Publications Abstracts

Preprints

• Brannan, Michael; Eifler, Kari; Voigt, Christian; Weber, M.
Quantum Cuntz-Krieger algebras
arXiv:2009.09466 [math.OA, math.QA], 40 pages (2020).  Motivated by the theory of Cuntz-Krieger algebras we define and study C*-algebras associated to directed quantum graphs. For classical graphs the C*-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to KK-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C*-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of KK-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
• Schmidt, Simon; Vogeli, Chase; Weber, M.
Uniformly vertex-transitive graphs
arXiv:1912.00060 [math.CO], 17 pages (2019).  We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph of a given graph to show that the uniform vertex-transitivity of the graph is equivalent to the existence of cliques of sufficient size in the derangement graph. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-transitivity in the situation of a graph with an imprimitive automorphism group. We classify the non-Cayley uniformly vertex-transitive graphs on less than 30 vertices outside of two complementary pairs of graphs.

• Eder, Christian; Levandovskyy, Viktor; Schanz, Julien; Schmidt, Simon; Steenpass, Andreas; Weber, M.
Existence of quantum symmetries for graphs on up to seven vertices: a computer based approach
arXiv:1906.12097 [math.QA, math.CO], 15 pages + appendix (2019).  The symmetries of a finite graph are described by its automorphism group; in the setting of Woronowicz's quantum groups, a notion of a quantum automorphism group has been defined by Banica capturing the quantum symmetries of the graph. In general, there are more quantum symmetries than symmetries and it is a non-trivial task to determine when this is the case for a given graph: The question is whether or not the algebra associated to the quantum automorphism group is commutative. We use Gröbner base computations in order to tackle this problem; the implementation uses GAP and the SINGULAR package LETTERPLACE. We determine the existence of quantum symmetries for all connected, undirected graphs without multiple edges and without self-edges, for up to seven vertices. As an outcome, we infer within our regime that a classical automorphism group of order one or two is an obstruction for the existence of quantum symmetries.

• Weber, M.
Partition C*-algebras
arXiv:1710.06199 [math.OA, math.CO, math.QA], 19 pages + 11 pages of appendix and references (2017).  We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain relations may be deduced from others and we explain a partition calculus for simplifying such computations. This article is a small note for C*-algebraists having no background in compact quantum groups, although our partition C*-algebras are motivated from those underlying Banica-Speicher quantum groups (also called easy quantum groups). We list many open questions about partition C*-algebras that may be tackled by purely C*-algebraic means, ranging from ideal structures and representations on Hilbert spaces to K-theory and isomorphism questions. In a follow up article, we deal with the quantum algebraic structure associated to partition C*-algebras.

• Weber, M.
Partition C*-algebras II - links to compact matrix quantum groups
arXiv:1710.08662 [math.OA, math.CO, math.QA], 27 pages (2017).  In a recent article, we gave a definition of partition C*-algebras. These are universal C*-algebras based on algebraic relations which are induced from partitions of sets. In this follow up article, we show that often we can associate a Hopf algebra structure to partition C*-algebras, and also a compact matrix quantum group structure. This follows the lines of Banica and Speicher's approach to quantum groups; however, we access them in a more algebraic way circumventing Tannaka-Krein duality. We give criteria when these quantum groups are quantum subgroups of Wang's free orthogonal quantum group. As a consequence, we see that even if we start with (generalized) categories of partitions which do not contain the pair partitions, in many cases we do not go beyond the class of Banica-Speicher quantum groups (aka easy quantum groups). However, we also discuss possible non-unitary Banica-Speicher quantum groups.

• Cébron, Guillaume; Weber, M.
Quantum groups based on spatial partitions
arXiv:1609.02321 [math.QA, math.OA], 32 pages (2016).  We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions - which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group On+ which do not contain the symmetric group Sn; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of partitions which do neither contain the pair partition nor the identity partition.

Monographs

• Weber, M.
Basiswissen Mathematik auf Arabisch und Deutsch
Springer Spektrum, 2018
168 pages
 Dieses Lehrbuch ist speziell für angehende Studierende mit arabischem Sprachhintergrund verfasst, die ein Studium im deutschen Sprachraum aufnehmen wollen. Um ihnen sowohl den sprachlichen als auch den fachlichen Einstieg zu erleichtern, ist die Gestaltung zweisprachig. Dies ermöglicht sowohl das Anknüpfen an bekannte Inhalte in der Muttersprache als auch das Erlernen der deutschen Begriffe. Inhaltlich frischt das Buch sehr konzentriert und konkret das nötigste mathematische Abiturwissen auf, das in Studiengängen wie Mathematik, Informatik, Natur- und Ingenieurwissenschaften vorausgesetzt wird. Das Buch ist grob in Analysis und Algebra gegliedert und beinhaltet möglichst wenige formale Definitionen, dafür aber viele anschauliche Beispiele und Verfahren sowie Beispielaufgaben.
• Voiculescu, Dan-Virgil; Stammeier, Nicolai; Weber, M. (eds)
Free probability and operator algebras
Münster Lecture Notes in Mathematics
European Mathematical Society (EMS)
132 pages
Zürich, 2016  Free probability is a probability theory dealing with variables having the highest degree of noncommutativity, an aspect found in many areas (quantum mechanics, free group algebras, random matrices etc). Thirty years after its foundation, it is a well-established and very active field of mathematics. Originating from Voiculescu's attempt to solve the free group factor problem in operator algebras, free probability has important connections with random matrix theory, combinatorics, harmonic analysis, representation theory of large groups, and wireless communication. These lecture notes arose from a masterclass in Münster, Germany and present the state of free probability from an operator algebraic perspective. This volume includes introductory lectures on random matrices and combinatorics of free probability (Speicher), free monotone transport (Shlyakhtenko), free group factors (Dykema), free convolution (Bercovici), easy quantum groups (Weber), and a historical review with an outlook (Voiculescu). In order to make it more accessible, the exposition features a chapter on basics in free probability, and exercises for each part. This book is aimed at master students to early career researchers familiar with basic notions and concepts from operator algebras.

Chapters in monographs

• Weber, M.
Basics in free probability, 6 pages
in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.

• Weber, M.
Easy quantum groups, 23 pages
in Free probability and operator algebras, ed. by Dan-V. Voiculescu, N. Stammeier, M. Weber, EMS, 2016.

In peer reviewed journals

Generating linear categories of partitions
to appear in Kyoto Journal of Mathematics, 2021
arXiv:1904.00166 [math.CT, math.QA], 19 pages (2019).  We present an algorithm for approximating linear categories of partitions (of sets). We report on concrete computer experiments based on this algorithm and how we found new examples of compact matrix quantum groups (so called non-easy'' quantum groups) with it. This also led to further theoretical insights regarding the representation theory of such quantum groups.

• Nechita, Ion; Schmidt, Simon; Weber, M.
Sinkhorn algorithm for quantum permutation groups
to appear in Experimental Mathematics, 2021
arXiv:1911.04912 [math.QA], 16 pages (2019).  We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.

• Mang, Alexander; Weber, M.
Non-Hyperoctahedral Categories of Two-Colored Partitions, Part II: All Possible Parameter Values
to appear in Applied Categorial Structures, 2021
arXiv:2003.00569 [math.CO, math.QA], 34 pages (2020).  This article is part of a series with the aim of classifying all non-hyperoctahedral categories of two-colored partitions. Those constitute by some Tannaka-Krein type result the co-representation categories of a specific class of quantum groups. However, our series of articles is purely combinatorial. In Part I we introduced a class of parameters which gave rise to many new non-hyperoctahedral categories of partitions. In the present article we show that this class actually contains all possible parameter values of all non-hyperoctahedral categories of partitions. This is an important step towards the classification of all non-hyperoctahedral categories.

• Mang, Alexander; Weber, M.
Non-hyperoctahedral categories of two-colored partitions, Part I: New categories
to appear in Journal of Algebraic Combinatorics
arXiv:1907.11417 [math.CO, math.QA], 30 pages (2019).  Compact quantum groups can be studied by investigating their co-representation categories in analogy to the Schur-Weyl/Tannaka-Krein approach. For the special class of (unitary) "easy" quantum groups these categories arise from a combinatorial structure: Rows of two-colored points form the objects, partitions of two such rows the morphisms; vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes O, B, S and H of such categories (inspired respectively by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups) we treat the first three - the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. The article is purely combinatorial in nature; The quantum group aspects are left out.

• Mang, Alexander; Weber, M.
Categories of two-colored pair partitions, Part II: Categories indexed by semigroups
Journal of Combinatorial Theory, Series A, Vol. 180, 105509, 43 pp., 2021
arXiv:1901.03266 [math.CO, math.QA], 37 pages (2019).  Within the framework of unitary easy quantum groups, we study an analogue of Brauer's Schur-Weyl approach to the representation theory of the orthogonal group. We consider concrete combinatorial categories whose morphisms are formed by partitions of finite sets into disjoint subsets of cardinality two; the points of these sets are colored black or white. These categories correspond to "half-liberated easy" interpolations between the unitary group and Wang's quantum counterpart. We complete the classification of all such categories demonstrating that the subcategories of a certain natural halfway point are equivalent to additive subsemigroups of the natural numbers; the categories above this halfway point have been classified in a preceding article. We achieve this using combinatorial means exclusively. Our work reveals that the half-liberation procedure is quite different from what was previously known from the orthogonal case.

New products and Z2-extensions of compact matrix quantum groups
to appear in Annales de l'Institut Fourier
arXiv:1907.08462 [math.QA, math.OA], 39 pages (2019).  There are two very natural products of compact matrix quantum groups: the tensor product G x H and the free product G * H. We define a number of further products interpolating these two. We focus more in detail to the case where G is an easy quantum group and H = Z2, the dual of the cyclic group of order two. We study subgroups of G * Z2 using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.

• Junk, Luca; Schmidt, Simon; Weber, M.
Almost all trees have quantum symmetry
Archiv der Mathematik, Vol. 115, 267-278, 2020.
arXiv:1911.02952 [math.CO, math.QA], 11 pages (2019).  From the work of Erdös and Renyi from 1963 it is known that almost all graphs have no symmetry. In 2017, Lupini, Mancinska and Roberson proved a quantum counterpart: Almost all graphs have no quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica's definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz's compact quantum groups. Now, Erdös and Renyi also proved a complementary result in 1963: Almost all trees do have symmetry. The crucial point is the almost sure existence of a cherry in a tree. But even more is true: We almost surely have two cherries in a tree - and we derive that almost all trees have quantum symmetry. We give an explicit proof of this quantum counterpart of Erdös and Renyi's result on trees.

• Jung, Stefan; Weber, M.
Models of quantum permutations. With an appendix by Alexandru Chirvasitu and Pawel Joziak
Journal of Functional Analysis, Vol. 279, Issue 2, 1 August 2020, 108516
See also: arXiv:1906.10409 [math.OA, math.QA], 24 pages.  For N greater or equal to 4 we present a series of *-homomorphisms from C(SN+) to Bn where SN+ is the quantum permutation group. They are not necessarily representations of the quantum group SN+ but they yield good and somewhat small'' operator algebraic models of quantum permutation matrices. In the inverse limit however they produce a quantum group which turns out to be isomorphic to SN+; the latter fact is thanks to an argument by Alexandru Chirvasitu and Pawel Joziak building on topological generation.

Intertwiner spaces of quantum group subrepresentations
Communications in Mathematical Physics, Vol. 376, 81-115, 2020.
See also: arXiv:1811.02821 [math.QA, math.OA], 38 pages.  We consider compact matrix quantum groups whose N-dimensional fundamental representation decomposes into an (N-1)-dimensional and a one-dimensional subrepresentation. Even if we know that the compact matrix quantum group associated to this (N-1)-dimensional subrepresentation is isomorphic to the given N-dimensional one, it is a priori not clear how the intertwiner spaces transform under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups but not as compact matrix quantum groups.

• Speicher, Roland; Weber, M.
Quantum groups with partial commutation relations
Indiana University Mathematics Journal, Vol. 68, 1849-1883, 2019.
See also: arXiv:1603.09192 [math.QA, math.OA], 44 pages (2016).  We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group: They are partially commutative in a way such that they do not interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon's quantum automorphism groups of graphs. They fit with the mixture of classical and free independence as recently defined by Speicher and Wysoczanski (rediscovering Lambda-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.

• Mang, Alexander; Weber, M.
Categories of two-colored pair partitions, Part I: Categories indexed by cyclic groups
The Ramanujan Journal, Vol. 53, 181-208, 2020.
arXiv:1809.06948 [math.CO, math.QA], 25 pages (2018).  We classify certain categories of partitions of finite sets subject to specific rules on the colorization of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka-Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group and Wang's free unitary quantum group. In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside.

• Weber, M.; Zhao, Mang
Factorization of Frieze patterns
Revista de la Union Matematica Argentina, Vol. 60 (2), 407-415, 2019
See also: arXiv:1809.00274 [math.CO], 9 pages.  In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator for quiddity cycles and its equivalence classes, respectively. We show that the sum is neither commutative nor associative, but we may circumvent this issue by passing to equivalence classes. We also address the question whether a decomposition of quiddity cycles into irreducible factors is unique and we answer it in the negative by giving counterexamples. We conclude that even under stronger assumptions, there is no canonical decomposition.

• Jung, Stefan; Weber, M.
Partition quantum spaces
Journal of Noncommutative Geometry, Vol. 14, Issue 3, 821-85, 2020
arXiv:1801.06376 [math.OA, math.FA], 35 pages (2018).  We propose a definition of partition quantum spaces. They are given by universal C*-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first d columns of easy quantum groups. However, we define them as universal C*-algebras rather than as C*-subalgebras of easy quantum groups. We also investigate the minimal number d needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, d takes the values one or two.

• Schmidt, Simon; Weber, M.
Quantum symmetries of graph C*-algebras
Canadian Mathematical Bulletin 61, 848-864, 2018
See also: arXiv:1706.08833 [math.OA, math.FA], 18 pages.  The study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

• Tarrago, Pierre; Weber, M.
The classification of tensor categories of two-colored noncrossing partitions
Journal of Combinatorial Theory, Series A, Vol. 154, Feb 2018, 464-506
See also: arXiv:1509.00988 [math.CO, math. QA], 40 pages.  Our basic objects are partitions of finite sets of points into disjoint subsets. We investigate sets of partitions which are closed under taking tensor products, composition and involution, and which contain certain base partitions. These so called categories of partitions are exactly the tensor categories being used in the theory of Banica and Speicher's orthogonal easy quantum groups. In our approach, we additionally allow a coloring of the points. This serves as the basis for the introduction of unitary easy quantum groups, which is done in a separate article. The present article however is purely combinatorial. We find all categories of two-colored noncrossing partitions. For doing so, we extract certain parameters with values in the natural numbers specifying the colorization of the categories on a global as well as on a local level. It turns out that there are ten series of categories, each indexed by one or two parameters from the natural numbers, plus two additional categories. This is just the beginning of the classification of categories of two-colored partitions and we point out open problems at the end of the article.

• Weber, M.
Introduction to compact (matrix) quantum groups and Banica-Speicher (easy) quantum groups
Notes of a lecture series at IMSc Chennai, India, 2015
Indian Academy of Sciences. Proceedings. Mathematical Sciences, Vol. 127, Issue 5, pp 881-933, Nov 2017.  This is a transcript of a series of eight lectures, 90 minutes each, held at IMSc Chennai, India from 05--24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz's quantum version of the Tannaka-Krein theorem. Building on this, we define Banica-Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica-Speicher quantum groups and we list some applications. We review the state of the art regarding Banica-Speicher quantum groups and we list some open problems.

• Mai, Tobias; Speicher, Roland; Weber, M.
Absence of algebraic relations and of zero divisors under the assumption of finite full
non-microstates free entropy dimension

Advances in Mathematics, Vol. 304, 2 January 2017, pages 1080-1107.
(we extended the former version arXiv:1407.5715 substantially)  We show that in a tracial and finitely generated W*-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the generators does not have atoms. Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.

• Tarrago, Pierre; Weber, M.
Unitary easy quantum groups: the free case and the group case
International Mathematics Research Notes, 18, 1 Sept 2017, 5710-5750.
See also: arXiv:1512.00195 [math. QA, math.OA], 39 pages.  Easy quantum groups have been studied intensively since the time they were introduced by Banica and Speicher in 2009. They arise as a subclass of (C*-algebraic) compact matrix quantum groups in the sense of Woronowicz. Due to some Tannaka-Krein type result, they are completely determined by the combinatorics of categories of (set theoretical) partitions. So far, only orthogonal easy quantum groups have been considered in order to understand quantum subgroups of the free orthogonal quantum group On+. We now give a definition of unitary easy quantum groups using colored partitions to tackle the problem of finding quantum subgroups of Un+. In the free case (i.e. restricting to noncrossing partitions), the corresponding categories of partitions have recently been classified by the authors by purely combinatorial means. There are ten series showing up each indexed by one or two discrete parameters, plus two additional quantum groups. We now present the quantum group picture of it and investigate them in detail. We show how they can be constructed from other known examples using generalizations of Banica's free complexification. For doing so, we introduce new kinds of products between quantum groups. We also study the notion of easy groups.

• Gabriel, Olivier; Weber, M.
Fixed point algebras for easy quantum groups
SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) 12 (2016), 097, 21 pages.
See also: arXiv:1606.00569 [math.OA, math.KT], 21 pages.  Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author, we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group Sn+, the free orthogonal quantum group On+ and the quantum reflection groups Hns+. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups - which are related to Hopf-Galois extensions.

• Freslon, Amaury; Weber, M.
On bi-free De Finetti theorems
Annales Mathématiques Blaise Pascal, 23(1), 21-51, 2016.
See also: arXiv:1501.05124 [math.PR, math.OA, math.QA], 16 pages.  We investigate possible generalizations of the de Finetti theorem to bi-free probability. We first introduce a twisted action of the quantum permutation groups corresponding to the combinatorics of bi-freeness. We then study properties of families of pairs of variables which are invariant under this action, both in the bi-noncommutative setting and in the usual noncommutative setting. We do not have a completely satisfying analogue of the de Finetti theorem, but we have partial results leading the way. We end with suggestions concerning the symmetries of a potential notion of n-freeness.

• Raum, Sven; Weber, M.
The full classification of orthogonal easy quantum groups
Communications in Mathematical Physics, 341(3), 751-779, Feb 2016.
See also: arXiv:1312.3857 [math.QA], 38 pages.  We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica-Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon's free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.

• Raum, Sven; Weber, M.
Easy quantum groups and quantum subgroups of a semi-direct product quantum group
Journal of Noncommutative Geometry, Vol. 9(4), 1261-1293, 2015
See also: arXiv:1311.7630 [math.QA], 26 pages.  We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.

• Freslon, Amaury; Weber, M.
On the representation theory of partition (easy) quantum groups
Journal für die reine und angewandte Mathematik [Crelle's Journal], Vol. 2016, Issue 720 (Nov 2016), 2016.
See also: arXiv:1308.6390 [math.QA], 42 pages.  Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group SN+, the free orthogonal quantum group ON+ as well as the hyperoctahedral quantum group HN+. We then extend our work to unitary easy quantum groups and link it with a ''freeness conjecture'' of T. Banica and R. Vergnioux.

• Raum, Sven; Weber, M.
The combinatorics of an algebraic class of easy quantum groups
Infinite Dimensional Analysis, Quantum Probability and related topics Vol 17, No. 3, 2014.
See also: arXiv:1312.1497 [math.QA], 16 pages.  Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum permutation group and his free orthogonal quantum group. In this article, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another article.

• Weber, M.
On the classification of easy quantum groups
Advances in Mathematics, Volume 245, 1 October 2013, pages 500-533.
See also: arXiv:1201.4723 [math.OA], 39 pages.  In 2009, Banica and Speicher began to study the compact quantum groups G with Sn c G c On+ whose intertwiner spaces are induced by some partitions. These so-called easy quantum groups have a deep connection to combinatorics. We continue their work on classifying these objects, by introducing some new examples of easy quantum groups. In particular, we show that the six easy groups On, Sn, Hn, Bn, Sn' and Bn' split into seven cases On+, Sn+, Hn+, Bn+, Sn'+, Bn'+ and Bn#+ on the side of free easy quantum groups. Also, we give a complete classification in the half-liberated and in the nonhyperoctahedral case.

• Weber, M.
On C*-Algebras Generated by Isometries with Twisted Commutation Relations
Journal of Functional Analysis, Volume 264, Issue 8, pages 1975-2004, 2013.
See also: arXiv:1207.3038 [math.OA], 35 pages.  In the theory of C*-algebras, interesting noncommutative structures arise as deformations of the tensor product, e.g. the rotation algebra Atheta as a deformation of C(S1)\otimes C(S1). We deform the tensor product of two Toeplitz algebras in the same way and study the universal C*-algebra T \otimesthetaT generated by two isometries u and v such that uv = e2 pi i thetavu and u*v = e-2 pi i thetavu*, for theta in R. Since the second relation implies the first one, we also consider the universal C*-algebra T *theta T generated by two isometries u and v with the weaker relation uv = e2 pi i thetavu. Such a "weaker case" does not exist in the case of unitaries, and it turns out to be much more interesting than the twisted "tensor product case" T \otimesthetaT. We show that T \otimesthetaT is nuclear, whereas T *theta T is not even exact. Also, we compute the K-groups and we obtain K0(T *theta T) = Z and K1(T *theta T) = Z, and the same K-groups for T \otimesthetaT.