( C N ) ⊗| w | | T u ⊗ w = u ⊗ w T } . LUING COMPACT MATRIX QUANTUM GROUPS 7
Such a collection of vector spaces forms a rigid monoidal ∗ -category in the followingsense(1) For T ∈ C G ( w , w ), T ′ ∈ C G ( w ′ , w ′ ), we have T ⊗ T ′ ∈ C G ( w w ′ , w w ′ ).(2) For T ∈ C G ( w , w ), S ∈ C G ( w , w ), we have ST ∈ C G ( w , w ).(3) For T ∈ C G ( w , w ), we have T ∗ ∈ C G ( w , w )(4) For every word w ∈ W , we have 1 ⊗| w | N ∈ C G ( w, w ).(5) There exist vectors ξ ∈ C G ( ∅ , ) and ξ ∈ C G ( ∅ , ) called the dualitymorphisms such that(2.1) ( ξ ∗ ⊗ C N )(1 C N ⊗ ξ ) = 1 C N , ( ξ ∗ ⊗ C N )(1 C N ⊗ ξ ) = 1 C N . For the last point, we can write explicit formulae(2.2) [ ξ ] ij = F ji , [ ξ ] ij = [ ¯ F − ] ji . Definition 2.1.
Consider a natural number N ∈ N . Let C ( w , w ) be a collectionof vector spaces of linear maps ( C N ) ⊗| w | → ( C N ) ⊗| w | satisfying the conditions(1)–(5) above. Then we call C a two-coloured representation category . For anycollection of sets C ( w , w ) of linear maps ( C N ) ⊗| w | → ( C N ) ⊗| w | satisfying (5),we denote by h C i the smallest category containing C . We say that C generates thiscategory.The justification for this name is given by the following formulation of theTannaka–Krein duality, which comes from [Fre17]. Theorem 2.2 (Woronowicz–Tannaka–Krein duality for CMQG) . Let C be a two-coloured representation category. Then there exists a unique compact matrix quan-tum group G such that C = C G . This quantum group is determined by the ideal I G = span { [ T u ⊗ w − u ⊗ w T ] ji | T ∈ C ( w , w ) } . We give only a sketch of proof here. For a detailed explanation, see [Gro18,Theorem 3.4.6]. The proof closely follows the proof formulated for orthogonalquantum groups in [Mal18].
Sketch of proof.
In order to give a sense to the formula for I G , we need to specifythe matrix F , so that the matrix u is well defined. We fix the duality morphismssatisfying Eqs. (2.3) and define F according to Eqs. (2.2). Note that the dualitymorphisms are not defined uniquely by Eqs. (2.3). Part of the “uniqueness” state-ment is that the resulting quantum group does not depend on the particular choiceof 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 [Mal18, Gro20] for details.)Now, let ˜ ξ , ˜ ξ be alternative solutions of (2.3) and let ˜ F be the alternativematrix and ˜ G the alternative resulting quantum group. Then we have( ˜ ξ ∗ ⊗ C N )(1 C N ⊗ ξ ) = F ˜ F − ∈ C G ( , ) = Mor( u , u ) . This means that u = ˜ F F − u F ˜ F − = ˜ F ¯ u ˜ F − and hence G ⊆ ˜ G . From symmetry,we have G = ˜ G .It remains to prove that we indeed have C G = C (from construction, we canactually easily see that C G ⊇ C ) and that G is a unique quantum group with thisproperty, that is, if C ˜ G = C for some quantum group ˜ G ⊆ U + ( F ), then surely DANIEL GROMADA G = ˜ G (again, from construction, we obviously have G ⊇ ˜ G ). This can be provenusing the double commutant theorem, see [Mal18, Gro20]. (cid:3) Proposition 2.3.
Let G ⊆ U + ( F ) be a compact matrix quantum group. Supposethat the associated category C G is generated by some collection C ( w , w ) . Then I G ⊆ C h x ij , x ∗ ij i as an ideal is generated by { [ T x ⊗ w − x ⊗ w T ] ji | T ∈ C ( w , w ) } . Proof.
Denote by I the ideal generated by C as formulated above. Obviously, wehave I ⊆ I G . To prove the opposite inclusion, it is enough to prove that C ( w , w ) := { T : ( C N ) ⊗| w | → ( C N ) ⊗| w | | T x ⊗ w − x ⊗ w T ∈ I } form a category. Then, since obviously C ⊆ C and hence C G ⊆ C , we must have I G ⊆ I .So, denote A := C h x ij , x ∗ ij i /I and by v ij denote the images of x ij by the naturalhomomorphism. Taking T ∈ C ( w , w ), T ∈ C ( w , w ). Then T T v ⊗ w = T v ⊗ w T = v ⊗ w T T , so T T ∈ C ( w , w ). For tensor product and involution, the proof is similar. (cid:3) Remark 2.4.
Recall the universal orthogonal quantum group O + ( F ) ⊆ U + ( F ),which is defined by the relation u = F ¯ uF − . The relation can also be written as u = u . Consequently, we have u ⊗ w = u ⊗| w | for any w ∈ W , so only the length ofthe word w matters. For any G ⊆ O + ( F ), we define C G ( k, l ) := Mor( u ⊗ k , u ⊗ l ) = C G ( w , w ) , where w , w are any words with | w | = k and | w | = l .2.2. Representation categories of quantum subgroups, intersections, andtopological generation.
In this section, we would like to briefly explain the con-cepts of the quantum group intersection and topological generation. Together withquantum subgroups, we relate those notions with the associated ideals of algebraicrelations I G and the representation categories C G . See also [Gro20, Sects. 2.3.4,2.5.4, 2.5.5, 3.4.5] for more detailed discussion. Proposition 2.5.
Consider
G, H ⊆ U + ( F ) . Then the following are equivalent. (1) H ⊆ G , (2) I H ⊇ I G , (3) C H ( w , w ) ⊇ C G ( w , w ) for all w , w ∈ W .Proof. The equivalence (1) ⇔ (2) follows directly from the definition of quan-tum subgroup. The equivalence (2) ⇔ (3) follows from Tannaka–Krein duality(Thm. 2.2). (cid:3) Consider H , H ⊆ U + ( F ). Their intersection H ∩ H is the largest compactmatrix quantum group contained in both H and H . This notion was recentlyheavily used especially in the work of Teodor Banica. See [Ban19]. Proposition 2.6.
Consider
G, H , H ⊆ U + ( F ) . Then the following are equiva-lent. (1) G = H ∩ H , (2) I G = I H + I H , LUING COMPACT MATRIX QUANTUM GROUPS 9 (3) C G = h C H , C H i .Proof. The equivalence follows from Proposition 2.5. G being the largest quantumgroup contained in H and H is equivalent to I G being the smallest ideal containing I H and I H , which is equivalent to C G being the smallest category containing C H and C H . (cid:3) Consider H , H ⊆ U + ( F ). The smallest quantum group G containing both H and H is said to be topologically generated by H and H . We denote it by G = h H , H i . This notion goes back to [Chi15, BCV17]. Proposition 2.7.
Consider
G, H , H ⊆ U + ( F ) . Then the following are equiva-lent. (1) G = h H , H i , (2) I G = I H ∩ I H , (3) C G ( w , w ) = C H ( w , w ) ∩ C H ( w , w ) for all w , w ∈ W .Proof. Again, the equivalence follows from Proposition 2.5. G being the smallestquantum group containing H and H is equivalent to I G being the largest idealcontained in I H and I H , which is equivalent to C G being the largest categorycontained in C H and C H . (cid:3) Frobenius reciprocity.
We define an involution on the set of two colours { , } mapping , . We extend this operation as a homomorphism on themonoid 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 wordbackwards).Consider a two-coloured representation category C and fix the duality morphisms ξ , ξ . We define a map Rrot : C ( w , w a ) → C ( w ¯ a, w ) for a ∈ { , } by T (1 ( C N ) | w | ⊗ ξ ∗ )( T ⊗ C N ) with ξ = ξ if a = and ξ = ξ if a = . We callthis map the right rotation .This map has an inverse Rrot − : C ( w a, w ) → C ( w , w ¯ a ) given by T ( T ⊗ C N )(1 ( C N ) ⊗| w | ⊗ ξ ). Similarly, we can define the left rotation Lrot : C ( aw , w ) → C ( w , ¯ aw ). As a consequence, we have the following. Proposition 2.8.
Let C be a two-coloured representation category. Then C isgenerated by the collection C ( ∅ , w ) with w running through W .Proof. We have C ( w , w ) = Rrot | w | C ( ∅ , w w ∗ ). (cid:3) Corollary 2.9.
Let G be a compact quantum group. Then the ideal I G is generatedby relations of the form u ⊗ w ξ = ξ . Some authors denote Fix( u ⊗ w ) := Mor(1 , u ⊗ w ) = C G ( ∅ , w ). Such particularintertwiners ξ ∈ Mor(1 , u ⊗ w ) satisfying u ⊗ w ξ = ξ are called the fixed points of u ⊗ w .Not only that the collection C ( ∅ , w ) already determines the whole category C .We can even characterize representation categories in terms of those fixed pointspaces by introducing some alternative operations. The following essentially refor-mulates the operations defined in [Gro18]. Definition 2.10.
Consider a two-coloured representation category C . Denote forsimplicity 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 :Π i : C ( ∅ , a · · · a k ) → C ( ∅ , a · · · a i − a i +2 · · · a k ) , Π i η := (1 N ⊗ · · · ⊗ N ⊗ ξ ∗ a i ⊗ N ⊗ · · · ⊗ N ) η. Pictorially, Π i η = η · · · ξ ∗ ai · · · . On elementary tensors, it acts asΠ i ( η ⊗ · · · ⊗ η k ) = ( η ti +1 F η i ) η ⊗ · · · ⊗ η i − ⊗ η i +2 ⊗ · · · ⊗ η k . • We define the rotation : R : C ( ∅ , a · · · a k ) → C ( ∅ , a k a · · · a k − ) , R := Lrot ◦ Rrot , so Rη = (1 N ⊗ · · · ⊗ N ⊗ ξ ∗ a k )(1 N ⊗ η ⊗ N ) ξ a k . Pictorially, Rη = η · · · ξ ∗ ak ξ ak . On elementary tensors, it acts as R ( η ⊗ · · · ⊗ η k ) = ( F ¯ F η k ) ⊗ η ⊗ · · · ⊗ η k − . Note that the rotation is obviously invertible with R − = Rrot − ◦ Lrot − : C ( ∅ , a · · · a k ) → C ( ∅ , a · · · a k a ). • We define the reflection : ⋆ : C ( ∅ , a · · · a k ) → C ( ∅ , a k · · · a ) η ⋆ := Rrot − k η ∗ = ( η ∗ ⊗ N ak ⊗ · · · ⊗ N a ) ξ a ··· a k , where ξ a ··· a k is the duality morphism associated to the object a · · · a k .Pictorially, Rη = η ∗ · · · · · · . On elementary tensors, it acts as( η ⊗ · · · ⊗ η k ) ⋆ = ( F a k ¯ η k ) ⊗ · · · ⊗ ( F a ¯ η ) . Proposition 2.11.
For any two-coloured representation category C , the collectionof 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 represen-tation categories, the sets C ( w , w ) := { Rrot | w | ξ | ξ ∈ C ( w w ∗ ) } = { Lrot −| w | p | p ∈ C ( w ∗ w ) } form a two-coloured representation category.Proof. The first part of the proposition follows from the fact that all the newoperations are defined using the category operations of tensor product, composition,and involution.
LUING COMPACT MATRIX QUANTUM GROUPS 11
The converse statement is proved by expressing the category operations in termsof the new operations:Lrot − k ξ ⊗ Rrot k ′ η = Lrot − k Rrot k ′ ( ξ ⊗ η ) , (Rrot k ξ ) ∗ = Rrot l ξ ⋆ , (Rrot l η )(Rrot k ξ ) = Rrot k Π m +1 Π m +2 · · · Π m + l ( η ⊗ ξ ) , where we assume that ξ ∈ C ( w w ) and η ∈ C ( w ′ w ′ ) (for the first row), resp. η ∈ C ( w w ) (for the last row). (cid:3) Free and tensor product.
The following two constructions were defined byWang.
Proposition 2.12 ([Wan95a]) . Let H = ( C ( H ) , v ) and H = ( C ( H ) , v ) becompact matrix quantum groups. Then H ∗ H := ( C ( H ) ∗ C C ( H ) , v ⊕ v ) is acompact matrix quantum group. For the co-multiplication we have that ∆ ∗ ([ v ] ij ) = ∆ H ([ v ] ij ) , ∆ ∗ ([ v ] kl ) = ∆ H ([ v ] kl ) . Proposition 2.13 ([Wan95b]) . Let H = ( C ( H ) , v ) and H = ( C ( H ) , v ) becompact matrix quantum groups. Then H × H := ( C ( H ) ⊗ max C ( H ) , v ⊕ v ) is a compact matrix quantum group. For the co-multiplication we have that ∆ × ([ v ] ij ⊗
1) = ∆ H ([ v ] ij ) , ∆ × (1 ⊗ [ v ] kl ) = ∆ H ([ v ] kl ) . The tensor product of quantum groups is a generalization of the group directproduct. The free product should also be seen as some free version of the directproduct. Note in particular that the free product of compact quantum groups doesnot generalize the group free product since the freeness occurs in the C*-algebramultiplication not in the quantum group comultiplication. For this reasons, someauthors call it rather dual free product .We will focus here mainly on products of quantum groups H = ( C ( H ) , v ) withcyclic group duals ˆ Z k . Denote by z the generator of the C*-algebra C ∗ ( Z k ), so thefundamental representation of H ∗ ˆ Z k or H × ˆ Z k is of the form u := v ⊕ z . Now, wewould like to describe the corresponding representation category. For our purposes,it will be convenient not to restrict only to the intertwiner spaces between tensorproducts of u and u , but to keep track of the blocks v and z .Consider G ⊆ U + ( F ) ∗ Z k for some k ∈ N with fundamental representation ofthe form u = v ⊕ z , where z is one-dimensional. Denote by N the size of v . Wedefine 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 incontrast with the overall length of w ).Then we can associate to G the following category C G ( w , w ) : = Mor( u ⊗ w , u ⊗ w )= { T : ( C N ) ⊗ [ w ] → ( C N ) ⊗ [ w ] | T u ⊗ w = u ⊗ w T } , where w , w ∈ W k .The axiomatization of a two-coloured representation category extends to thiscase as follows. Definition 2.14.
Consider a natural number N ∈ N and k ∈ N . A collection C ( w , w ), w , w ∈ W k of vector spaces of linear maps ( C N ) ⊗ [ w ] → ( C N ) ⊗ [ w ] iscalled a Z k -extended representation category if it satisfies the following(1) For T ∈ C G ( w , w ), T ′ ∈ C G ( w ′ , w ′ ), we have T ⊗ T ′ ∈ C G ( w w ′ , w w ′ ).(2) For T ∈ C G ( w , w ), S ∈ C G ( w , w ), we have ST ∈ C G ( w , w ).(3) For T ∈ C G ( w , w ), we have T ∗ ∈ C G ( w , w )(4) For every word w ∈ W k , we have 1 ⊗ [ w ] N ∈ C G ( w, w ).(5) There exist vectors ξ ∈ C G ( ∅ , ) and ξ ∈ C G ( ∅ , ) called the dualitymorphisms such that(2.3) ( ξ ∗ ⊗ C N )(1 C N ⊗ ξ ) = 1 C N , ( ξ ∗ ⊗ C N )(1 C N ⊗ ξ ) = 1 C N . 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 itis not necessary to assume the existence of duality morphisms for the trianglessince we automatically have C ( ∅ , ) = C ( ∅ , ) = C ( ∅ , ∅ ) ∋ Z k -extended case. Inparticular, the Tannaka–Krein duality associates a compact matrix quantum group G ⊆ U + ( F ) ∗ ˆ Z k to any Z k -extended representation category.Also the Frobenius reciprocity holds for Z k -extended representation categoriesand the operations on the fixed point spaces can be defined in a similar way. In par-ticular, the contraction C ( ∅ , w w ) → C ( ∅ , w w ) or C ( ∅ , w w ) → C ( ∅ , w w )is defined the same way as in Sect. 2.3. On the other hand the contraction C ( ∅ , w w ) → C ( ∅ , w w ) is simply the identity since already on the level ofobjects we have w w = w w . Similarly, the rotation of squares is defined thesame way as the rotation of circles in Sect. 2.3 and will be denoted by R . Therotation of triangles is then simply the identity. Consequently, we have R [ w ] : C ( ∅ , w w ) → C ( ∅ , w w ).A way how to model Z -extended representation categories using partitions isdescribed in [GW19]. Many new examples of quantum groups were obtained usingthis approach. In particular, several new product constructions interpolating thefree and tensor product. 3. Glued products
In [TW17, Gro18, GW19], the representation categories of glued products oforthogonal easy quantum groups with cyclic groups were studied. Here, we revisitthe theory dropping the easiness assumption.3.1.
Gluing procedure.Definition 3.1.
Let G be a compact matrix quantum group with fundamentalrepresentation of the form u = v ⊕ v . Denote by ˜ A the C*-subalgebra of C ( G )generated by elements of the form [ v ] ij [ v ] kl . Then ˜ G := ( ˜ A, v ⊗ v ) is a compactmatrix quantum group called the glued version of G .A particular example of the gluing procedure is the glued free product H ˜ ∗ H defined as the glued version of the free product H ∗ H and the glued tensor product H ˜ × H defined as the glued version of the tensor product H × H . Those twodefinitions were first formulated in [TW17]. LUING COMPACT MATRIX QUANTUM GROUPS 13
Again, we will be interested in particular in the case when u = v ⊕ z , where z is one-dimensional. The glued free product H ˜ ∗ ˆ Z k is also called the free k -complexification of H and the glued tensor product H ˜ × ˆ Z k is called the tensor k -complexification . The free complexification was studied already by Banica in[Ban99, Ban08]. Remark 3.2.
The glued version ˜ G of a quantum group G is by definition a quotientof G . It may happen that the elements [ v ] ij [ v ] kl already generate the whole C*-algebra C ( G ), so C ( ˜ G ) = C ( G ). In this case, we have that ˜ G is isomorphic to G .However, the ˜ G and G are still distinct as compact matrix quantum groups sincetheir fundamental representations are different.The same holds in particular for the glued products and for the complexifications.Considering G = ( C ( G ) , v ) and ˆ Z k = ( z, C ∗ ( Z k )), we have that G ˜ × ˆ Z k ≃ G × ˆ Z k or G ˜ ∗ ˆ Z k ≃ G ∗ ˆ Z k if the elements v ij z actually generate the whole algebra C ( G ) ⊗ max C ∗ ( Z k ), resp. C ( G ) ∗ C C ∗ ( Z k ).Now, we are going to characterize the representation categories of the gluedversions. Definition 3.3.
Let us fix k ∈ N . Then for any word w ∈ W we associate its glued version ˜ w ∈ W k mapping , . Proposition 3.4.
Consider G ⊆ U + ( F ) ∗ ˆ Z k with fundamental representation u = v ⊕ z . Let C G be the associated Z k -extended representation category. Let ˜ G bethe glued version of G . Then C ˜ G ( w , w ) = C G ( ˜ w , ˜ w ) for every w , w ∈ W . That is C ˜ G is a full subcategory of C G given by consideringthe glued words only. The ideal associated to ˜ G can be described as I ˜ G = { f ∈ C h ˜ x ij , ˜ x ∗ ij i | f ( x ij z, z ∗ x ∗ ij ) ∈ I G } ≃ I G ∩ C h x ij z, z ∗ x ∗ ij i . Proof.
Denote by ˜ v = vz the fundamental representation of ˜ G . Consider a word w and its glued version ˜ w . Directly from the definitions of ˜ v and ˜ w , we have˜ v ⊗ w = u ⊗ ˜ w . So, C ˜ G ( w , w ) = Mor(˜ v ⊗ w , ˜ v ⊗ w ) = Mor( u ⊗ ˜ w , u ⊗ ˜ w ) = C G ( ˜ w , ˜ w ) . For the ideal, we have I ˜ G = { f ∈ C h ˜ x ij , ˜ x ∗ ij i | f (˜ v ij , ˜ v ∗ ij ) = f ( v ij z, z ∗ v ∗ ij ) in C ( ˜ G ) ⊆ C ( G ) } = { f ∈ C h ˜ x ij , ˜ x ∗ ij i | f ( x ij z, z ∗ x ∗ ij ) ∈ I G } . (cid:3) Glued version and projective version.
As a side remark, let us relate thenew notion of glued version with already existing notion of projective version . Definition 3.5 ([BV10]) . Let G ⊆ U + ( F ) be a compact matrix quantum group.We define its projective version as P G := ( C ( P G ) , u ⊗ u ), where C ( P G ) is theC*-subalgebra of C ( G ) generated by the elements u ij u kl . Proposition 3.6.
Consider a compact matrix quantum group G with fundamentalrepresentation of the form v ⊕ v . Denote by G ′ := ( C ( G ) , v ⊕ v ) ≃ G . Let ˜ G bethe glued version of G . Then ˜ G ≃ P G ′ . 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 form ( v ⊗ v ) ⊕ ( v ⊗ v ) ⊕ ( v ⊗ v ) ⊕ ( v ⊗ v ). In contrast, C ( ˜ G ) ⊆ C ( G ) is generated only by v ⊗ v . We need to show that C ( ˜ G ) and C ( P G ′ ) coincide as subalgebras of C ( G ). We can express v ⊗ v = ( v ⊗ v ) ,v ⊗ v = (id ⊗ ξ (2) ∗ ⊗ id)( v ⊗ v ⊗ v ⊗ v )(id ⊗ ξ (2) ⊗ id) / k ξ (2) k ,v ⊗ v = (id ⊗ ξ (1) ∗ ⊗ id)( v ⊗ v ⊗ v ⊗ v )(id ⊗ ξ (1) ⊗ id) / k ξ (1) k , where ξ = ξ (1) ⊕ ξ (2) and ξ = ξ (1) ⊕ ξ (2) are the duality morphisms correspond-ing to v ⊕ v . (cid:3) Remark 3.7.
Often it happens that G and G ′ are identical as compact matrixquantum groups. For instance, in the case of tensor product or free product withˆ Z k . So, we can write H ˜ × ˆ Z k ≃ P ( H × ˆ Z k ) , H ˜ ∗ ˆ Z k ≃ P ( H ∗ ˆ Z k ) . Degree of reflection.
We characterize the notion of the degree of reflection,which was introduced in [TW18] in the categorical language and in [GW19] forgeneral compact matrix quantum groups.
Definition 3.8 ([TW18]) . For a word w ∈ W , we define c ( w ) to be the number ofwhite circles in w minus the number of black circles in w .Recall that given a quantum group G = ( C ( G ) , u ), we can construct a quantumsubgroup of G – so called diagonal subgroup – imposing the relation u ij = 0 forall i = j . If we, in addition, impose the relation u ii = u jj for all i and j , weget a quantum group corresponding to a C*-algebra generated by a single unitary.Therefore, it must be a dual of some cyclic group. Definition 3.9 ([GW19]) . Let G be a quantum group and denote by ˆΓ the quantumsubgroup of G given by u ij = 0, u ii = u jj for all i = j . The order of the cyclicgroup Γ is called the degree of reflection of G . If the order is infinite, we set thedegree of reflection to zero. Lemma 3.10.
Let G = ( C ( G ) , u ) be a compact matrix quantum group, k ∈ N .The following are equivalent. (0) The number k divides the degree of reflection of G . (1) The mapping u ij δ ij z extends to a ∗ -homomorphism ϕ : C ( G ) → C ∗ (ˆ Z k ) . (2) For any w ∈ W , Mor(1 , u ⊗ w ) = { } only if c ( w ) is a multiple of k . (3) The ideal I G is Z k -homogeneous. (4) We have G = G ˜ × ˆ Z k .Proof. (0) ⇔ (1): If k is the degree of reflection of G , then directly by the definitionthere is a ∗ -homomorphism ϕ : C ( G ) → C ∗ (ˆ Z k ). Such an homomorphism obviouslyexists also if k is a divisor of k . By definition, ˆ Z k is the largest group with thisproperty, so k must be a divisor of k .(1) ⇒ (2): Take any ξ ∈ Mor(1 , u ⊗ w ), so u ⊗ w ξ = ξ . Applying the homomor-phism ϕ , we get z c ( w ) ξ = ξ . If ξ = 0, we must have z c ( w ) = 1, so c ( w ) is a multipleof k . LUING COMPACT MATRIX QUANTUM GROUPS 15 (2) ⇒ (3): By Corollary 2.9, 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 ˜ × ˆ Z k ). To prove that it extends to a homomorphism, take any f ∈ I G .Suppose f is Z k -homogeneous 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 := ( ε ⊗ id) α , where ε is the counit of G and α isthe ∗ -homomorphism C ( G ) → C ( G ) ⊗ C ∗ (ˆ Z k ). Then indeed ϕ ( u ij ) = ε ( u ij ) z = δ ij z . (cid:3) As a consequence, we have the following four equivalent characterizations of thedegree of reflection.
Proposition 3.11.
Let G be a compact matrix quantum group, k ∈ N . Thefollowing are equivalent. (1) The number k is the degree of reflection of G . (2) We have { c ( w ) − c ( w ) | C G ( w , w ) = { }} = k Z . (3) The number k is the largest such that I G is Z k -homogeneous. (4) The number k is the largest such that G = G ˜ × ˆ Z k .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 theequivalent conditions in Lemma 3.10. For (2) note that the set { c ( w ) − c ( w ) | C G ( w , w ) = { }} is indeed a subgroup of Z . The fact that { c ( w ) − c ( w ) } is closed under addition follows from C G being closed under the tensor product.The fact that { c ( w ) − c ( w ) } is closed under subtraction follows from C G beingclosed under the involution. The statement (2) in Lemma 3.10 can be formulatedas { c ( w ) − c ( w ) | C G ( w , w ) = { }} ⊆ k Z . Taking the maximal k , we gain theequality. (cid:3) Proposition 3.12.
Consider G ⊆ O + ( F ) . Then one of the following is true. (1) The degree of reflection of G is one and C G (0 , k ) = { } for some odd k ∈ N . (2) The degree of reflection of G is two and C G ( k, l ) = { } for every k + l odd.Proof. Recall from Remark 2.4 that the intertwiner spaces C G ( w , w ) depend onlyon the length of the words w and w for G ⊆ O + ( F ). This allowed us to introducethe notation C G ( k, l ). Now the proposition follows from Proposition 3.11. First ofall, the degree of reflection must be a divisor of two (that is, either one or two)since we have C G ( ∅ , ) = C G ( ∅ , ) ∋ ξ = 0. Then G has degree of reflection oneif and only if C G ( ∅ , w ) = C G (0 , | w | ) = { } for some word w with c ( w ) = 1. Sucha word with c ( w ) = 1 must be of odd length. (cid:3) Global colourization.
The notion of globally-colourized categories was in-troduced in [TW18] and studied in more detail in [Gro18]. Here, we reformulatethe results in the non-easy case.
Definition 3.13.
A compact matrix quantum group G = ( C ( G ) , u ) is called glob-ally colourized if the following holds in C ( G )(3.1) u ij u ∗ kl = u ∗ kl u ij . for all possible indices i, j, k, l .Assuming G ⊆ U + ( F ), this can be equivalently expressed using the entries ofthe unitary representations u = u and u = F ¯ uF − as(3.2) u ij u kl = u ij u kl . Proposition 3.14.
A compact matrix quantum group G = ( C ( G ) , u ) is globallycolourized if and only if for every w , w , w ′ , w ′ ∈ W satisfying | w ′ | = | w | , | w ′ | = | w | , c ( w ′ ) − c ( w ′ ) = c ( w ) − c ( w ) we have Mor( u ⊗ w ′ , u ⊗ w ′ ) = Mor( u ⊗ w , u ⊗ w ) . Proof.
The equality (3.2) can be also expressed as u ⊗ u = u ⊗ u , so it is equivalentto 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 showthe equality for w = w ′ = ∅ . 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 inMor( u ⊗ w , u ⊗ w ′ ), and Mor( u ⊗ w ′ , u ⊗ w ), which implies the desired equality. (cid:3) Consider H ⊆ O + N ( F ). It is easy to check that the tensor complexification H ˜ × ˆ Z k is a globally colourized quantum group with degree of reflection k for every k ∈ N .In the following theorem, we prove the converse for k = 0. Theorem 3.15.
Consider G ⊆ U + ( F ) with F ¯ F = c N , c ∈ R . Then G isglobally colourized with zero degree of reflection if and only if G = H ˜ × ˆ Z , where H = G ∩ O + ( F ) .Proof. We denote by u , v , z the fundamental representations of G , H , and ˆ Z k ,respectively. The quantum group H is the quantum subgroup of G defined by therelation v = v . As mentioned above, the right-left implication is clear since v ij commute with z , so u ij u kl = v ij zz ∗ v kl = z ∗ v ij v kl z = u ij u kl . Now, let us prove the left-right implication. First, we show that there is a sur-jective ∗ -homomorphism α : C ( G ) → C ( H ˜ × ˆ Z ) ⊆ C ( H ) ⊗ max C ∗ ( Z )mapping u ij u ′ ij := v ij z . 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 f ( u ′ ij ) = f ( v ij z ) = f ( v ij ) z l = 0. This proves the existence of such a homo-morphism. Its surjectivity is obvious.Now it remains to prove that α is injective and hence is a ∗ -isomorphism. Denoteby ξ ∈ Mor(1 , u ⊗ u ) ⊆ C N ⊗ C N the tensor with entries ξ ij = √ Tr( F ∗ F ) F ji , whichis normalized so that ξ ∗ ξ = 1. We construct a ∗ -homomorphism β : C ( H ) ⊗ max C ∗ ( Z ) → M ( C ( G )) LUING COMPACT MATRIX QUANTUM GROUPS 17 mapping z z ′ := (cid:18) y (cid:19) , v ij v ′ ij := (cid:18) u ij u ij (cid:19) , y := ξ ∗ ( u ⊗ u ) ξ. To prove the existence of such a homomorphism, we need the following.Using the fact that ξξ ∗ ∈ Mor( u ⊗ u , u ⊗ u ) = Mor( u ⊗ u, u ⊗ u ) (the equalityfollows from global colourization thanks to Proposition 3.14), we derive yy ∗ = ξ ∗ ( u ⊗ u ) ξξ ∗ ( u ∗ ⊗ u ∗ ) ξ = ξ ∗ ξξ ∗ ( uu ∗ ⊗ uu ∗ ) ξ = 1and similarly y ∗ y = 1. From this, we can also deduce z ′ z ′∗ = z ′∗ z ′ = 1.Using the fact that 1 N ⊗ ξ ∗ ∈ Mor( u ⊗ u ⊗ u , u ) = Mor( u ⊗ u ⊗ u , u ), wederive u y = (1 N ⊗ ξ ∗ )( u ⊗ u ⊗ u )(1 N ⊗ ξ ) = u and similarly yu = u . This allows us to see that v ′ ij z ′ = z ′ v ′ ij = u ij .Now, it only remains to show that all relations of the generators v ij are satisfiedby 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 F ¯ F = c N to derive v ′ := (1 ⊗ F ) (cid:18) u ¯ u (cid:19) (1 ⊗ F − ) = (cid:18) F ¯ F u ¯ F − F − F ¯ uF − (cid:19) = (cid:18) uu (cid:19) = v ′ . For the second part, take any f ∈ I G . Assume it is Z -homogeneous of degree i .Then we have f ( v ′ ij ) = f ( u ij z ′∗ ) = f ( u ij ) z ′− i = 0 . This concludes the proof of existence of β . Now, noticing that β ◦ α is the embed-ding of C ( G ) into diagonal matrices over C ( G ), we see that α must be injective. (cid:3) Remark 3.16.
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 ˜ × ˆ Z k ⊆ G ⊆ H ˜ × ˆ Z . However, we were unable to prove the inclusion G ⊆ H ˜ × ˆ Z k . This problem isactually equivalent to proving a stronger version of Proposition 3.14: Consider G globally colourized with degree of reflection k . Taking w , w ∈ W with | w | = | w | ,does it hold that Mor(1 , u ⊗ w ) = Mor(1 , u ⊗ w ) whenever c ( w ) ≡ c ( w ) modulo k ?3.5. Tensor complexification.
In this section, we study the tensor complexifica-tion with respect to representation categories and algebraic relations.
Theorem 3.17.
Consider a compact matrix quantum group G = ( C ( G ) , v ) , k ∈ N .Denote by z the generator of C ∗ ( Z k ) and by u := vz the fundamental representationof G ˜ × ˆ Z k . We have the following characterizations of G ˜ × ˆ Z k . (1) The ideal I G ˜ × ˆ Z k is the Z k -homogeneous part of I G . That is, I G ˜ × ˆ Z k = { f ∈ I G | f i ∈ I G for every i ∈ Z k } . (2) The representation category of G ˜ × ˆ Z k looks as follows Mor( u ⊗ w , u ⊗ w ) = ( Mor( v ⊗ w , v ⊗ w ) if c ( w ) − c ( w ) is a multiple of k, { } otherwise. (3) G ˜ × ˆ Z k is topologically generated by G and ˆ Z k . More precisely, G ˜ × ˆ Z k = h G, E ˜ × ˆ Z k i , where E denotes the trivial group of the appropriate size, so E ˜ × ˆ Z k is the quantum group ˆ Z k with the representation z ⊕ · · · ⊕ z = z N .Proof. For (1), we can express f ( u ij , u ∗ ij ) = f ( v ij z, z ∗ v ∗ ij ) = X l ∈ Z k f l ( v ij z, z ∗ v ∗ ij ) = X l ∈ Z k f l ( v ij , v ∗ ij ) z l , where f = P l f l is the decomposition of into the homogeneous components f l ofdegree 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 ˜ × ˆ Z k .Conversely, if there is some l ∈ Z k such that f l I G , then f ( v ij z, z ∗ v ∗ ij ) = 0 andhence f I G ˜ × ˆ Z k .For (2), first we prove that Mor(1 , u ⊗ w ) = Mor(1 , v ⊗ w ) if c ( w ) ∈ k Z . Indeed, wehave u ⊗ w = ( vz ) ⊗ w = z c ( w ) v ⊗ w = v ⊗ w . Secondly, the fact that Mor(1 , u ⊗ w ) = { } if c ( w ) k Z follows from Lemma 3.10.To prove (3), note that the category corresponding to G ˜ × ˆ Z k (given by (2)) isindeed the intersection of the category C G and the category C E ˜ × ˆ Z k , whose morphismspaces are given byMor(( z N ) ⊗ w , ( z N ) ⊗ w ) = ( C N if c ( w ) − c ( w ) is a multiple of k, { } otherwise . (cid:3) Remark 3.18.
An alternative proof of the proposition above could go as follows.One can easily see that the Z k -extended category associated to G × ˆ Z k looks asfollows C G × ˆ Z k ( w , w ) = ( C G ( w ′ , w ′ ) if t ( w ) − t ( w ) is a multiple of k, { } otherwise,where w ′ , w ′ ∈ W are created from w , w ∈ W k mapping , ,
7→ ∅ ,
7→ ∅ and by t ( w ) we mean the number of white triangles minus the number ofblack triangles in w (which is a well-defined element of Z k ). The item (2) of theproposition then follows from Proposition 3.4. Remark 3.19.
As a consequence of Theorem 3.17, we have that( G ˜ × ˆ Z k ) ˜ × ˆ Z l = G ˜ × ˆ Z lcm( k,l ) A direct proof of this statement was formulated already in [TW17, Proposition 8.2].
Lemma 3.20.
Let G ⊆ U + ( F ) be a quantum group with degree of reflection k .Denote by v its fundamental representation. Consider l ∈ N and denote by z thegenerator of C ∗ ( Z l ) . Then z k ∈ C ( G ˜ × ˆ Z l ) for every l . Consequently, z nk ∈ C ( G ˜ × ˆ Z l ) for every n ∈ N , where k := gcd( k, l ) .Proof. From Proposition 3.11, we can find a vector ξ ∈ Mor(1 , v ⊗ w ) with c ( w ) = k and k ξ k = 1. Recall that C ( G ˜ × ˆ Z l ) is generated by the elements v ij z and that v ij commute with z , so C ( G ˜ × ˆ Z l ) ∋ ξ ∗ ( vz ) ⊗ w ξ = ξ ∗ v ⊗ w ξ z c ( w ) = z k . Consequently, z nk ∈ C ( G ˜ × ˆ Z k ) for every n and obviously { z nk } n ∈ N = { z nk } n ∈ N . (cid:3) LUING COMPACT MATRIX QUANTUM GROUPS 19
Proposition 3.21.
Let G ⊆ U + ( F ) be a quantum group with degree of reflection k .Consider a number l ∈ N . Then G ˜ × Z l ≃ G × Z l if and only if k is coprime with l .Proof. Assume we have G ˜ × ˆ Z l ≃ G × ˆ Z l . Suppose d is a divisor of both k and l .Then we must have also G ˜ × ˆ Z d ≃ G × ˆ Z d . But from Lemma 3.10, we have that G ˜ × ˆ Z 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 ˜ × ˆ Z l ) ⊆ C ( G ) ⊗ max C ∗ ( Z l ) since this already implies the equality of the C*-algebras. Thisfollows directly from Lemma 3.20. (cid:3) Remark 3.22. If l is not coprime with k , but l := l/ gcd( k, l ) is coprime with k ,we can use Remark 3.19, Lemma 3.10, and Proposition 3.21 to obtain G ˜ × Z l = ( G ˜ × Z gcd( k,l ) ) ˜ × Z l = G ˜ × Z l ≃ G × Z l . Finally, we are going to characterize irreducible representations of the tensorcomplexification. Note that the irreducibles of the standard tensor product G × ˆ Z l (or G × H in general) was obtained already by Wang in [Wan95b]. Proposition 3.23.
Let G ⊆ U + ( F ) be a quantum group with degree of reflection k .Consider arbitrary l ∈ N . Then G ˜ × ˆ Z l has the following complete set of mutuallyinequivalent irreducible representations (3.3) { u α z ki + d α | α ∈ Irr
H, i = 0 , . . . , l − } , where l = 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 Propo-sition 3.11. This means that the algebra O ( G ) is Z k -graded assigning degree oneto v ij and degree minus one to v ∗ ij , where v is the fundamental representationof G . Consequently, the entries of any irreducible representation u α , α ∈ Irr G are Z k -homogeneous of some degree d α (recall Sect. 1.4).By Theorem 3.17, I G ˜ × ˆ Z l is the Z l -homogeneous part of I G . Consequently, I G ˜ × ˆ Z l is Z lcm( k,l ) -homogeneous and O ( G ˜ × ˆ Z l ) is Z lcm( k,l ) -graded. However, this time thedegree is computed with respect to the variables u ij := v ij z .The irreducible representations of the standard tensor product G × ˆ Z l were de-scribed by Wang in [Wan95b]. 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 elementsof C ( G ˜ × ˆ Z l ) ⊆ C ( G × ˆ Z l ). That is, we need to determine all the pairs ( α, n ) suchthat u α z n is a matrix with entries in C ( G ˜ × ˆ Z l ) ⊆ C ( G × ˆ Z l ).We first prove that every irreducible of G ˜ × ˆ Z l is equivalent to one from Eq. (3.3).As we just mentioned, it must be of the form u α z n for some α, n . Since it isa representation of G ˜ × ˆ Z 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 := gcd( k, l ). Thus, we must have n = k i + d α for some i ∈ Z .Obviously, { z k i + d α } i ∈ Z = { z ki + d α } l i =1 .For the converse inclusion, we need to show that the entries of u α z ki + d α areelements of C ( G ˜ × ˆ Z 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 ) isa subrepresentation of u ⊗ w = v ⊗ w z c ( w ) . Hence, it is a representation of G ˜ × ˆ Z l . From Lemma 3.20, it follows that also u α z ki + c ( w ) is a representation of G ˜ × ˆ Z l . Since d α ≡ c ( w ) modulo k , this is equivalent to considering representations u α u ki + d α . (cid:3) Free complexification.
The goal of this section is to characterize the rep-resentation categories of the free complexifications, that is, the quantum groups H ˜ ∗ ˆ Z l . For the free complexification, we do not have many results yet even in theeasy case. In [TW17], the two-coloured categories corresponding to free complexifi-cations of free orthogonal easy quantum groups are provided. For us, the motivatingresult is [GW19, Proposition 4.21] linking the free complexification by Z with thecategory Alt C generated by alternating coloured partitions. This proposition wasproven with the help of categories of partitions with extra singletons describing thefree product with Z and the functor F describing the gluing procedure. Also here,we will make use of the Z l -extended representation categories describing the freeproduct H ∗ ˆ Z l and then we will glue the factors and apply Proposition 3.4 to findthe corresponding representation category. An interesting result is that the freecomplexification H ˜ ∗ ˆ Z l actually does not depend on the number l unless the degreeof reflection of H equals to one. Definition 3.24.
A monomial of even length of the form x i j x ∗ i j x i j x ∗ i j · · · ∈ C h x ij , x ∗ ij i , where the variables with and without star alternate, is called alternat-ing . A linear combination of alternating monomials, where either all start withnon-star variable or all start with star variable, is called an alternating polyno-mial . A quantum group G is called alternating if I G is generated by alternatingpolynomials.Considering a compact matrix quantum group G ⊆ U + ( F ) with unitary funda-mental representation u , recall the notation u := u , u := F ¯ uF − . So, the relationsof 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 formalso using the alternative ideal. Lemma 3.25.
Let H be a compact matrix quantum group, k, l ∈ N such that gcd( k, l ) = 1 (using the convention gcd(0 , k ) = k ). Then we have H ˜ ∗ ˆ Z = ( H ˜ × ˆ Z k ) ˜ ∗ ˆ Z l . Proof.
We denote by v , z , r , s the fundamental representations of H , ˆ Z , ˆ Z k , and ˆ Z l ,respectively. We need to find a ∗ -isomorphism C ( H ˜ ∗ ˆ Z ) → C (( H ˜ × ˆ Z k )˜ ∗ ˆ Z l ) mapping v ij z v ij sr .First, we see that there exists a ∗ -homomorphism α : C ( H ) ∗ C C ∗ ( Z ) → ( C ( H ) ⊗ max C ∗ ( Z k )) ∗ C C ∗ ( Z l )mapping v ij v ij , z sr. since sr is a unitary.This ∗ -homomorphism then restricts to a surjective ∗ -homomorphism of the formwe are looking for. It remains to prove that it is injective. To prove this, weconstruct a ∗ -homomorphism β : ( C ( H ) ⊗ max C ∗ ( Z k )) ∗ C C ∗ ( Z l ) → M d (cid:0) C ( H ) ∗ C C ∗ ( Z ) (cid:1) LUING COMPACT MATRIX QUANTUM GROUPS 21 mapping v ij v ij
0. . .0 v ij , s
11 . . . 1 , r z ∗ z , where d = 1 is some common divisor of k and l .We can check that the images satisfy all the defining relations between the gen-erators, so such a homomorphism indeed exists. Now, we can see that β ◦ α isinjective, so α must be injective. Thus, the restriction of α we are interested in isalso injective. (cid:3) Proposition 3.26.
Let H be a compact matrix quantum group and l ∈ N . Thenthe Z l -extended category C H ∗ ˆ Z l is generated by the collection C ( ι ( w ) , ι ( w )) := C H ( w , w ) , where ι : W → W l is the injective homomorphism mapping , . Moreover, we have the following inductive description. If w ∈ W k containsno triangles, i.e. w = ι ( w ′ ) for some w ′ ∈ W , then C H ∗ ˆ Z l ( ∅ , w ) = C H ( ∅ , w ′ ) . Otherwise, C H ∗ ˆ Z l ( ∅ , w ) = R [ w ] ( ξ ⊗ · · · ⊗ ξ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w = w w · · · w l ξ i ∈ C H ∗ ˆ Z l ( ∅ , w i ) , i = 1 , . . . , l − ξ l ∈ C H ∗ ˆ Z l ( ∅ , w l w ) . Proof.
Let C be the Z l -extended category generated by C . Then the associatedquantum group G = ( C ( G ) , v ⊕ z ) is a quantum subgroup of U + ( F ) ∗ ˆ Z l defined bythe relations of H for v and no relations for z (except for zz ∗ = z ∗ z = 1 = z l ). Butthis is exactly the free product H ∗ ˆ Z l . Now, it remains to prove that C is given bythe above described recursion.The inclusion ⊇ follows from the fact that C has to be closed under the categoryoperations. To check the inclusion ⊆ , it is enough to check that the right-hand sidedefines 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 2.3and 2.4. Checking this is straightforward using induction. Nevertheless it maybecome a bit lengthy to check all the details. We will do it here for the rotationand tensor product.So, denote the sets given by the inductive description by ˜ C . If we take words w without triangles, that is, w = ι ( w ′ ), then the sets ˜ C ( ∅ , w ) = C H ( ∅ , w ′ ) are closedunder all the operations since C H is a category. To show closedness under rotationsin general, we do an induction on the length of the word w . So, consider an element ξ ∈ ˜ C ( ∅ , w ) with w = w w · · · w l , so it is of the form ξ = R [ w ] ( ξ ⊗ · · · ⊗ ξ l ).First, suppose that w l is not empty and denote by x its last letter. Then wedirectly have Rξ = R [ w ]+1 ( ξ ⊗ · · · ⊗ ξ l ) ∈ ˜ C ( ∅ , Rw ). For the case w l = ∅ , the lastletter of w is a triangle. So, we need to check that ˜ C ( ∅ , w w · · · w l − ) ∋ ξ =( R [ w ] ξ l ) ⊗ ξ ⊗ · · · ⊗ ξ l − . This is true thanks to the fact that R [ w ] ξ l ∈ ˜ C ( ∅ , w )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: η ⊗ ξ = R [ w ′ ] ( ξ ⊗ R − [ w ′ ] η ).So, assume w = w w · · · w l , so ξ is of the form ξ = R [ w ] ( ξ ⊗ · · · ⊗ ξ l ) ∈ ˜ C ( ∅ , w w · · · w l ). Then we have ξ l ∈ ˜ C ( ∅ , w l w ) by assumption, R [ w ] ξ l ∈ ˜ C ( ∅ , w w l ) ˜ C closed u. rotations, R [ w ] ξ l ⊗ η ∈ ˜ C ( ∅ , w w l w ′ ) by induction,˜ ξ l = R − [ w ] ( R [ w ] ξ l ⊗ η ) ∈ ˜ C ( ∅ , w l w ′ w ) ˜ C closed u. inv. rot., ξ ⊗ η = R [ w ] ( ξ ⊗ · · · ⊗ ξ l − ⊗ ˜ ξ l ) ∈ ˜ C ( ∅ , w w · · · w l w ′ ) by definition of ˜ C . (cid:3) Lemma 3.27.
We can arrange the recursion of the above proposition in such away that the words w , . . . , w l − contain no triangles, so we have C H ∗ ˆ Z l ( ∅ , w ) = R [ w ] ( ξ ⊗ · · · ⊗ ξ l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w = w w · · · w l ξ i ∈ C H ( ∅ , w ′ i ) , i = 1 , . . . , l − ξ l ∈ C H ∗ ˆ Z l ( ∅ , w l w ) . Proof.
We prove this by induction. Take an arbitrary word w ∈ W l and supposethat the above description works for any shorter word. Now consider an element ξ ∈ C H ∗ ˆ Z l ( ∅ , w ), so it is of the form ξ = R [ w ] ( ξ ⊗ · · · ⊗ ξ l ) corresponding to thedecomposition w = w w · · · w l . Suppose now that w i contains some trianglesfor some i ∈ { , . . . , l − } . By induction hypothesis, we can write ξ i = R [ a ] ( η ⊗· · ·⊗ η l ) corresponding to w i = a a · · · a l , where a , . . . , a l − contain no triangles,so η i ∈ C H ( ∅ , a ′ i ). But this means that we can write also ξ = R [ w ··· w i − a ] ( η ⊗ · · · ⊗ η l − ⊗ ˜ η l ) , where ˜ η l = R − [ a ] ( R [ a ] η l ⊗ ξ i +1 ⊗ · · · ⊗ ξ l ⊗ R − [ w ] ξ ⊗ · · · ⊗ R − [ w i − ] ξ i − ) ∈ C H ∗ ˆ Z l ( ∅ , a l w i +1 · · · w l w w · · · w i − a ) . (cid:3) In the following theorem, we describe the representation category of the freecomplexification. In the formulation, we use the following notation. Given anelement w ∈ W or w ∈ W l , we use negative powers to indicate the colour inversion,that is, w − j = ¯ w j . For example, ( ) − = ( ) = . Theorem 3.28.
Let H be a compact matrix quantum group with degree of reflection k = 1 . Then all H ˜ ∗ ˆ Z l coincide for all l ∈ N \ { } . The ideal I H ˜ ∗ ˆ Z l is generated bythe alternating polynomials in I H . The representation category C H ˜ ∗ ˆ Z l is a (wide)subcategory of the representation category C H generated by the sets C ( ∅ , ( ) j ) := C H ( ∅ , ( ) j ) , j ∈ Z . This also holds if k = 1 and l = 0 .Proof. Let I ⊆ C h x ij , x ∗ ij i be the ideal generated by the alternating polynomialsin I H . Denote by u ij = v ij z the fundamental representation of H ˜ ∗ ˆ Z l . To provethat I ⊆ I H ˜ ∗ ˆ Z l , take any alternating polynomial f ∈ I H . If all monomials in f start with a non-star variable, we have f ( v ij z ) = f ( v ij ) = 0; if all monomials startwith a star variable, then f ( v ij z ) = z ∗ f ( v ij ) z = 0. In both cases, we have proventhat f ∈ I H ˜ ∗ ˆ Z l . The opposite inclusion I ⊇ I H ˜ ∗ ˆ Z l will follow from the statement LUING COMPACT MATRIX QUANTUM GROUPS 23 about representation categories as all the relations corresponding to the elementsof C are alternating.Note that it is enough to prove the statement for k = 1 and l = 0. Indeed, for k = 1 and l = 0, we have by Lemma 3.25 that H ˜ ∗ ˆ Z = H ˜ ∗ ˆ Z k . For k = 1, l = 0we use Lemma 3.25 to express H ˜ ∗ ˆ Z = ( H ˜ × ˆ Z ) ˜ ∗ Z l . Since c (( ) j ) = 0 ∈ Z forevery j , we have C H ( ∅ , ( ) j ) = C H ˜ × ˆ Z ( ∅ , ( ) j ).So, let C be the two-coloured representation category generated by C . We needto prove that C H ˜ ∗ ˆ Z l ( ∅ , w ) = C ( ∅ , w ) for every w ∈ W . In order to do that, we willuse Proposition 3.4, whose statement can be, in this case, formulated as(3.4) C H ˜ ∗ ˆ Z l ( ∅ , w ) = C H ∗ ˆ Z l ( ∅ , ˜ w ) , where ˜ w ∈ W l is the glued version of w ∈ W .Let us start with the easier inclusion ⊇ . Since C H ˜ ∗ ˆ Z l is a category, it is enoughto show that C H ˜ ∗ ˆ Z l ( ∅ , w ) ⊇ C ( ∅ , w ) for every w = ( ) j , j ∈ Z . Note that the gluedversion of w is in this case ˜ w = ( ) j = ( ) j . Combining Proposition 3.26 andEquation (3.4), we have C ( ∅ , ( ) j ) = C H ( ∅ , ( ) j ) = C H ∗ ˆ Z k ( ∅ , ( ) j ) = C H ˜ ∗ ˆ Z l ( ∅ , ( ) j ) . We will prove the opposite inclusion ⊆ by induction on the length of w . Takesome ξ ∈ C H ˜ ∗ ˆ Z l ( ∅ , w ) = C H ∗ ˆ Z l ( ∅ , ˜ w ) . Suppose ξ = 0. According to Lemma 3.27, we can assume that ˜ w = w w · · · w l ,where w , . . . , w l − contain no triangles, and then ξ = R w ( ξ ⊗ · · · ⊗ ξ l ) with ξ i ∈ C H ( ∅ , w ′ i ) and ξ l ∈ C H ∗ ˆ Z l ( ∅ , w l w ). Since ˜ w is the glued version of w , this meansthat in all the words w , . . . , w l − the colours alternate (two consecutive whitesquares would necessarily have a white triangle between them, two consecutiveblack squares would have = l − between them). Moreover, since we assume ξ i = 0, we must have c ( w ′ i ) ∈ k Z and, since k = 1, this means that the w i ’s are ofeven length. So, w i = ( ) j i , w ′ i = ( ) j i .Finally note that if we delete or from some word w , its glued version willbe given by deleting resp. . In particular, denote by ˆ w the element w afterdeleting all the subwords w ′ , . . . , w ′ l − . Its glued version is then w l w l = w w l .Using the induction hypothesis, this finishes the proof as we have ξ = R w ( ξ ⊗ · · · ⊗ ξ l )with ξ i ∈ C H ( ∅ , w ′ i ) = C H ( ∅ , ( ) j i ) = C ( ∅ , ( ) j i ) for i = 1 , . . . , l − R w ξ l ∈ C H ∗ ˆ Z ( ∅ , w w l ) = C H ˜ ∗ ˆ Z ( ∅ , ˆ w ) = C ( ∅ , ˆ w ) . (cid:3) We may ask what happens if we iterate those free complexifications. The follow-ing statement was again already formulated in [TW17]; however, without a proof.(Note that it generalizes Lemma 3.25 dropping the assumption gcd( k, l ) = 1.) Proposition 3.29.
Let H be a compact matrix quantum group, k, l ∈ N \ { } .Then ( H ˜ × ˆ Z k ) ˜ ∗ ˆ Z l = ( H ˜ ∗ ˆ Z k ) ˜ ∗ ˆ Z l = H ˜ ∗ Z . Proof.
The second equality follows directly from Theorem 3.28 – we see that iter-ating the operation on the categories for the second time cannot change it since C H ( ∅ , ( ) j ) = C H ˜ ∗ ˆ Z k ( ∅ , ( ) j ). For the first equality, we use, in addition, Theo-rem 3.17. Since c (( ) j ) = 0, we have C H ˜ × ˆ Z k ( ∅ , ( ) j ) = C H ( ∅ , ( ) j ). (cid:3) Again, we can ask in what situations does it happen that the glued free product H ˜ ∗ ˆ Z l is isomorphic to the standard one. Obviously, the necessary condition isthat H has degree of reflection one since H ˜ ∗ ˆ Z l ≃ H ∗ ˆ Z l implies H ˜ × ˆ Z l ≃ H × ˆ Z l and here we can use Proposition 3.21. We can formulate the converse in the caseof globally-colourized quantum groups H (in particular, if H ⊆ O + ( F )). Proposition 3.30.
Let H be a globally colourized compact matrix quantum groupwith degree of reflection one. Then H ˜ ∗ ˆ Z k ≃ H ∗ ˆ Z k for every k ∈ N .Proof. Denote by v the fundamental representation of H and by z the generatorof C ∗ ( Z k ). Again, it is enough to show that we have z ∈ C ( H ˜ × ˆ Z k ) ⊆ C ( H ) ⊗ max C ∗ ( Z k ) since this already implies the equality of the C*-algebras. From Proposi-tion 3.11, we can find a vector ξ ∈ Mor(1 , v ⊗ w ) with c ( w ) = 1 and k ξ k = 1. Since H is globally colourized, we have Mor(1 , v ⊗ w ) = Mor(1 , v ⊗ ˜ w ), where ˜ w = . . . , | ˜ w | = | w | . For such a word, we have ( vz ) ⊗ ˜ w = ( v z )( z ∗ v )( v z ) · · · ( v z ) = v ⊗ ˜ w z , so ξ ∗ ( vz ) ⊗ ˜ w ξ = ξ ∗ ( v ⊗ ˜ w z ) ξ = ξ ∗ ξ z = z. (cid:3) Free complexification of orthogonal quantum groups.
In this section,we will study more in detail the free complexification H ∗ ˆ Z k with H ⊆ O + ( F ).Recall that we define O + ( F ) ⊆ U + ( F ) only for F satisfying F ¯ F = c N for some c ∈ R . We will use this assumption in the whole section. Definition 3.31.
A quantum group G = ( C ( G ) , u ) ⊆ U + ( F ) with F ¯ F = c N iscalled invariant with respect to the colour inversion if the map u ij [ F ¯ uF − ] ij extends to a ∗ -isomorphism.Let us explain a bit this definition. First of all, note that the required ∗ -homo-morphism maps u ij u ij , u ij u ij . Indeed, the first assignment is exactly the definition. For the second one, we have u ij = [ F ¯ uF − ] ij [ F ¯ F u ¯ F − F − ] ij = u ij thanks to the assumption F ¯ F = c N . In the Kac case F = 1 N , the homomor-phism maps u ij u ∗ ij . But let us stress that for general elements of C ( G ) thehomomorphism does not coincide with the ∗ -operation (simply because the ∗ is nota homomorphism).Secondly, we have the following alternative formulations. Proposition 3.32.
Consider G = ( C ( G ) , u ) ⊆ U + ( F ) with F ¯ F = c N . Then thefollowing are equivalent. (1) C ( G ) has an automorphism u ij ↔ u ij . That is, G is invariant w.r.t. thecolour inversion. (2) I G is invariant w.r.t. x ij ↔ x ij . More precisely, we mean one of the fol-lowing equivalent conditions. (a) I G is invariant w.r.t. the ∗ -homomorphism mapping x ij x ij LUING COMPACT MATRIX QUANTUM GROUPS 25 (b) I G is invariant w.r.t. the homomorphism mapping x ij x ij and x ij x ij . (3) C G is invariant w.r.t. ↔ . That is, C G ( ¯ w , ¯ w ) = C G ( w , w ) .Proof. The equivalence (1) ⇔ (2a) follows from the universal property of C ( G ).For (1) ⇒ (3), take T ∈ C G ( w , w ), so T u ⊗ w = u ⊗ w T . Applying the auto-morphism, we get T u ⊗ ¯ w = u ⊗ ¯ w T , so T ∈ C G ( ¯ w , ¯ w ).For (3) ⇒ (2b), we use the Tannaka–Krein, namely the fact that I G is spannedby the relations of the form T x ⊗ w = x ⊗ w T . Those relations are invariant withrespect to the homomorphism x x , x x since this homomorphism maps x w x ¯ w . Consequently, the whole ideal I G must be invariant with respect to thishomomorphism.The implication (2b) ⇒ (1) again follows from the universal property of C ( G ).We get that u ij u ij , u ij u ij extends to a homomorphism C ( G ) → C ( G ). Usingthe assumption F ¯ F = c N , we can show this actually must be a ∗ -homomorphism. (cid:3) As an example, note that all the universal unitary quantum groups U + ( F ) with F ¯ F = c N have this property. In addition, any quantum group G ⊆ O + ( F ) hasthis property. Theorem 3.33.
Consider G ⊆ U + ( F ) with F ¯ F = c N . Then G is alternatingand invariant with respect to the colour inversion if and only if it is of the form G = H ˜ ∗ ˆ Z , where H = G ∩ O + ( F ) .Proof. The right-left implication follows from Theorem 3.28: The fact that H ˜ ∗ ˆ Z is alternating is precisely the statement of Theorem 3.28. As we mentioned above, H ⊆ O + ( F ) is surely invariant with respect to the colour inversion. Accord-ing to Proposition 3.32, 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 ˜ ∗ ˆ Z according to Theo-rem 3.28. Consequently, also H ˜ ∗ ˆ Z must be invariant with respect to the colourinversion.In order to prove the left-right implication, we construct a surjective ∗ -homo-morphism α : C ( G ) → C ( H ˜ ∗ Z )mapping u ij u ′ ij := v ij z . To prove that such a homomorphism exists, takeany alternating element f ∈ I G . Since H ⊆ G , we have f ( v ij ) = 0. We needto prove that f ( u ′ ij ) = 0. If all terms of f start with a non-star variable, then f ( u ′ ij ) = f ( v ij z ) = f ( v ij ) = 0; if all terms start with a star variable, then f ( u ′ ij ) = z ∗ f ( v ij ) z = 0.It remains to prove that α is injective. To do that, we define a ∗ -homomorphism β : C ( H ) ∗ C C ∗ ( Z ) → M ( C ( G ))mapping v ij v ′ ij := (cid:18) u ij u ij (cid:19) , z z ′ := (cid:18) (cid:19) . We immediately see that indeed z ′ z ′∗ = z ′∗ z ′ = 1. In exactly the same way as in theproof of Theorem 3.15, we also prove that v ′ := (1 ⊗ F )¯ v ′ (1 ⊗ F − ) = v ′ =: v ′ .Finally, take f ∈ I G and, for convenience, use the representation in variables u ij and u ij . Suppose f ( x ij , x ij ) is alternating such that all variables start with x ij .We have f ( v ′ ij , v ′ ij ) = (cid:18) f ( u ij , u ij ) 00 f ( u ij , u ij ) (cid:19) = 0 , where f ( u ij , u ij ) = 0 directly by f ∈ I G and f ( u ij , u ij ) by invariance under thecolour inversion.Since obviously β ◦ α is injective, we have proven that α is a ∗ -isomorphism. (cid:3) Considering a quantum group H = ( C ( H ) , v ) ⊆ O + ( F ), we have H ˜ × ˆ Z k ⊆ H ˜ × ˆ Z ⊆ H ˜ ∗ ˆ Z . 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 ˜ ∗ ˆ Z ) as we may not have z ∈ C ( H ˜ ∗ ˆ Z ). Proposition 3.34.
Consider H ⊆ O + ( F ) . Then H ˜ × ˆ Z is a quantum subgroup of H ˜ ∗ ˆ Z given by the relation (3.5) u ij u kl = u ij u kl . For k ∈ N , H ˜ × ˆ Z k is a quantum subgroup of H ˜ × ˆ Z with respect to the relation (3.6) u i j · · · u i k j k = u i j · · · u i k j k . Before proving the statement, note that Relations (3.5) and (3.6) correspond tothe two-coloured partitions ⊗ and ⊗ k , respectively. Hence, those are exactlythe same relations that were used to construct the quantum groups H ˜ × ˆ Z and H ˜ × ˆ Z k in [Gro18, Theorem 5.1]. Proof.
Relation (3.5) is the relation of the global colourization (see Def. 3.13) andit is obviously satisfied in H ˜ × ˆ Z . We just need to show that imposing this relationis enough. By Corollary 2.9 and Theorem 3.17, the ideal I H ˜ × ˆ Z is generated byrelations of the form u ⊗ w ξ = ξ , where ξ ∈ C H ˜ × ˆ Z ( ∅ , w ) = C H ( ∅ , l ), c ( w ) = 0, l := | w | /
2. In H ˜ ∗ ˆ Z , we have a relation of the form u ⊗ ( ) l ξ = ξ . The formerrelation can surely be derived from the latter one and Relation (3.5) since it isobtained 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 , thenRelations (3.6) say v i j · · · v i k j k z k = v i j · · · v i k j k z − k , which is surely satisfied in H ˜ × ˆ Z k . For the converse, the ideal I H ˜ × ˆ Z k is generated by relations of the form u ⊗ w ξ = ξ , where ξ ∈ C H ˜ × ˆ Z ( ∅ , w ) = C H (0 , l ), where c ( w ) is a multiple of 2 k and l := | w | /
2. Again, this relation can be derived from u ⊗ ( ) l ξ = ξ using Rel. (3.5)to permute colours and Rel. (3.6) to swap colours of k consecutive white points toblack or vice versa. (cid:3) Free complexification and partition categories.
In sections 3.4–3.5, wegeneralized results that were formulated in the language of categories of partitionsin [Gro18]. In contrast, Sections 3.6–3.7 were rather new. Hence, it is interestingto 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 originaldefinition of categories of partitions as defined in [BS09], we refer to the survey[Web17]. See also [TW18] for the definition of two-coloured categories of partitionsand [GW20] for the definition of linear categories of partitions. Also see [GW19] for
LUING COMPACT MATRIX QUANTUM GROUPS 27 the definition of the two-coloured category Alt C associated to a given non-colouredcategory C . Alternatively, everything is summarized in the thesis [Gro20]. Proposition 3.35.
Let K ⊆ P N -lin be a linear category of partitions and denoteby H ⊆ O + N the corresponding quantum group. Then H ˜ ∗ ˆ Z ⊆ U + N corresponds tothe category Alt K . Moreover, the following holds. (1) If = p ∈ K (0 , l ) for some l odd, then H ˜ ∗ ˆ Z k corresponds to the category h Alt K , ˜ p ⊗ k i , where ˜ p is the partition p with colour pattern · · · . (2) If K (0 , l ) = { } for all l odd, then H ˜ ∗ ˆ Z k = H ˜ ∗ ˆ Z for all k ∈ N .Proof. The base statement that H ˜ ∗ ˆ Z corresponds to Alt K follows directly fromTheorem 3.28. By Proposition 3.12, the distinction of the cases correspond to thesituation that either (1) H has degree of reflection one or (2) H has degree ofreflection two. So, item (2) is also contained in Theorem 3.28.For item (1), denote by w := · · · the word of length l with alternating colours.Since H has degree of reflection one, we have H ˜ ∗ ˆ Z m ≃ H ∗ ˆ Z m for every m ∈ N by Proposition 3.30. We can actually prove this directly repeating the proofof Prop. 3.30: Denote by v the fundamental representation of H and by z thefundamental representation of ˆ Z m . Then since we have v ⊗ l ξ p = ξ p , we must have C ( H ˜ × ˆ Z m ) ∋ ξ ∗ p u ⊗ w ξ p = ξ ∗ p ( v ⊗ l z ) ξ p = z k ξ p k . Now, H ˜ ∗ ˆ Z k is just a quantum subgroup of H ˜ ∗ ˆ Z with respect to the relation z k = 1. Note that u ⊗ w k ξ ⊗ kp = ( v ⊗ kl z k ) ξ ⊗ kp = ξ ⊗ kp z k . So, the relation z k = 1 can be written as u ⊗ w k ξ ⊗ kp = ξ ⊗ kp , which is exactly therelation corresponding to ˜ p . (cid:3) Proposition 3.36.
Let C ⊆ P be a category of partitions and denote by G ⊆ O + N the corresponding easy quantum group. Then G ˜ ∗ ˆ Z is a unitary easy quantum groupcorresponding to the category Alt C . (1) If ↑ 6∈ C , then G ˜ ∗ ˆ Z k = G ˜ ∗ ˆ Z for all k ∈ N and it corresponds to thecategory Alt C . (2) If ↑ ∈ C , then G ˜ ∗ ˆ Z k corresponds to the category h Alt C , ⊗ k i .Proof. This is just a reformulation of Proposition 3.35 to the easy case. Note that C contains some partition of odd length if and only if it contains the singleton ↑ . (cid:3) Ungluing
The purpose of this section is to reverse the gluing procedure from Definition 3.1.The motivating result is the one-to-one correspondence formulated in terms of par-tition categories in [GW19, Theorem 4.10]. According to [GW19, Proposition 4.15],the functor F providing this correspondence translates to the quantum group lan-guage exactly in terms of gluing.Recall from Def. 3.1 that given a quantum group G ⊆ U + ( F ) ∗ ˆ Z k with funda-mental representation u = v ⊕ z , we define its glued version to be the quantum group˜ G ⊆ U + ( F ) with fundamental representation ˜ u := vz and underlying C*-algebra C ( ˜ G ) ⊆ C ( G ) generated by the elements v ij z . Definition 4.1.
Consider ˜ G ⊆ U + ( F ), k ∈ N . Then any G ⊆ U + ( F ) ∗ ˆ Z k suchthat ˜ G is a glued version of G is called a Z k -ungluing of ˜ G .In Section 4.1, we are going to study the ungluings in general and show thatthey always exist. Unsurprisingly, an ungluing of a quantum group is not defineduniquely. The ungluings introduced in Section 4.1 are universal, but not particularlyinteresting. In Section 4.2, we are going to study more interesting ungluings of theform G ⊆ O + ( F ) ∗ ˆ Z , which allow us to generalize the above mentioned one-to-onecorrespondence. We formulate the result as Theorem 4.13, which constitutes themain result of this section.4.1. General ungluings.Proposition 4.2.
There exists a Z k -ungluing for every quantum group ˜ G and forevery k ∈ N . Namely, we have the trivial Z k -ungluing ˜ G × E , where E ⊆ ˆ Z k is thetrivial group. Moreover, ˜ G × ˆ Z k is an ungluing of ˜ G whenever k divides the degreeof reflection of G .Proof. The first statement is obvious as we have ˜ G ˜ × E = ˜ G . The second followsfrom Lemma 3.10 as we have ˜ G ˜ × ˆ Z k = ˜ G . (cid:3) Let us denote by ι : C h ˜ x ij , ˜ x ∗ ij i → C h x ij , x ∗ ij , z, z ∗ i the embedding ˜ x ij x ij z .Consider ˜ G ⊆ U + ( F ) and G its ungluing. The fact that ˜ G is a glued version of G is, according to Proposition 3.4, characterized by the equality ˜ I ˜ G := ι ( I ˜ G ) = I G ∩ ι ( C h ˜ x ij , ˜ x ∗ ij i ). Consequently, we have I G ⊇ ˜ I ˜ G for every ungluing G of a quantumgroup ˜ G . Definition 4.3.
Consider ˜ G ⊆ U + ( F ), k ∈ N . Let I G ⊆ C h x ij , x ∗ ij , z, z ∗ i bethe ∗ -ideal generated by ˜ I ˜ G . Put C ( G ) := C ∗ ( C h x ij , x ∗ ij , z, z ∗ i /I G ). Then G :=( C ( G ) , v ⊕ z ) is called the maximal Z k -ungluing of ˜ G . Proposition 4.4.
The maximal Z k -ungluing always exists. That is, consideringthe notation of Definition 4.3, G is indeed a compact matrix quantum group and ˜ G is indeed its glued version.Proof. First of all, note that ˜ I ˜ G contains the relations vv ∗ = v ∗ v = 1 N and v v ∗ = v ∗ v = 1 N , where v = F ¯ vF − . So, if G is well defined, then we must have G ⊆ U + ( F ) ∗ ˆ Z k .To prove that G is well defined, we need to check that I G /I U + ( F ) is a Hopf ∗ -ideal. Since I ˜ G /I U + ( F ) is a Hopf ∗ -ideal, we have that ˜ I ˜ G /I U + ( F ) is a coidealinvariant under the antipode. Consequently, the ideal generated by ˜ I ˜ G /I U + ( F ) must be a Hopf ∗ -ideal.From the construction, it is clear that, if ˜ G has some Z k -ungluing, then G mustbe the maximal one (since we take the smallest possible ideal I G ). But everyquantum group has the trivial ungluing as mentioned in Prop. 4.2. (cid:3) Remark 4.5.
We do not have to know explicitly the whole ideal I ˜ G to computethe maximal ungluing. Consider ˜ G ⊆ U + ( F ) and suppose that it is determined bya set of relations ˜ R . That is, I ˜ G is generated by the coideal ˜ R . Then the maximal Z k -ungluing G of ˜ G is defined by the relations R := ι ( ˜ R ). That is, taking thegenerating relations for ˜ G and exchanging ˜ v ij for v ij z and ˜ v ∗ ij for z ∗ v ∗ ij . LUING COMPACT MATRIX QUANTUM GROUPS 29
Alternatively, we can describe the maximal ungluing by its representation cate-gory. Recall the definition of the gluing homomorphism mapping , .Given a word w ∈ W , the image ˜ w under this homomorphism is called the gluedversion of w by Definition 3.3. If G is a quantum group and ˜ G its glued version,then C ˜ G is a full subcategory of C G according to Proposition 3.4. The full embed-ding is given exactly by the above mentioned gluing homomorphism. Consequently,the maximal ungluing G of some ˜ G should be a quantum group with the minimalrepresentation category containing C ˜ G as a full subcategory. Proposition 4.6.
Consider ˜ G ⊆ U + ( F ) and G ⊆ U + ( F ) ∗ ˆ Z k its maximal Z k -ungluing. Then the Z k -extended representation category C G is generated by thesets C ( ˜ w , ˜ w ) = C ˜ G ( w , w ) , where ˜ w , ˜ w are glued versions of w , w ∈ W . Inaddition, if C G is generated by some ˜ C , then C ˜ G is generated by C ( ˜ w , ˜ w ) =˜ C ( w , w ) .Proof. By Tannaka–Krein duality, I ˜ G is linearly spanned by relations of the form[ T ˜ v ⊗ w − v ⊗ w T ] ji . By definition of the maximal ungluing, the ideal I G is generatedby elements of ˜ I ˜ G , which are exactly the relations [ T u ⊗ ˜ w − u ⊗ ˜ w T ] ji correspondingto the set C .For the second statement, notice that if ˜ C generates C ˜ G , then C must generate C . This follows simply from the fact that gluing of words is a monoid homo-morphism (see also Prop. 3.4). Consequently, by what was already proven, C generates C G . (cid:3) Example 4.7.
As an example, consider the quantum group ˜ G k := O + ( F ) ˜ × ˆ Z k , k ∈ N . By Proposition 3.34, it is the quantum subgroup of O + ( F ) ˜ ∗ ˆ Z = U + ( F )with respect to the relation˜ v i j ˜ v i j · · · ˜ v i k j k = ˜ v i j ˜ v i j · · · ˜ v i k j k , where ˜ v denotes the fundamental representation of ˜ G . We can also take ˜ G := O + ( F ) ˜ × ˆ Z , which is a quantum subgroup of U + ( F ) given by˜ v ij ˜ v kl = ˜ v ij ˜ v kl . Now, take arbitrary l ∈ N . The maximal Z l -ungluing of ˜ G k is a quantum group G k ⊆ U + ( F ) ∗ ˆ Z l with fundamental representation of the form v ⊕ z given by thesame relations if we substitute ˜ v ij by v ij z . v i j zv i j z · · · v i k j k z = z ∗ v i j z ∗ v i j · · · z ∗ v i k j k . The ungluing G is defined by the relation v ij v kl = z ∗ v ij v kl z. We can also represent the relations diagrammatically. The defining relations of O + ( F ) ˜ × ˆ Z k and O + ( F ) ˜ × ˆ Z are ( ) ⊗ k and respectively (see also [TW17, Gro18]).To obtain the maximal ungluing, we can have to put a white triangle after everywhite circle and a black triangle in front of every black circle . So, the definingrelation for G k corresponds to ( ) ⊗ k , for G , we have .We can see that the result is quite something different than simply O + ( F ) × ˆ Z k as we may have hoped. In general, the maximal ungluing never provides any useful decomposition intosmaller pieces since we always have the trivial decomposition inside the maximalone ˜ G × E ⊆ G .4.2. Orthogonal ungluings.
As just mentioned, constructing large unitary unglu-ings G ⊆ U + ( F ) ∗ ˆ Z k may not be very useful. In this subsection, we study ungluingsthat are orthogonal, that is, of the form G ⊆ O + ( F ) ∗ ˆ Z . For the rest of this sub-section, we assume F ¯ F = c N .For a given quantum group G ⊆ O + ( F ) ∗ ˆ Z , we will denote by I G the correspond-ing ideal inside A := C h x ij i ∗ CZ (instead of taking C h x ij , r i or C h x ij , x ∗ ij , r, r ∗ i ).The generator of CZ will be denoted by r . Note that we have to consider the non-standard involution x ∗ ij = [ F − xF ] ij on A . The algebra A is Z -graded (assigningall variables x ij and r degree one). We will denote by ˜ A the even part of A . Then˜ Ar is the odd part of A . Lemma 4.8.
The mapping x ij x ij r extends to an injective ∗ -homomorphism ι A : C h x ij , x ∗ ij i → A . Its image ι A ( A ) equals to ˜ A – the even part of A .Proof. The existence of the ∗ -homomorphism ι A follows from the fact that its do-main is a free algebra. Obviously, for any monomial f ∈ C h x ij , x ∗ ij i , the image ι A ( f ) has even degree. It remains to show that, for any monomial of even degree˜ f ∈ ˜ A , there exists a unique monomial f ∈ C h x ij , x ∗ ij i such that ˜ f = ι A ( f ).This is done easily by induction on the “length” of ˜ f measured by the numberof variables x ij or x ∗ ij (ignoring the r ’s). Suppose ˜ f is in the reduced form, that is,the variable r does not appear twice consecutively. If ˜ f starts with the variable x ij ,we can write ˜ f ( x ij , r ) = x ij r ˜ f ( x ij , r ) for some f ∈ ˜ A , so ι − A ( ˜ f )( x ij , x ∗ ij ) = x ij ι − A ( ˜ f )( x ij , x ∗ ij ). If ˜ f starts with r followed by x ij , so ˜ f ( x ij , r ) = rx ij ˜ f ( x ij , r )for some f ∈ ˜ A , then ι − A ( ˜ f )( x ij , x ∗ ij ) = x ∗ ij ι − A ( ˜ f )( x ij , x ∗ ij ). (cid:3) Remark 4.9. ˜ A is the ∗ -subalgebra of A generated by the elements x ij r . Conse-quently, for any G ⊆ O + ( F ) ∗ ˆ Z , we can express the coordinate algebra associatedto its glued version ˜ G as O ( ˜ G ) = { f ( v ij , r ) | f ∈ ˜ A } ⊆ O ( G ) . In addition, we can rephrase Proposition 3.4 by saying that a quantum group˜ G ⊆ U + ( F ) is a glued version of G ⊆ O + ( F ) ∗ ˆ Z if and only if we have ˜ I ˜ G := ι A ( I ˜ G ) = I G ∩ ˜ A .Recall the definition of quantum groups ˜ G ⊆ U + ( F ) invariant with respect tothe colour inversion from Def. 3.31. Proposition 4.10.
Consider a compact matrix quantum group ˜ G ⊆ U + ( F ) with F ¯ F = c N invariant with respect to the colour inversion. Let G ′ be its maximal Z -ungluing. Then G := G ′ ∩ ( O + ( F ) ∗ ˆ Z ) is also a Z -ungluing. Definition 4.11.
The quantum group G from Proposition 4.10 will be called the canonical Z -ungluing of ˜ G .Before proving the proposition, recall that I G ′ ⊆ C h x ij , x ∗ ij , r, r ∗ i is defined asthe smallest ideal containing ι ( I ˜ G ). Consequently, I G = I G ′ / ( x = x , r = 1) is thesmallest ideal of the algebra A = C h x ij , x ∗ ij , r, r ∗ i / ( x = x , r = 1) containing ˜ I ˜ G = LUING COMPACT MATRIX QUANTUM GROUPS 31 ι ( I ˜ G ) / ( x = x , r = 1). In other words, the canonical Z -ungluing is determinedby relations of the form f ( x ij r, rx ij ) with f ∈ I ˜ G ⊆ C h x ij , x ∗ ij i . Proof.
Adopting the notation introduced above, we need to prove that I G ∩ ˜ A = ˜ I ˜ G .We prove that I G = ˜ I ˜ G + ˜ I ˜ G r = span { f, f r | f ∈ ˜ I ˜ G } . Then it will be clear that I G ∩ ˜ A = ( ˜ I ˜ G + ˜ I ˜ G r ) ∩ ˜ A = ˜ I ˜ G since ˜ I ˜ G ⊆ ˜ A , so ˜ I ˜ G r ∩ ˜ A = ∅ .To prove the equality, it is enough to show that the right-hand side is an idealsince then it must be the smallest one containing ˜ I ˜ G , which is exactly I G . So, denotethe right-hand side by I . By Proposition 3.32, G being invariant with respect to thecolour inversion means that I G is invariant with respect to interchanging x ij ↔ x ij .Applying ι A , we get that ˜ I ˜ G is invariant with respect to x ij r ↔ rx ij , so ˜ I ˜ G isinvariant with respect to conjugation by r , that is, x rxr . We use that to provethat I is an ideal. The subspace I is obviously invariant under right multiplicationby r . For the left multiplication, we can write rx = ( rxr ) r . For multiplicationby x ij , we can write xx ij = ( xx ij r ) r and x ij x = r (( x ij r ) x ). (cid:3) Proposition 4.12.
Consider G ⊆ O + ( F ) ∗ ˆ Z and denote by k its degree of reflec-tion. Then exactly one of the following situations occurs. (1) If k = 1 , then C ( ˜ G ) = C ( G ) and hence ˜ G ≃ G . (2) If k = 2 , then C ( G ) is Z -graded and C ( ˜ G ) is its even part.Proof. Since G is orthogonal, its degree of reflection must be either one or two byProposition 3.12. 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 isa vector ξ ∈ Mor(1 , u ⊗ k ), k ξ k = 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, itmust be an element of C ( ˜ G ) (see Remark 4.9).If the degree of reflection equals two, then by Proposition 3.11 the Z -gradingof A passes to O ( G ) and hence also C ( G ). As mentioned in Remark 4.9, O ( ˜ G ) con-sists of even polynomials in the generators v ij and r and hence is the even partof O ( G ). (cid:3) The following theorem provides a non-easy counterpart of Theorem [GW19, The-orem 4.10].
Theorem 4.13.
There is a one-to-one correspondence between (1) quantum groups G ⊆ O + ( F ) ∗ ˆ Z with degree of reflection two and (2) quantum groups ˜ G ⊆ U + ( F ) that are invariant with respect to the colourinversion.This correspondence is provided by gluing and canonical Z -ungluing.Proof. Almost everything follows from Proposition 4.10. The only remaining thingto prove is that, given G ⊆ O + ( F ) ∗ ˆ Z and ˜ G its glued version, then ˜ G is invariantwith respect to the colour inversion and G is its canonical Z -ungluing. The firstassertion follows from the fact that I G and hence also ˜ I := I G ∩ ˜ A = ˜ I ˜ G is invariantwith respect to conjugation by r . For the second assertion, we need to prove that I G = ˜ I + ˜ Ir (see the proof of Proposition 4.10). Since G has degree of reflectiontwo, we have that I G is Z -graded and hence it decomposes into an even and oddpart, which is precisely ˜ I and ˜ Ir . (cid:3) In [GW19], we introduced the Z -extensions H × k ˆ Z exactly to be the canonical Z -ungluings of H ˜ × ˆ Z k . In the following, we recall the definition of those productsand provide a direct proof for the fact that H × k ˆ Z are canonical Z -ungluings of H ˜ × ˆ Z k for arbitrary H ⊆ O + ( F ). Definition 4.14 ([GW19]) . Let G and H be compact matrix quantum groups anddenote by u and v their respective fundamental representations. We define thefollowing quantum subgroups of G ∗ H . The product G ×× H is defined by takingthe quotient of C ( G ∗ H ) by the relations(4.1) ab ∗ x = xab ∗ , a ∗ bx = xa ∗ b the product G ×× H is defined by the relations(4.2) ax ∗ y = x ∗ ya, axy ∗ = xy ∗ a the product G × H by the combination of the both pairs of relations and, finally,given k ∈ N , the product G × k H is defined by the relations(4.3) a x · · · a k x k = x a · · · x k a k , where a, b, a , . . . , a k ∈ { u ij } and x, y, x , . . . , x k ∈ { v ij } . (Equivalently, we canassume a, b, a , . . . , a k ∈ span { u ij } and x, y, x , . . . , x k ∈ span { v ij } .) Proposition 4.15.
Consider H ⊆ O + ( F ) with degree of reflection two, k ∈ N .Then the canonical Z -ungluing of H ˜ × ˆ Z k is H × k ˆ Z .Proof. From Proposition 3.34, we can express H ˜ × ˆ Z k as a quantum subgroupof H ˜ ∗ ˆ Z = H ˜ ∗ ˆ Z given by certain relations. Denoting by ˜ v the fundamentalrepresentation of H ˜ × ˆ Z k , we just have to “unglue” the relations substituting ˜ v ij by v ij r . For H ˜ × ˆ Z , we get v ij v kl = rv ij v ij r, which is obviously equivalent to v ij v kl r = rv ij v kl – the defining relation for H ×× ˆ Z = H × ˆ Z . For H ˜ × ˆ Z k , k >
0, we get v i j zv i j z · · · v i k j k z = zv i j zv i j · · · zv i k j k , which is exactly the defining relation for H × k ˆ Z . (cid:3) Irreducibles and coamenability for ungluings and Z -extensions. Inthis section, we will study properties of the gluing procedure and canonical Z -ungluing. In many cases, we can view the Z -extensions H × k ˆ Z as a motivatingexample. It remains an open question, whether one can generalize the statementsfor arbitrary products H × k H .For this section, we will assume that G ⊆ O + N ∗ ˆ Z has degree of reflection two.For quantum groups G and H , the property O ( H ) ⊆ O ( G ) can be understoodeither as H being a quotient of G or the discrete dual ˆ H being a quantum subgroupof ˆ G . In that case, we can study the homogeneous space ˆ G/ ˆ H by defining l ∞ ( ˆ G/ ˆ H ) := { x ∈ l ∞ ( ˆ G ) | x ( ab ) = x ( b ) for all a ∈ O ( H ) and b ∈ O ( G ) } , where l ∞ ( ˆ G ) is the space of all bounded functionals on O (0) G . Proposition 4.16.
Consider G ⊆ O + N ∗ Z with degree of reflection two. Then ˆ G/ ˆ˜ G consists of two points. More precisely, l ∞ ( ˆ G/ ˆ˜ G ) = { x ∈ l ∞ ( ˆ G ) constant on the Z -homogeneous parts of O ( G ) } ≃ C . LUING COMPACT MATRIX QUANTUM GROUPS 33
Proof.
Consider x ∈ l ∞ ( ˆ G ). By Proposition 4.12, O ( G ) is Z -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. (cid:3) 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 4.17.
Consider G ⊆ O + N ∗ ˆ Z and ˜ G ⊆ U + N its glued version. Then G is coamenable if and only if ˜ G is coamenable.Proof. It is easy to see that the Haar state on ˜ G is given just by restriction of theHaar state h on G . We also have that all positive elements of C ( G ) are containedin C ( ˜ G ). Hence, the faithfulness of h on C ( G ) is equivalent to the faithfulness on C ( ˜ G ). (cid:3) Note that it is well known that the coamenability is preserved under tensorproduct 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 interpolatingproducts from Def. 4.14.Let us also recall another well known fact that can be seen directly from the def-inition of coamenability: Let G be a coamenable compact quantum group. Thenevery its quotient, that is, every quantum group H with C ( H ) ⊆ C ( G ) is coa-menable. Proposition 4.18.
Consider H ⊆ O + N , k ∈ N . Then H × k ˆ Z is co-amenable ifand only if H is co-amenable.Proof. The left-right implication follows from the fact that H is a quotient of H × k ˆ Z . Now, let us prove the right-left implication. If the degree of reflection of H isone, then by [GW19, Theorem 5.5], the new product actually coincides with thetensor product H × k ˆ Z = H × ˆ Z , so it indeed preserves the coamenability.Now suppose H has degree of reflection two. If H is coamenable, then H × ˆ Z k must be coamenable. Consequently, its glued version H ˜ × ˆ Z k is coamenable since itis a quotient quantum group. Finally, H × k ˆ Z is coamenable by Proposition 4.17. (cid:3) Now, we are going to look on the irreducible representations of the ungluings.
Proposition 4.19.
Consider G ⊆ O + ( F ) ∗ ˆ Z with degree of reflection two andfundamental representation v ⊕ r . Let ˜ G ⊆ U + N be its glued version. Then theirreducibles of G are given by { u α , u α r | α ∈ Irr ˜ G } . Proof.
First, we prove that all the matrices are indeed representations of G . Surelyall u α are representations. The r is also a representation. Hence u α r = u α ⊗ r mustalso be representations.Secondly, we prove that the representations are mutually inequivalent. Therepresentations u α are mutually inequivalent by definition. From this, it followsthat the representations u α r are mutually inequivalent. Since G has degree ofreflection two, we have that O ( G ) is graded with O ( ˜ G ) being its even part. So,the entries of u α are even, whereas the entries of u α r are odd, so they cannot beequivalent. Finally, we need to prove that those are all the representations. This can beproven using the fact that entries of irreducible representations form a basis of thepolynomial algebra. If we prove that the entries of the representations span thewhole O ( G ), we are sure that we have all the irreducibles. This is indeed true:span { u αij , u αij r | α ∈ Irr ˜ G } = O ( ˜ G ) + O ( ˜ G ) r = O ( G ) . (cid:3) Proposition 4.20.
Consider H ⊆ O + ( F ) with degree of reflection two, k ∈ N .Then the complete set of mutually inequivalent irreducible representations of H × k ˆ Z is given by (4.4) u α,i,η = u α s i r η , α ∈ Irr
H, i ∈ { , . . . , k − } , η ∈ { , } , where (4.5) s = X l v il rv ∗ il r = X k v ∗ kj rv kj r for any i, j = 1 , . . . , N .Here v ⊕ r denotes the fundamental representation of H × k ˆ Z .Proof. Denote ˜ v := vr the fundamental representation of the glued version of H × k ˆ Z , which can be identified with H ˜ × ˆ Z k by Proposition 4.15. Thus, we can alsowrite ˜ v ij = v ij z ∈ C ( H ˜ × ˆ Z k ) ⊆ C ( H × ˆ Z k ). We can also express s = X l v il rv ∗ il r = [ vrvr ∗ ] ii = [ vr ( F − vrF ) t ] ii = [˜ v ( F − ˜ vF ) t ] ii = [ v ( F − vF ) t z ] ii = [ vv ∗ ] ii z = z (4.6)and similarly for the second expression in Eq. (4.5). In particular, we have that s = z is a representation of H ˜ × ˆ Z k and hence also of H × k Z .According to Proposition 3.23, we know that irreducibles of H ˜ × ˆ Z k are of theform u α,i := u α z i + d α , α ∈ Irr H , i = 0 , . . . , k − d α ∈ { , } is the degree of α .According to Proposition 4.19, we have that the set of irreducibles of H × k ˆ Z k is u α,i r η . Now the only point is to express these in terms of u α , s , and r .Suppose first that d α = 0. In this case, the situation is simple since we can usethe above mentioned fact that z = s to derive u α,i r η = u α z i r η = u α s i r η . In the situation d α = 1, we need to prove that u α z = u α r . The left hand side isa representation of H ˜ × ˆ Z k – the glued version of H × ˆ Z k – and the right-handside is a representation of the glued version of H × k ˆ Z . As we already mentioned,these quantum groups coincide, so the equality makes sense. (Note that it is notpossible to show that z = r . Not only that this is not true, the equality doesnot even make sense since s and r are not elements of a common algebra.) Since u α z is a representation of H ˜ × ˆ Z k , we have, in particular, that the entries of therepresentation u αab z are elements of O ( H ˜ × ˆ Z k ). That is, there are polynomials f αab ∈ C h x ij , x ij i of degree one such that f αab (˜ v ij , ˜ v ij ) = f αab ( v ij z, z ∗ v ij ) = u αab z .Since v ij commute with z , we can arrange the “ -pattern” of the monomials in f αab in an arbitrary way if we keep the property that they have degree one. In particular,we can say that every monomials of f αab have an alternating colour pattern, that is,they are of the form x i j x i j · · · x i n j n . Then we can express u αab z = f αab ( v ij z, z ∗ v ij ) = f αab (˜ v ij , ˜ v ij ) = f αab ( v ij r, rv ij ) = u αab r. LUING COMPACT MATRIX QUANTUM GROUPS 35
This is exactly what we wanted to prove. Now we have u α,i r η = u α zz i r η = u α rs i r η To obtain the form in the statement, note just that rsr = s ∗ = s k − . (cid:3) References [Ban96] Teodor Banica. Th´eorie des repr´esentations du groupe quantique compact libre O ( n ). Comptes rendus de l’Acad´emie des sciences. S´erie 1, Math´ematique , 322:241–244, 1996.[Ban99] Teodor Banica. Representations of compact quantum groups and subfac-tors.
Journal f¨ur die reine und angewandte Mathematik , 509:167–198, 1999.doi:10.1515/crll.1999.509.167.[Ban08] Teodor Banica. A note on free quantum groups.
Annales Math´ematiques Blaise Pascal ,15(2):135–146, 2008. doi:10.5802/ambp.243.[Ban18] Teodor Banica. Super-easy quantum groups: definition and examples.
Bulletin of thePolish Academy of Sciences. Mathematics , 66:57–68, 2018. doi:10.4064/ba8133-2-2018.[Ban19] Teodor Banica.
Free quantum groups and related topics , 2019. URL https://banica.u-cergy.fr/a3.pdf . Visited on 9 June 2020.[BCV17] Michael Brannan, Benoˆıt Collins, and Roland Vergnioux. The connes embedding prop-erty for quantum group von Neumann algebras.
Transactions of the American Mathe-matical Society , 369:3799–3819, 2017. doi:10.1090/tran/6752.[BS09] Teodor Banica and Roland Speicher. Liberation of orthogonal Lie groups.
Advances inMathematics , 222(4):1461–1501, 2009. doi:10.1016/j.aim.2009.06.009.[BV10] Teodor Banica and Roland Vergnioux. Invariants of the half-liberated orthogonal group.
Annales de l’Institut Fourier , 60(6):2137–2164, 2010. doi:10.5802/aif.2579.[Chi15] Alexandru Chirvasitu. Residually finite quantum group algebras.
Journal of FunctionalAnalysis , 268(11):3508–3533, 2015. doi:10.1016/j.jfa.2015.01.013.[CW16] Guillaume C´ebron and Moritz Weber. Quantum groups based on spatial partitions.arXiv:1609.02321, 2016.[Fre17] Amaury Freslon. On the partition approach to schur-weyl duality andfree quantum groups.
Transformation Groups , 22(3):707–751, Sep 2017.doi:10.1007/s00031-016-9410-9.[FW16] Amaury Freslon and Moritz Weber. On the representation theory of partition (easy)quantum groups.
Journal f¨ur die reine und angewandte Mathematik , 720:155–197, 2016.doi:doi:10.1515/crelle-2014-0049.[Gro18] Daniel Gromada. Classification of globally colorized categories of partitions.
InfiniteDimensional Analysis, Quantum Probability and Related Topics , 21(04):1850029, 2018.doi:10.1142/S0219025718500297.[Gro20] Daniel Gromada.
Compact matrix quantum groups and their representation categories .PhD thesis, Saarland University, 2020.[GW19] Daniel Gromada and Moritz Weber. New products and Z -extensions of compact matrixquantum groups. arXiv:1907.08462, 2019.[GW20] Daniel Gromada and Moritz Weber. Intertwiner spaces of quantum group sub-representations. Communications in Mathematical Physics , 376:81–115, 2020.doi:10.1007/s00220-019-03463-y.[Mal18] Sara Malacarne. Woronowicz tannaka-krein duality and free orthogo-nal quantum groups.
Mathematica Scandinavica , 122(1):151–160, 2018.doi:10.7146/math.scand.a-97320.[NT13] Sergey Neshveyev and Lars Tuset.
Compact Quantum Groups and Their RepresentationCategories . Soci´et´e Math´ematique de France, Paris, 2013.[RW15] Sven Raum and Moritz Weber. Easy quantum groups and quantum subgroups of a semi-direct product quantum group.
Journal of Noncommutative Geometry , 9(4):1261–1293,2015. doi:10.4171/JNCG/223.[RW16] Sven Raum and Moritz Weber. The full classification of orthogonal easy quan-tum groups.
Communications in Mathematical Physics , 341(3):751–779, 2016.doi:10.1007/s00220-015-2537-z.[Tim08] Thomas Timmermann.
An Invitation to Quantum Groups and Duality . EuropeanMathematical Society, Z¨urich, 2008. [TW17] Pierre Tarrago and Moritz Weber. Unitary easy quantum groups: The free case andthe group case.
International Mathematics Research Notices , 2017(18):5710–5750, 2017.doi:10.1093/imrn/rnw185.[TW18] Pierre Tarrago and Moritz Weber. The classification of tensor categories of two-colorednoncrossing partitions.
Journal of Combinatorial Theory, Series A , 154:464–506, 2018.doi:10.1016/j.jcta.2017.09.003.[VDW96] Alfons Van Daele and Shuzhou Wang. Universal quantum groups.
International Journalof Mathematics , 07(02):255–263, 1996. doi:10.1142/S0129167X96000153.[Wan95a] Shuzhou Wang. Free products of compact quantum groups.
Communications in Math-ematical Physics , 167(3):671–692, 1995. doi:10.1007/BF02101540.[Wan95b] Shuzhou Wang. Tensor products and crossed products of compact quantumgroups.
Proceedings of the London Mathematical Society , s3-71(3):695–720, 1995.doi:10.1112/plms/s3-71.3.695.[Web17] Moritz Weber. Introduction to compact (matrix) quantum groups and Banica–Speicher(easy) quantum groups.
Proceedings – Mathematical Sciences , 127(5):881–933, 2017.doi:10.1007/s12044-017-0362-3.[Wor87] Stanis law L. Woronowicz. Compact matrix pseudogroups.
Communications in Mathe-matical Physics , 111(4):613–665, Dec 1987. doi:10.1007/BF01219077.[Wor88] Stanis law L. Woronowicz. Tannaka-Krein duality for compact matrix pseu-dogroups. Twisted SU ( N ) groups. Inventiones mathematicae , 93(1):35–76, 1988.doi:10.1007/BF01393687.
Saarland University, Fachbereich Mathematik, Postfach 151150, 66041 Saarbr¨ucken,Germany
E-mail address ::