compact matrix quantum groups
Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades
einer Doktorin der Naturwissenschaften genehmigte Dissertation
vorgelegt von
Laura Maaßen, M.Sc.
aus
Grevenbroich
Berichter: Univ.-Prof. Dr. rer. nat. Gerhard Hiß Univ.-Prof. Dr. rer. nat. Moritz Weber
Priv.-Doz. Dr. rer. nat. Amaury Freslon, Ph.D.
Tag der mündlichen Prüfung: 29. Juni 2021
Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek verfügbar.
Ich, Laura Maaßen, erkläre hiermit, dass diese Dissertation und die darin dargelegten Inhalte die eigenen sind und selbstständig, als Ergebnis der eigenen originären Forschung, generiert wurden. Hiermit erkläre ich an Eides statt:
1. Diese Arbeit wurde vollständig oder größtenteils in der Phase als Doktorand dieser Fakultät und Universität angefertigt;
2. Sofern irgendein Bestandteil dieser Dissertation zuvor für einen akademischen Ab- schluss oder eine andere Qualifikation an dieser oder einer anderen Institution ver- wendet wurde, wurde dies klar angezeigt;
3. Wenn immer andere eigene- oder Veröffentlichungen Dritter herangezogen wurden, wurden diese klar benannt;
4. Wenn aus anderen eigenen- oder Veröffentlichungen Dritter zitiert wurde, wurde stets die Quelle hierfür angegeben. Diese Dissertation ist vollständig meine eigene Arbeit, mit der Ausnahme solcher Zitate;
5. Alle wesentlichen Quellen von Unterstützung wurden benannt;
6. Wenn immer ein Teil dieser Dissertation auf der Zusammenarbeit mit anderen basiert, wurde von mir klar gekennzeichnet, was von anderen und was von mir selbst erarbeitet wurde;
7. Teile dieser Arbeit wurden zuvor veröffentlicht und zwar in:
• L. Maassen. “The intertwiner spaces of non-easy group-theoretical quantum groups”. In: J. Noncommut. Geom. 14.3 (2020), pp. 987–1017. doi: 10 . 4171/JNCG/384
• L. Maassen. “Deligne categories and easy quantum groups”. In: Report on the Oberwolfach Mini-Workshop: Operator Algebraic Quantum Groups. Vol. 16.
4. Mathematisches Forschungsinstitut Oberwolfach. Zurich: EMS Publishing House, 2019, pp. 2842–2845. doi: 10.4171/OWR/2019/45
• J. Flake and L. Maassen. Semisimplicity and Indecomposable Objects in Inter- polating Partition Categories. 2020. eprint: arXiv:2003.13798
Datum, Unterschrift
One key result obtained from the investigation of compact matrix quantum groups is a Tannaka-Krein type duality, by which any compact matrix quantum group can be fully recovered from its representation category. Following this idea, easy quantum groups are defined through a combinatorial description of their representation categories. In this thesis, we study the representation categories of so-called group-theoretical quantum groups and show that they can be described by a combinatorial calculus similar to that used for easy quantum groups. Furthermore, we analyse the structure of abstract tensor categories that interpolate the representation categories of easy quantum groups.
This thesis thus concerns research problems at the intersection of the theory of compact quantum groups, combinatorics and category theory with links to group theory.
The first part of this thesis concerns group-theoretical quantum groups. We define an analogue of orthogonal group-theoretical quantum groups in the unitary setting and show that their description as semi-direct product quantum groups can be generalised. We describe their representation categories, both in the easy and the non-easy case. For this purpose, we introduce modified versions of categories of partitions, which model the ’group-theoretical structure’ of the diagonal subgroups of group-theoretical quantum groups. Moreover, we define a modified fiber functor linked with the classical fiber functor via Moebius inversion. Subsequently, we show that the application of the Tannaka-Krein duality yields the desired description of the representation categories of group-theoretical quantum groups.
Next, we restrict our attention to the orthogonal case. Although it is known that uncount- ably many orthogonal group-theoretical easy quantum groups exist, almost no concrete examples have been studied. We compute various examples with small generators, in- cluding in particular a new series of easy quantum groups between the hyperoctahedral series and higher hyperoctahedral series. We conclude our analysis of orthogonal group- theoretical quantum groups by an improved version of a de Finetti theorem by Raum and Weber.
In the second part of this thesis, we study interpolating partition categories in the frame- work of Deligne’s interpolation categories. Interpolating partition categories are the cat- egorial abstraction of categories of partitions together with a complex interpolation para- meter. We explain that their semisimplifications interpolate the representation categories of easy quantum groups. Next, we show that the semisimplicity of an interpolating parti- tion category is encoded in the determinants of certain Gram matrices. We compute the set of interpolation parameters yielding semisimple interpolating partition categories for all group-theoretical easy quantum groups. Moreover, we parametrise the indecompos- able objects in all interpolating partition categories by an explicitly constructible system of finite groups and exhibit their Grothendieck rings as filtered deformations. We apply these results to orthogonal easy groups and free orthogonal easy quantum groups.
I would like to express my sincere gratitude to my supervisors Moritz Weber and Gerhard Hiß for their patient support and guidance throughout this project. I thank Moritz Weber for always making time for me, even when he did not actually have any. The countless discussions with him as well as his feedback on my work, my articles and my talks were invaluable. I would also like to thank him for the good mood he spreads, which I could use so well during the last months. Moreover, I greatly appreciated all the invitations to Saarbrücken and the opportunity to take part in so many conferences. I am very grateful to Gerhard Hiß for taking me to my very first conference while I was writing my bachelor thesis and thereby awakening my interest in mathematical research. I would also like to thank him for suggesting to work on the topic of quantum groups. Especially his idea to work on Deligne’s interpolation categories gave me new motivation during my PhD project. Furthermore, I greatly appreciate his detailed feedback on my work and that he was always available for a conversation.
I would like to thank Amaury Freslon for kindly agreeing to review this thesis. Sincere thanks go to Sven Raum for inviting me to Stockholm and for the inspiring discussions on group-theoretical quantum groups. Moreover, I would like to express my gratitude to Roland Speicher for the opportunity to take part in the one-month program in Montreal.
I would also like to thank Johannes Flake for the fruitful collaboration on interpolating partitions categories and for sharing his expertise on Deligne’s interpolation categories.
I wish to express my gratitude to all my colleagues at the chair of algebra and number theory and at the chair of algebra and representation theory. I would particularly like to thank my friends and office mates Christoph Schönnenbeck, Melanie Harms and Dominik Bernhardt for turning days from exhausting into decent and from good into great. Espe- cially during the time in home office, I realised how much I miss the coffee breaks with them. A special thanks goes to my friend Andrea Thevis, with whom I have been fight- ing through the world of algebra together for so long and who is always there for me. I would also like to thank all the members of Moritz Weber and Roland Speicher’s working group, who always made me feel very welcome in Saarbrücken. In particular, I would like to thank Simon Schmidt, Felix Leid and Daniel Gromada for the lovely time at various conferences. I am immeasurably grateful to all my friends for their consistent support.
Moreover, I cannot express how grateful I am to my parents, my sister and my boyfriend, who always believed in me and without whom I could not have completed this thesis.
Finally, I would like to acknowledge the support I received from theRWTH Scholarship for Doctoral Studentsand thank the SFB-TRR 195Symbolic Tools in Mathematics and their Application for accepting me as a member of the Integrated Research Training Group.
Introduction 13
I. Preliminaries 21
1. Compact quantum groups 25
1.1. C∗-algebras . . . 25
1.1.1. Algebraic definition . . . 26
1.1.2. Universal C∗-algebras . . . 28
1.1.3. Group C∗-algebras of discrete groups . . . 29
1.1.4. C∗-algebras as bounded operators on Hilbert spaces . . . 30
1.1.5. Tensor products . . . 31
1.2. Compact (matrix) quantum groups . . . 32
1.2.1. Compact quantum groups – Generalising compact groups . . . 33
1.2.2. Compact matrix quantum groups – Generalising compact matrix groups . . . 34
1.2.3. Morphisms . . . 37
1.3. Discrete quantum groups . . . 39
1.3.1. Quantum groups associated to discrete groups . . . 39
1.3.2. Diagonal subgroups . . . 41
1.4. Products of compact matrix quantum groups . . . 42
1.4.1. The direct product and the glued direct product . . . 42
1.4.2. The tensor (k)-complexification . . . 44
2. Representation theory of compact quantum groups 47 2.1. Representations and intertwiners . . . 47
2.1.1. Basic definitions . . . 48
2.1.2. Decomposing representations . . . 49
2.2. Category theory – basic concepts . . . 51
2.2.1. Additive, pseudoabelian and abelian categories . . . 51
2.2.2. Monoidal categories . . . 56
2.3. C*-tensor categories . . . 63
2.3.1. *-categories and C*-categories . . . 63
2.3.2. Monoidal *-categories . . . 67
2.4. A Tannaka-Krein type duality . . . 68
3. Easy quantum groups 73
3.1. Categories of partitions . . . 73
3.1.1. Partition diagrams and operations . . . 74
3.1.2. Categories of partitions - Definition and examples . . . 77
3.1.3. Categories of partitions on one row . . . 79
3.1.4. Categories of non-coloured partitions . . . 81
3.1.5. Classification results for categories of partitions . . . 83
3.2. Associating compact matrix quantum groups to categories of partitions . . 86
3.2.1. A fiber functor . . . 87
3.2.2. Definition of easy quantum groups . . . 88
3.2.3. Relations associated to partitions . . . 89
3.2.4. Examples . . . 91
3.3. A categorial point of view . . . 95
II. Group-theoretical quantum groups 99
4. Unitary group-theoretical quantum groups 103 4.1. Definition and examples . . . 1044.2. The semi-direct product structure . . . 109
4.2.1. Semi-direct product quantum groups . . . 109
4.2.2. A semi-direct product with the diagonal subgroup . . . 111
4.3. n-restricted skew categories of partitions . . . 117
4.3.1. Partitions represented by tuples . . . 117
4.3.2. Modified operations on partitions . . . 118
4.3.3. Definition of n-restricted skew categories of partitions and basic properties . . . 121
4.3.4. n-restricted skew categories of partitions on one row . . . 123
4.3.5. The group-theoretical structure . . . 126
4.4. A modified fiber functor . . . 132
4.4.1. Associating linear maps to partitions . . . 132
4.4.2. Respecting the modified operations . . . 135
4.4.3. Applying the Tannaka-Krein duality . . . 138
4.4.4. Modified relations associated to partitions . . . 140
4.4.5. The refinement order on partitions and the Moebius function . . . . 142
4.4.6. Relations with the ’classical’ fiber functor . . . 145
4.5. Three types of group-theoretical quantum groups . . . 146
4.5.1. Description by n-restricted skew categories of partitions . . . 146
4.5.2. Description by skew categories of partitions - skew easy quantum groups . . . 148
4.5.3. Description by group-theoretical categories of partitions - group- theoretical easy quantum groups . . . 152
4.5.4. A comparison of all three types . . . 159
4.6. A categorial point of view . . . 162
4.6.1. (n-restricted) skew categories as monoidal categories . . . 163
4.6.2. An equivalence of monoidal categories . . . 169
5. Orthogonal group-theoretical quantum groups 185
5.1. The structure of orthogonal group-theoretical quantum groups . . . 185
5.2. New examples of orthogonal group-theoretical easy quantum groups . . . . 192
5.2.1. Known examples . . . 192
5.2.2. Computing new examples . . . 195
5.2.3. A new series between the hyperoctahedral and the higher hyperoc- tahedral series . . . 208
5.2.4. Summary . . . 214
5.3. A de Finetti theorem . . . 216
5.3.1. Introduction to free probability theory . . . 216
5.3.2. Quantum group action on sequences of random variables . . . 219
5.3.3. Alternative de Finetti theorems for orthogonal group-theoretical quantum groups . . . 220
III. Interpolating partition categories 227
6. Introduction to interpolating partition categories 231 7. Semisimplicity of interpolating partition categories 237 7.1. Semisimple algebras and semisimple categories . . . 2387.1.1. Semisimple algebras . . . 238
7.1.2. Semisimple categories . . . 240
7.2. Conditions for semisimplicity . . . 242
7.2.1. Negligible Morphisms . . . 243
7.2.2. Associated Gram matrices . . . 246
7.2.3. Semisimplification . . . 249
7.3. Results for special cases . . . 250
7.4. The group-theoretical case . . . 252
7.4.1. The semilattice structure . . . 253
7.4.2. Computing determinants . . . 255
7.4.3. Semisimplicity of interpolating partition categories corresponding to group-theoretical categories of partitions . . . 262
8. Indecomposable objects in interpolating partition categories 265 8.1. Introduction . . . 266
8.1.1. Indecomposable objects and the Grothendieck ring . . . 266
8.1.2. Primitive idempotents . . . 269
8.2. Projective partitions . . . 270
8.2.1. Through-blocks . . . 271
8.2.2. Projective partitions and their through-block decomposition . . . . 273
8.2.3. Associating finite groups . . . 274
8.2.4. Lifting idempotents . . . 277
8.2.5. An equivalence relation . . . 279
8.3. A parametrisation of the indecomposable objects . . . 283
8.4. The Grothendieck ring . . . 285
8.4.1. A filtration on the Grothendieck ring . . . 285
8.4.2. A description of the associated graded ring of the Grothendieck ring 287 8.5. Concrete examples . . . 291 8.5.1. The group and the free case . . . 292 8.5.2. Temperley–Lieb categories as a special case . . . 301
9. Open questions 311
Appendix Computing examples of group-theoretical quantum groups 321
Bibliography 327
Groups are one of the most fundamental structures in mathematics and they arise nat- urally in the study of symmetries. With the emergence of quantum physics and mod- ern mathematics, an interest in symmetries in non-commutative settings, which can no longer be adequately described by classical groups, has evolved. This led to the the- ory of quantum groups, which goes back to Drinfel’d [Dri87] and Jimbo [Jim85]. Here, quantum groups were introduced as Hopf algebra deformations of universal enveloping algebras of semisimple Lie algebras. But the theory of quantum groups has been de- veloped in different directions, and we follow the topological approach by Woronowicz [Wor87a]. The philosophy of Woronowicz’s approach is based on the Gelfand–Naimark duality, which identifies function algebras over compact topological spaces with commut- ative C∗-algebras. Hence, non-commutative C∗-algebras can be interpreted as function algebras over non-commutative topological spaces and they serve as the underlying algeb- ras for topological quantum groups.
To be more precise, the algebra of complex-valued continuous functions C(G) over a compact group G inherits a dualised group multiplication ∆ : C(G) → C(G)⊗C(G).
Woronowicz axiomatised the properties of this comultiplication, which allowed him to replace the function algebraC(G) by an arbitrary C∗-algebra A. The resulting structures are called compact quantum groups; see [Wor98]. Although strictly speaking, a compact quantum group is not a group, we can think of it as the virtual object whose function algebra is the – not necessarily commutative – C∗-algebra A. In this thesis, we mostly consider the special case of so-called compact matrix quantum groups (originally called compact matrix pseudogroups), which generalise function algebras over compact mat- rix groups; see [Wor87a]. In contrast to compact quantum groups, a compact matrix quantum group is defined via a fixed matrix representationu∈An×n, called fundamental representation.
Two fundamental techniques of constructing examples of compact matrix quantum groups are deformations, based on the approach of Drinfel’d and Jimbo (see for instance [Wor87b;
Ros90; LS91]), and liberations, based on the work of Wang [Wan95; Wan98]. In this thesis, we consider compact matrix quantum groups which arise from the latter approach. A lib- eration of a compact matrix groupG⊆Un is a quantum group G+, which is obtained by completely dropping the commutativity relations in a suitable universal presentation of the function algebra C(G). Free quantum groups and variations of Wang’s constructions have been studied (see for instance [Ban96; Ban97; Ban99b; Bic04; BBC07a; BBC07b]), and a connection between free quantum groups and Voiculescu’s free probability theory [VDN92a] was established (see for instance [KS09]). This eventually led to the intro-
duction of (orthogonal) easy quantum groups by Banica and Speicher [BS09], which have various connections to free probability theory; see for instance [Web17], as well as to non-commutative geometry; see for instance [Ban16]. Easy quantum groups are a com- binatorial class of compact matrix quantum groups and are one of the main objects of interest in this thesis.
The key ingredient for the definition of easy quantum groups is a Tannaka-Krein type duality due to Woronowicz [Wor88], by which any compact quantum group can be re- covered from its representation category. In fact, for an (orthogonal) compact matrix quantum group the knowledge of the intertwiner spaces of tensor powers of the fun- damental representation HomG(u⊗k, u⊗l) = {T : (Cn)⊗k → (Cn)⊗l | T u⊗k = u⊗lT} is sufficient. Easy quantum groups are defined via a combinatorial description by set- partitions of these intertwiner spaces. More precisely, we consider usual set-partitions of finite sets of the form {1, . . . , k,10, . . . , l0}, which can be pictured by k upper and l lower points, connected according to the blocks of the partition. For example, the par- tition p = {{1,2,20},{3,4,50},{10,30},{40}} ∈ P(4,5) is represented by the following diagram:
p= .
To any such partition p, it is possible to associate linear maps Tp(n) : (Cn)⊗k → (Cn)⊗l along which the composition, the tensor product and the adjoint of linear maps can be traced back to diagrammatic operations on partitions. Note that similar diagrammatic constructions appear in various settings in the literature, for instance in the well-known Schur-Weyl duality and its variants. Banica and Speicher defined categories of partitions as collections of partitions C(k, l)k,l∈N0 that are closed under those diagrammatic opera- tions and contain the partitions ∈P(1,1) and ∈P(2,0).
By the Tannaka-Krein duality, any category of partitions C defines a series of compact matrix quantum groupsGn(C), n∈N,with HomGn(C)(u⊗k, u⊗l) = span{Tp(n)|p∈ C(k, l)}.
Compact matrix quantum groups that arise in this way are called (orthogonal) easy quantum groups. The adjunct ’orthogonal’ refers to the fact that the fundamental rep- resentation is orthogonal. However, Tarrago and Weber [TW17] generalised the above construction by considering two-coloured partitions and introducedunitary easy quantum groups. There exists a full classification of orthogonal easy quantum groups, which has been completed by Raum and Weber [RW16], while the classification of unitary easy quantum groups is still an open problem. However, many partial results are available for the unitary case; see [TW18; Gro18; MW20a; MW21; MW19; MW20b].
With the long-term goal of understanding the structure of all compact matrix quantum groups, the following natural question arises: To what extent do other compact matrix quantum groups have a combinatorial structure similar to that of easy quantum groups?
Over the past years, several generalisation and variations of easy quantum groups have been studied; see amongst others [CW16; Fre17; Ban18; FS18; GW19; GW20]. In Part II of this thesis, we contribute to this program by introducing and investigating a combinat- orial framework for an uncountably large class of compact matrix quantum groups, called group-theoretical quantum groups.
Orthogonal group-theoretical quantum groups have been introduced by Raum and Weber [RW15] in the course of the classification of orthogonal easy quantum groups. To be precise, Raum and Weber defined the notion ’group-theoretical’ only for orthogonal easy quantum groups, but their definition can naturally be extended to arbitrary orthogonal compact matrix quantum groups. The term ’group-theoretical’ refers to a correspondence between categories of partitions that induce orthogonal group-theoretical easy quantum groups and words in certain normal subgroups ofZ∗n2 . This correspondence is built on a de- scription of orthogonal group-theoretical quantum groups as semi-direct product quantum groups. We define an analogue of orthogonal group-theoretical quantum groups in the unitary setting and show that their description as semi-direct product quantum groups can be generalised.
We next develop a combinatorial description of the intertwiner spaces of – not necessarily easy – group-theoretical quantum groups, using modified versions of categories of parti- tions as well as a modified fiber functorTb(n). In addition, we linkTb(n)to the classical fiber functor T(n) via Moebius inversion, which yields a description of the intertwiner spaces by linear combinations of partitions. We show that our combinatorial description is quite similar to that of easy quantum groups, if the group-theoretical quantum groups satisfy a certain closure property. Parts of these results have already been published in [Maa20].
It is known that there exist uncountably many group-theoretical quantum groups; see [RW15, Sec. 5.1]. However, almost no concrete examples have been studied and hence we go on to compute various examples. In particular, we find a new series of easy quantum groups between the hyperoctahedral series and higher hyperoctahedral series. We con- clude our discussion on group-theoretical quantum groups by an improved version of the de Finetti theorem for orthogonal easy group-theoretical quantum groups by Raum and Weber [RW15, Thm. 5.12.].
In Part III of this thesis, we consider the combinatorial framework of easy quantum groups from a more abstract point of view. To any category of partitions C and non-negative integer n ∈ N0, one can naturally associate an abstract tensor category Rep(C, n) such that the linear mapsTp(n), p∈ C,induce a functor T(n): Rep(C, n)→fdHilb into the cat- egory of finite-dimensional Hilbert spaces fdHilb, which factors through the representation category of the corresponding easy quantum group Rep(Gn(C)). Here the parameter n appears in a prefactor of the composition of partitions assuring the functoriality of T(n). Deligne [Del07] considered these tensor categories for the special case of the category of all partitions C = P, corresponding to the symmetric groups Sn = Gn(P), and re- placed the parameter n ∈N0 by an arbitrary complex number t ∈ C. This construction yields (non-abelian) tensor categories, which interpolate the representation categories of the symmetric groups and are interesting in their own right. Deligne’s approach initiated a fruitful area of research over the past years, concered with the categorial structure of these interpolation categories and analogues for other groups like orthogonal groups, gen- eral linear groups or symplectic groups; see among others [CO11; Mor12; CO14; CH17;
BEH19; EHS20]. Since we can consider such interpolation categories for all easy quantum groups, it seems naturally to start a systematic investigation, building a link between the (representation) theory of easy quantum groups and the theory of Deligne’s interpolation categories. We hope to start such a program with this thesis and [FM20].
We begin by studying these interpolating partition categories Rep(C, t) with respect to semisimplicity. We show that the set of interpolation parameters t, for which Rep(C, t) is not semisimple, is always countable and encoded in the determinants of certain Gram matrices. Using an abstract analysis of certain subobject lattices developed by Knop [Kno07], we compute these parameters for all group-theoretical easy quantum groups.
We next develop a parametrisation of the indecomposable objects in all interpolating par- titions categories via a set of finite groups associated to certain ’representing partitions’.
This approach is based on the results of Freslon and Weber [FW16] on the representation theory of easy quantum groups. However, since parts of these results rely on the operator algebraic structure of easy quantum groups, we have to consider suitable abstractions using only algebraic techniques. In particular, this reveals the significance of a certain filtration on the Grothendieck ring and we show that our parametrisation of the indecom- posable objects actually describes the associated graded ring. We conclude with applying these results to orthogonal easy groups and free orthogonal easy quantum groups.
In the following paragraphs, we explain this thesis’ structure and its main results. This thesis is divided into three parts, Part I on preliminaries, Part II on group-theoretical quantum groups and Part III on interpolating partition categories. We begin the prelim- inary part by an introduction to compact quantum groups; see Chapter 1. In Chapter 2 we present the basics on the representation theory of compact quantum groups, focus- sing on the structure of their representation categories. Subsequently, we introduce easy quantum groups in Chapter 3, first with a purely combinatorial part on categories of partitions, before we then turn to the corresponding easy quantum groups.
In Part II we investigate the structure of group-theoretical quantum groups. In Chapter 4 we develop a combinatorial description of their intertwiner spaces. We begin by introdu- cing unitary group-theoretical quantum groups; see Definition 4.1.1. We show that they can be described as semi-direct product quantum groups with their diagonal subgroups ΓG, similar to the orthogonal case.
Theorem A (Theorem 4.2.6)Let G= (A, u, n) be a unitary group-theoretical quantum group with Sn⊆G. Then there exists an Sn-invariant normal subgroupNZ∗n such that
ΓG∼=Z∗n/N and G∼=dΓG 1Sn.
In order to describe the intertwiner spaces of group-theoretical quantum groups, we define n-restricted skew categories of partitions for anyn ∈N∪ {∞}; see Definition 4.3.10, and show that they are in one-to-one-correspondence with Sn-invariant normal subgroups of Z∗n; see Corollary 4.3.24. We call ∞-restricted skew categories of partitions just skew categories of partitions. Next, we introduce a modified fiber functor Tb(n); see Defini- tion 4.4.4. We show that Tb(n) is related to the classical fiber functor T(n) via Moebius inversion on partitions; see Proposition 4.4.40, and that Tb(n) respects the modified oper- ations of n-restricted skew categories of partitions; see Theorem 4.4.13. By applying the Tannaka-Krein duality within this modified combinatorial framework, we can describe the intertwiner spaces of group-theoretical quantum groups as follows.
Theorem B (Theorem 4.5.1, Theorem 4.5.5, Theorem 4.5.19)
(i) Letn ∈N. For anyn-restricted skew category of partitions R, there exists a unique group-theoretical quantum group G with
HomG(uk, ul) = spanC{Tbp(n)|p∈ R(k,l)}.
Vice versa, for any group-theoretical quantum group G =Z\∗n/N 1 Sn, there exists an n-restricted skew category of partitions R that satisfies the above condition.
(ii) For any skew category of partitions R, there exists a unique series of group- theoretical quantum groups Gn(R), n∈N, with
HomGn(R)(uk, ul) = spanC{Tbp(n)|p∈ R(k,l)}.
Vice versa, for any group-theoretical quantum group G = Z\∗n/N 1 Sn with N =N∞∩Z∗n, there exists a skew category of partitions R that satisfies the above condition.
(iii) A group-theoretical quantum group G = Z\∗n/N 1 Sn is easy if and only if N is sSn-invariant. In this case, we have N = N∞ ∩Z∗n and the corresponding skew category of partitions and the corresponding category of partitions coincide.
Here N∞ denotes the completion of N as an S∞-invariant normal subgroup of Z∗∞ and sSn denotes the strongly symmetric reflection group; see Definition 4.5.9.
Note that description (ii) by skew categories is quite similar to that for easy quantum groups, since any skew category of partitions also induces a whole series of quantum groups. This means that we may think of (non-easy) group-theoretical quantum groups whose corresponding normal subgroups satisfyN =N∞∩Z∗n as combinatorial quantum groups, close to the easy case. We conclude our discussion on the intertwiner spaces of group-theoretical quantum groups by an analysis of the abstract tensor categories which can be associated to (n-restricted) skew categories of partitions. In particular, we prove that, for a group-theoretical easy quantum group, the corresponding category of partitions and the corresponding skew category of partitions are equivalent; see Theorem 4.6.14.
In Chapter 5 we restrict our attention to the orthogonal case. We observe that any finitely generated group-theoretical category of partitions is of the form h , pi for a partition p∈P(0, k) on kpoints and we compute all such group-theoretical categories of partitions with k ≤ 13; see Theorem 5.2.17. In particular, we find a new series of easy quantum groups H(s)n ⊆ Gn(Ks) ⊆ H[s]n , s ∈ 4N, between the hyperoctahedral series H(s)n and the higher hyperoctahedral series H[s]n (see Lemma 5.2.24), which is defined via the following categories of partitions:
Ks =h , psi with ps = · · · ∈P(0,2s+ 2).
The categories of partitions Ks, s∈4N,can equivalently be described as
Ks =h , · · · , i,
where · · · ∈ P(0,2s) are the partitions used to introduce the (higher) hyperoctahedral series; see Lemma 5.2.24. The chapter closes with an improved version of a de Finetti theorem for orthogonal group-theoretical easy quantum groups by Raum and Weber; see Theorem 5.3.23.
Part III of this thesis concerns the interpolating partition categories Rep(C, t) associated to categories of partitions C and complex interpolation parameterst ∈C. We begin with a rather short Chapter 6, introducing interpolating partition categories and explaining their categorial properties. In Chapter 7 we analyse interpolating partition categories with respect to semisimplicity. We find that Rep(C, t) is semisimple if and only if all neg- ligible endomorphisms are trivial, which is encoded in the determinants of certain Gram matrices; see Proposition 7.2.15. In particular, the set{t ∈C|Rep(C, t) not semisimple}
is countable. Next, we use the refinement order on partitions together with a technique developed by Knop [Kno07] to compute these determinants for all group-theoretical cat- egories of partitions. We obtain the following result.
Theorem C (Theorem 7.4.27)LetC be a group-theoretical category of partitions. Then Rep(C, t) is semisimple if and only if t /∈N0.
In Chapter 8 we investigate the structure of indecomposable objects for all interpolating partition categories Rep(C, t) witht 6= 0. This means we have to analyse primitive idem- potent morphisms in these categories. For this purpose, we consider certain ’symmetric’
idempotents, called projective partitions (Definition 8.2.6), together with an equivalence relation (Definition 8.2.30). Furthermore, we associate a finite group SC(p) to any such equivalence class [p]; see Definition 8.2.18. These definitions go back to Freslon and Weber [FW16]. We show that the group algebraCSC(p) of a projective partitionpis isomorphic to a certain quotient of an endomorphisms algebra of the given interpolating partition category (see Lemma 8.2.25) and that we can lift idempotents from this quotient; see Proposition 8.2.28. These and some auxiliary results give the following parametrisation of the indecomposable objects in Rep(C, t).
Theorem D (Theorem 8.3.3) Let C be a category of partitions and t ∈ C\{0}. Then transferring and lifting idempotents yields a bijection
L: G
p∈Proj¯ C/∼
Irr(SC(p))←→
( isomorphism classes of non-zero indecomposable objects in Rep(C, t)
)
.
Subsequently, we consider a filtration on the Grothendieck ringK(Rep(C, t)) of Rep(C, t), induced by the number of through-blocks of a partition; see Lemma 8.4.6. We can describe the associated graded ring as follows.
Theorem E (Proposition 8.4.13)LetC be a category of partitions andt ∈C\{0}. For any projective partitionp∈ProjC, we consider the Grothendieck group K(Rep(SC(p))) of the representation categoryRep(SC(p)). Then
R:= M
¯
p∈ProjC/∼
K(Rep(SC(p))) is a ring with multiplication given by
[ρ1]·[ρ2] := [IndSSC(p⊗q)
C(p)×SC(q)(ρ1ρ2)]
Then R is ring isomorphic to the associated graded ring grK(Rep(C, t)).
We apply the above results on indecomposable objects to all orthogonal easy groups and free orthogonal easy quantum groups. Moreover, we compute the indecomposable objects for the interpolating partition categories Rep(S+t ) of the free symmetric quantum groups explicitly; see Section 8.5.
Finally, we discuss some open questions together with some conjectures in Chapter 9.
Furthermore, this thesis is supplemented by an appendix on the algorithm used to compute examples of orthogonal group-theoretical easy quantum groups; see Chapter 9.
Preliminaries
groups in this thesis, we also introduce the more general concept of compact quantum groups. This allows a more differentiated look at some properties and constructions, as we can highlight in which way the ’matrix representation’ is involved.
In Chapter 1 we provide a brief introduction to C∗-algebras before we introduce compact (matrix) quantum groups. We consider in particular compact matrix quantum groups associated to discrete groups as well as the direct product and the glued direct product of compact matrix quantum groups. In Chapter 2 we consider representation categories of compact (matrix) quantum groups and recall a Tannaka-Krein type duality by Woronow- icz. For this purpose we also provide some background in category theory. In Chapter 3 we give an introduction to categories of partitions and easy quantum groups.
In this preliminary part, we only give proofs for facts that might not be standard or involve essential concepts that will be used again later.
Let us fix some notations and conventions.
Notation 0.0.1 We denote by
N ={1,2,3. . .}, the natural numbers,
N0 ={0,1,2. . .}=N∪ {0}, the natural numbers with 0, Z ={. . .−3,−2,−1,0,1,2,3. . .}, the integers,
Z× ={. . .−3,−2,−1,1,2,3. . .}=Z\{0}, the non-zero integers, and for any n ∈N we set
n :={1,2, . . . , n}
±n:={−n, . . . ,−2,−1,1,2, . . . , n}.
Notation 0.0.2 Let S be a set and k, l ∈ N0. We refer to elements of the cartesian product Sk as tuples (over S), where S0 contains only the empty tuple (). For any two tuples i= (i1, . . . , ik)∈Sk and j = (j1, . . . , jl)∈Sl we denote their concatenation by
ij= (i1, . . . , ik, j1, . . . , jl)∈Sk+l.
Moreover, for any tuple i= (i1, . . . , ik)∈Sk and m∈N we denote the m-fold concatena- tion ofi with itself by
im = (i1, . . . , ik, i1, . . . , ik, . . . , i1, . . . , ik,)∈Skm.
Convention Throughout this thesis we assume rings, including algebras, to be unital and associative. Moreover, we set 00 := 1.
Compact quantum groups
In this chapter we give an introduction to compact quantum groups. The term ’quantum group’ is used in various settings. It was originally introduced by Drinfeld [Dri87]
and Jimbo [Jim85] as Hopf algebra deformations of the universal enveloping algebras of semisimple Lie algebras. However, we follow the approach of Woronowicz [Wor87a]
and consider topological quantum groups, which can be understood as liberations of the algebras of complex-valued continuous functions on compact groups.
We start by recalling some basics on C∗-algebras, which constitute the essential underlying structure of compact quantum groups; see Section 1.1. In Section 1.2, we introduce compact quantum groups and compact matrix quantum groups as well as morphisms between them. In particular, we explain that compact (matrix) quantum groups with a commutative underlying C∗-algebra correspond to the classical case, i.e. to function algebras of compact (matrix) groups. In Section 1.3, we introduce compact quantum groups associated to group algebras of discrete groups, which are a key ingredient for the structure description of group-theoretical quantum groups in Part II. We conclude this chapter with some constructions for compact matrix quantum groups, more precisely the direct product and the glued direct product of compact matrix quantum groups; see Section 1.4.
1.1. C
∗-algebras
There are two different equivalent definitions for C∗-algebras, either by introducing them as certain abstract Banach ∗-algebras or as norm closed ∗-subalgebras of the algebra of bounded linear operators on a Hilbert space. We use the abstract approach in this thesis, as we mainly consider universal C∗-algebras, which can be treated in a purely abstract and algebraic way.
1.1.1. Algebraic definition
In this subsection give the algebraic definition of C∗-algebras and consider the most im- portant examples of C∗-algebras. We follow [Bla06, Ch. II.1].
Definition 1.1.1 A∗-algebra is an associative algebra Aover the complex numbers with an involutive antiautomorphism ∗ :A→A such that
(λa)∗ =λa∗ for all λ∈C, a∈A.
A C∗-(semi)norm ρ:A→C on a ∗-algebra A is a (semi)norm that satisfies ρ(a∗a) =ρ(a)2 and ρ(ab)≤ρ(a)ρ(b) for all a, b∈A.
A ∗-algebra which is also a Banach space with respect to a C∗-norm, i.e. complete with respect to this norm, is called C∗-algebra.
Remark 1.1.2 Note that, for any C∗-(semi)norm ρ:A →C, we have ρ(a∗) =ρ(a) for alla ∈A,
since ρ(a)2 = ρ(a∗a)≤ ρ(a∗)ρ(a) and ρ(a∗)2 =ρ(aa∗) ≤ ρ(a)ρ(a∗) implyρ(a)≤ ρ(a∗)≤ ρ(a).
Definition 1.1.3An algebra homomorphismϕ:A→B between two ∗-algebras is called
∗-homomorphism if ϕ(a∗) =ϕ(a)∗ for all a∈A.
Remark 1.1.4One can show that any∗-homomorphism between two C∗-algebras is norm- decreasing and hence continuous; see [Bla06, Cor. II.1.6.6.]. Thus ∗-homomorphisms are also the natural choice of morphisms between C∗-algebras. Moreover, this means that any
∗-isomorphism between two C∗-algebras is an isometry.
Example 1.1.5
(i) Consider the continuous complex-valued functions
C(X) = {φ:X →C|φ continous}
on a compact Hausdorff spaceX. Then C(X) is a commutative algebra with point- wise addition and multiplication. Moreover, point-wise complex conjugation turns C(X) into a∗-algebra and together with the supremum norm, which is well-defined since X is compact, it is a C∗-algebra.
(ii) We consider the matrix algebra
Mn(C) =Cn×n
for some n∈N. For any matrix A∈Cn×nthe adjoint matrix is the transposed and entry-wise conjugate matrix A∗ = (Aji)i,j∈n ∈ Cn×n. This defines a ∗-structure on Mn(C) and one can check that the operator norm on Mn(C), given by
kAk= max
x∈Cn,kxk2=1kAxk2 for all A∈Mn(C), is a C∗-norm. Hence Mn(C) is a C∗-algebra.
(iii) More generally we can consider the operator algebra of bounded linear operators L(H) ={T :H →H |T linear, bounded}
on a Hilbert spaceH. Any operatorT :H →Hhas anadjoint operatorT∗ :H →H with hh1, T h2i=hT∗h1, h2i, whereh., .idenotes the inner product onH. This turns L(H) into a∗-algebra and again one can show thatL(H) together with the operator norm, given by
kTk= inf{C ≥0| kT hk ≤Ckhk ∀h∈H} for all T ∈ L(H),
is a C∗-algebra. Note that we recover the matrix algebra Mn(C) = L(Cn) as a special case.
In fact, the previous examples are generic in the following sense. In Subsection 1.1.4 we show that any C∗-algebra is∗-isomorphic to a norm closed ∗-subalgebra ofL(H) for some Hilbert spaceH. This can be seen as an alternative definition of C∗-algebras.
Furthermore, for any compact Hausdorff space X the function algebra C(X) is a com- mutative C∗-algebra and vice versa the well-known Gelfand–Naimark theorem says that any commutative C∗-algebra is of this form; see for instance [Bla06, Thm. II.2.2.4].
Theorem 1.1.6 (Gelfand–Naimark) Let A be a commutative C*-algebra. Then X :={φ :A→C|φ non-zero ∗-homomorphism}
is a compact Hausdorff space (with respect to the weak ∗-topology) and A→C(X), a7→ˆa with ˆa:X →C, φ7→φ(a) is a ∗-isomorphism.
Thus we can identify commutative C∗-algebras with function algebras over compact Haus- dorff spaces and hence we can think of non-commutative C∗-algebras as function algebras over ’non-commutative topological spaces’.
The following notions are motivated by the interpretation of C∗-algebras as operator algebras.
Definition 1.1.7 LetA be a ∗-algebra. An element a∈A is called
• self-adjoint if a =a∗,
• unitary if aa∗ =a∗a= 1,
• normal if aa∗ =a∗a,
• projection if a =a∗ =a2,
• isometry if a∗a = 1,
• partial isometry if a∗a is a projection,
• positive if a=b∗b for some b∈A.
1.1.2. Universal C
∗-algebras
We introduce universal C∗-algebras, which are C∗-algebras that can be obtained by certain universal algebraic constructions. We follow [Cun93, Sec. 1].
We start with the definition of the universal C∗-enveloping algebra of a ∗-algebra, which can be viewed as the ’smallest’ C∗-algebra completion of a given∗-algebra.
Definition 1.1.8 Let A be a∗-algebra. We set
kak:= sup{ρ(a)|ρ is a C∗-seminorm on A}
for all a∈ A. If kak<∞ for all a ∈A, we define the universal C∗-enveloping algebra Aˆ of A as the completion of
A.
{a∈A| kak= 0}.
Remark 1.1.9 The universal C∗-enveloping algebra ˆA of A has the following universal property. Let ι : A → Aˆ denote the canonical ∗-homomorphism. If B is any C∗-algebra with a ∗-homomorphism ϕ0 : A → B, then there exists a unique ∗-homomorphism ϕ : Aˆ→B with ϕ◦ι=ϕ0.
A Aˆ
B
ϕ0
ι
ϕ
The abstract definition of a universal C∗-enveloping algebra allows us to consider abstract C∗-algebras given by generators and relations.
Definition 1.1.10 LetE ={xi |i∈I}be a set of indeterminates andE∗ ={x∗i |i∈I}
a set of indeterminates in bijection with E but disjoint. The ∗-algebra of polynomials in the (non-commuting) variables E∪E∗ is the algebra
P(E) := ChE∪E∗i,
where we extend ∗ : E∪E∗ → E ∪E∗, xi 7→ x∗i, x∗i 7→ xi on P(E), such that P(E) is a ∗-algebra. Let R ⊆ P(E) and consider the two-sided ideal J(R)P(E) given by the
∗-closure of the ideal generated by R. We put A(E, R) :=P(E).
J(R).
If the universal C∗-enveloping algebra of A(E, R) exists, we denote it by C∗(E |R) and call it the universal C∗-algebra with generators E and relations R.
Remark 1.1.11 Note that the universal C∗-algebra C∗(E |R) only exists if the relations impose a uniform bound on the norm of the generators. As a consequence of Remark 1.1.9,
the universal C∗-algebra C∗(E | R) has the following universal property. If B is any C∗- algebra with elements{yi ∈B |i∈I}fulfilling the relationsR, then there exists a unique
∗-homomorphismϕ:C∗(E |R)→B with ϕ(xi) = yi for all i∈I.
A(E, R) C∗(E |R)
B
xi7→yi ϕ
Example 1.1.12
(i) Consider the universal C∗-algebra
C∗(x|xx∗ =x∗x= 1),
which exists as ρ(x)2 =ρ(xx∗) = ρ(1) = 1. One can check that this C∗-algebra is isomorphic to the continuous functions on the sphere S1 ={z ∈C| |z|= 1} via
C∗(x|xx∗ =x∗x= 1)→C(S1), x7→idS1.
(ii) There does not exists a universal C∗-algebra C∗(x | x = x∗) since ρ(x) can be arbitrarily large.
1.1.3. Group C
∗-algebras of discrete groups
We introduce another important class of C∗-algebras, the maximal group C∗-algebras of discrete groups. We follow [Web17].
For any discrete group Γ the group algebra CΓ, given by CΓ ={X
g∈Γ
αgg |αg ∈C, αg 6= 0 only for finitely manyg ∈Γ}, with
X
g∈Γ
αgg
X
h∈Γ
βhh
= X
g,h∈Γ
αgβhgh,
has the following natural ∗-structure:
X
g∈Γ
αgg
∗
=X
g∈Γ
αgg−1.
Note that any element g ∈ Γ is unitary in CΓ and hence ρ(g) = qρ(g∗g) = 1 for any C∗-seminorm ρonCΓ. It follows that the universal C∗-enveloping algebra of CΓ exists.
Definition 1.1.13 Let Γ be a discrete group. The maximal group C∗-algebra C∗(Γ) is the universal C∗-enveloping algebra ofCΓ.
Remark 1.1.14 It is straightforward to check that the maximal group C∗-algebra C∗(Γ) of a discrete group Γ can also be written as a universal C∗-algebra as follows. Let e ∈Γ be the neutral element of Γ. Then we have
C∗(Γ)∼=C∗(ug, g ∈Γ|ue = 1, uguh =ugh, ug−1 =u∗g for all g, h∈Γ).
Moreover, if Γ is finite, then the group algebra CΓ is already a finite-dimensional C∗- algebra and we have C∗(Γ) =CΓ.
Example 1.1.15
(i) The maximal group C∗-algebra of the infinite cyclic group Z is given by C∗(Z) =C∗(z |zz∗ =z∗z = 1)
and hence it is ∗-isomorphic to C(S1); see Example 1.1.12(i).
(ii) Fork ∈Nthe maximal group C∗-algebra of the cyclic groupZk :=Z/kZof order k is given by
C∗(Zk) = C∗(z |zz∗ =z∗z = 1, zk = 1).
Note that we can also consider Zk as the compact subgroup of S1 that consists of allk-th roots of unity,
Zk ∼={e2πimk |m ∈k} ⊆S1,
and hence C(Zk) is a C∗-algebra by Example 1.1.5(i). One can check that the maximal group C∗-algebraC∗(Zk) and the function algebraC(Zk) are∗-isomorphic.
1.1.4. C
∗-algebras as bounded operators on Hilbert spaces
In this subsection, we recall the GNS-construction, which shows that any C∗-algebra can be realised concretely as an algebra of bounded linear operators on some Hilbert space.
We follow [Bla06, Ch. II.6].
Let us introduce representations of C∗-algebras.
Definition 1.1.16 Arepresentation of a C∗-algebra is a∗-homomorphism π:A→ L(H) for some Hilbert space H. We call π cyclic if there exists an element ξ ∈ H, the cyclic vector, such thatπ(A)ξ is dense inH. Two representationsπ1 :A→ L(H1) andπ2 :A→ L(H2) are calledunitary equivalent if there exists a unitary operator U ∈ L(H1, H2) with U π1(a) =π2(a)U for all a∈A.
The GNS-construction, which was independently discovered by Gelfand and Naimark [GN43] and Segal [Seg47], provides a method to construct representations of C∗-algebras.
For the following description of the GNS-construction see [Bla06, p. II.6.4].
Theorem 1.1.17 (GNS-construction) Let A be a C∗-algebra and φ : A → C a pos- itive linear functional on A, i.e. φ(a∗a) ≥ 0 for all a ∈ A. Then there exists a cyclic representation
πφ:A→ L(Hφ), which is unique up to unitary equivalence.
In general a GNS-representation does not have to be faithful. However, one can consider the universal GNS-representation of a C∗-algebra A, which is obtained by taking the direct sum of all GNS-representations
M
φ
πφ:A→ L(M
φ
Hφ);
see for instance [Tim08, Ch. 12.1.]. Gelfand and Naimark showed that the universal GNS- representation is always faithful and hence any C∗-algebra can be realised as a concrete operator algebra; see for instance [Bla06, Cor. II.6.4.10].
Theorem 1.1.18 (Gelfand–Naimark theorem) Any C∗-algebra A admits a faithful representation π : A → L(H). In particular, A is ∗-isomorphic to a norm closed ∗- subalgebra ofL(H).
1.1.5. Tensor products
In this subsection we introduce tensor product constructions for C∗-algebras. We follow [Bla06, Ch. II.9.].
Consider the algebraic tensor product A ⊗B of two C∗-algebras A and B. Then the operation (a⊗b)∗ = a∗ ⊗b∗ turns A⊗B into a ∗-algebra but it does not need to be a C∗-algebra. In fact, there may exist several norms such that the completions with respect to such a norm results in a C∗-algebra. For any C∗-normγ onA⊗B, we denote byA⊗γB the completion of A⊗B with respect to γ. We take a look at the two ’extremal’ cases, the maximal and the minimal tensor product.
Let us start with the minimal tensor product. Let A and B be C∗-algebras and let πA : A → L(HA), πB : B → L(HB) be the (faithful) universal GNS-representations, respectively. Then they induce a faithful representation
πA⊗πB :A⊗B → L(HA)⊗ L(HB)∼=L(HA⊗HB), a⊗b7→πA(a)⊗πB(b) onHA⊗HB; see for instance [Bla06, p. II.9.1.3.]. Hence
ka⊗bkmin :=kπA(a)⊗πB(b)k
is a C∗-norm on A ⊗B and the completion A⊗min B can be identified with the C∗- subalgebra of L(HA⊗HB) generated by all operators of the form πA(a)⊗πB(b) with a∈A, b∈B.
Definition 1.1.19 Let A and B be C∗-algebras. The completion A⊗minB of A⊗B with respect to k.kmin is called the minimal tensor product of A and B.
Remark 1.1.20 It turns out that k.kmin is indeed the smallest C∗-norm on A⊗B, i.e.
for all C∗-norms γ on A⊗ B, we have ka ⊗bkmin ≤ ka ⊗bkγ and hence a surjective
∗-homomorphism A⊗γB A⊗minB; see for instance [Bla06, p. II.9.5.1.].
Now let us take a look at the maximal tensor product. One can show that γ(a⊗b) ≤ kakAkbkB < ∞ for any C∗-seminorm γ on A⊗B; see [Bla06, Cor. II.9.2.2.]. Thus the universal C∗-enveloping algebra of A⊗B (recall Definition 1.1.8) exists, since
ka⊗bkmax := sup{ρ(a⊗b)|ρ is a C∗-seminorm on A⊗B}
exists for all a⊗b. Moreover, k.kmax is a C∗-norm (not only a C∗-seminorm) on A⊗B, since ka⊗bkmax≥ ka⊗bkmin >0 for all 06=a∈A and 06=b ∈B.
Definition 1.1.21 Let A and B be C∗-algebras. The universal C∗-enveloping algebra A⊗maxB of A⊗B is called the maximal tensor product of A and B.
Remark 1.1.22By definitionk.kmaxis the largest C∗-norm onA⊗B and by Remark 1.1.9 there exists a surjective∗-homomorphismA⊗maxB A⊗γB for all C∗-normsγ onA⊗B.
Remark 1.1.23 Note that in general k.kmin 6= k.kmax. However, if we consider finite- dimensional C∗-algebras A and B, then A⊗B is finite-dimensional and hence already a C∗-algebra. Since there exists only one C∗-norm on a C∗-algebra, we havek.kmin =k.kmax in this case.
Consider compact Hausdorff spacesX1 and X2. Then
C(X1)⊗C(X2)→C(X1×X2), f⊗g 7→(X1×X2 →C, (x, y)7→f(x)g(y)) is an isomorphism; see [Bla06, Thm. II.9.4.4.]. Thus C(X1)⊗C(X2) is again already a C∗-algebra and we have k.kmin =k.kmax.
1.2. Compact (matrix) quantum groups
For any compact groupGthe continuous complex-valued functionsC(G) form a commut- ative C∗-algebra; see Example 1.1.5(i), on which the group structure of Ginduces a dual structure. Woronowicz [Wor87a] axiomatised this dual structure and then replacedC(G) by a not necessarily commutative C∗-algebra, yielding the notion of compact quantum groups. Hence we can think of compact quantum groups with non-commutative under- lying C∗-algebras as (function algebras over) some kind of ’non-commutative topological groups’. Furthermore, ifGis a matrix group, thenC(G) has a canonical matrix represent- ation. This can be generalised to the notion of compact matrix quantum groups. In this section we explain these concepts in detail, provide some standard examples and discuss morphisms between compact (matrix) quantum groups. We mainly follow [Wor87a] and [Wor98] in this section.
