Abstracts

Benjamin Anderson-Sackaney: Amenability of Fusion Modules and Quasi-regular Representations
Our talk will be about a notion of amenability for fusion modules and some characterizations in the setting of compact quantum groups. Coamenability of a compact quantum group G is characterized by a certain constraint on the classical dimension function of the fusion ring C[Rep(G)] generated by the unitary finite dimensional representations Rep(G). A fusion ring together with a dimension function is called amenable if it satisfies an anologue of this constraint. The category of unital G-C*-algebras with unital G-equivariant *-homomorphisms as morphisms is known to be in bijection with certain fusion modules over C[Rep(G)]. This can be refined further for coideals. The fusion modules over C[Rep(G)] naturally come equipped with an analogue of the classical dimension function. In this talk we will introduce amenability of a fusion module. In the case of a coideal that admits a quasi-regular representation on the universal C*-algebra C_u(G), amenability of its corresponding fusion module characterizes a certain coamenability property of the quasi-regular representation. This coamenability property recovers coamenability of an inclusion of groups in the sense of Popa and Monod. We will discuss these things as well as some additional interesting results, including a duality result for our coamenability property of certain quasi-regular representations.
This is joint work with Leonid Vainerman.

Yuki Arano: Actions of tensor categories on C*-algebras
Tensor categories arise naturally from subfactors, quantum groups and many other mathematical objects. In particular, the classification of subfactors is closely related to actions of tensor categories on a factor. We use this machinery to C*-algebras and overview a possible direction toward classifying C*-subalgebras of finite index.

Kenny De Commer: (TBA)

Joel Right Dzokou Talla: Invariant integrals on coideals and their Drinfeld doubles
Let A be a CQG Hopf *-algebra, i.e. a Hopf *-algebra with a positive invariant state. Given a *-invariant right coideal subalgebra B of A, we provide conditions for the existence of a quasi-invariant integral on the orthogonal coideal B^{\perp} inside the dual discrete multiplier Hopf *-algebra of A.  Given such a quasi-invariant integral, we show how it can be extended to a quasi-invariant integral on the Drinfeld double coideal. As an application, we determine the quasi-invariant integral for the coideal *-algebra U_q(2,R) constructed from the Podleś spheres. Based on joint work with Kenny De Commer.

Uwe Franz: Brownian motions on Universal Quantum Groups
I will describe the Gaussian generating functionals in the sense of Schürmann on the universal unitary and orthogonal quantum groups of
Wang and Van Daele. Then I will discuss properties of their Gaussian parts, i.e., of their minimal quantum subgroups through which all these generating functionals factorize. Based on joint work with Amaury Freslon and Adam Skalski.

Amaury Freslon: Classical actions of quantum permutations
Quantum permutation groups can act non-trivially, and even ergodically, on finite spaces. This is, in view of many quantum rigidity results, an exception and it is natural to wonder whether there are other classical spaces on which quantum permutations can act. H. Huang constructed a family of such spaces, and we will show that these are the only possibilities. This is a joint work with F. Taipe and S. Wang.

Erik Habbestad: Braided quantum groups over abelian groups
We apply Majid’s transmutation procedure to Hopf algebra maps H -> C[T], where T is a compact abelian group, and explain how this construction gives rise to braided Hopf algebras over quotients of T by subgroups that are cocentral in H. This allows us to unify and generalize a number of recent constructions of braided compact quantum groups, starting from the braided SU_q(2) quantum group, and describe their bosonizations. This is joint work with Sergey Neshveyev.

Lukas Hataishi: Injectivity and crossed products
In this talk I shall discuss a strategy to proving that the crossed product functor, from discrete quantum group actions to actions of the dual compact quantum group, preserves injectivity. The main tool in our strategy is an operator system theoretic version of the crossed product, the Fubini crossed product, and its interpretation as a cotensor product. It allows us to generalize and simplify proofs of several claims on injectivity, proven first in the classical case by Hamana. This talk will be based on a joint work with Joeri Ludo de Ro.

Pawel Kasprzak: Quantum Mycielski Graphs
Mycielskian of a graph G = (V,E) is a  graph M(G) with vertex space  containing  two copies of V and a singleton; the edge space of M(G) are chosen in a way that chromatic number of M(G) is one larger then the chromatic number of G and clique numbers of G and M(G) are the same. In my talk I will discuss Mycielski construction for quantum graphs and discuss the behaviour of chromatic and clique number in this case. Joint work with Arek Bochniak. 

Kan Kitamura: Discrete quantum subgroups of complex semisimple quantum groups
In this talk, I will examine discrete quantum subgroups of the quantum double of a compact quantum group. Brannan-Chirvasitu-Viselter proposed a quantum version of lattices in locally compact quantum groups by considering the invariant state on the corresponding homogeneous space. It will be observed that at least discrete quantum subgroups do not yield such homogeneous spaces, when the original compact quantum group is not of Kac type. In the case of the quantum double of q-deformation of a compact semisimple Lie group regarded as its complexification, we can also obtain a classification of its discrete quantum subgroups.

Marco Matassa: Crystal limits of compact semisimple quantum groups as higher-rank graph algebras
I will discuss some results concerning the crystal limit of q-deformations of compact semisimple Lie groups, that is the limit when the parameter q tends to zero. I will discuss how the resulting objects can be described as higher-rank algebras, a notion originally introduced by Kumjan-Pask. The construction makes use of the theory of crystal bases by Kashiwara and Lusztig. Based on joint work with Robert Yuncken.

J. P. McCarthy: Another look at idempotent states on quantum permutation groups
Woronowicz proved the existence of the Haar state for compact quantum groups under a separability assumption later removed by Van Daele in a new existence proof. A minor adaptation of Van Daele's proof yields an idempotent state in any non-empty weak*-compact convolution-closed convex subset of the state space. Such subsets, and their associated idempotent states, are studied in the case of quantum permutation groups.

Sergey Neshveyev: Dual cocycles and quantization of a class of locally compact groups
We discuss the problem of quantization and compare three approaches that give a solution for a class of locally compact groups - via reflecting across the Hopf-Galois object defined by an irreducible (possibly projective) representation, via cocycle twisting defined by an explicit dual 2-cocycle, and via a bicrossed product defined by a matched pair of subgroups. (Joint work in progress with Pierre Bieliavsky, Victor Gayral and Lars Tuset.)

Sergio Pacheco: Adapting C*-classification techniques equivariantly with respect to actions of tensor categories
In this talk I will discuss an adaptation of approximate intertwining arguments equivariant with respect to actions of unitary tensor categories.  As an application of these intertwining techniques, I will discuss a characterisation of equivariant D-stability, for strongly self-absorbing C*-algebras D, which generalises a classical result of Toms and Winter. Time permitting I may discuss how through this characterisation one may access the question of preservence of Jiang-Su stability for finite index inclusions.

Lukas Rollier: Generalised Tannaka-Krein duality for proper actions of algebraic quantum groups on discrete spaces
Woronowicz’s famous version of the Tannaka-Krein duality establishes a bridge between compact quantum groups on the one hand, and their representation categories on the other. In recent work, Vaes and myself used a ‘similar construction’ to obtain quantum automorphism groups of connected locally finite graphs as algebraic quantum groups. In this presentation, I will solidify what this ‘similar construction’ entails, and show that in general, a proper action of an algebraic quantum group on a discrete space may be built from any unitary 2-category that admits a sufficiently well-behaved functor into the category of finite type pre-Hilbert bimodules over that space. This is ongoing work in collaboration with Stefaan Vaes.

Alexander Shapiro: Positive and Gelfand-Tsetlin representations of quantum groups
I will give a cluster-algebraic description of the Gelfand-Tsetlin subalgebra of a quantum group in type A, and discuss a relation between Gelfand-Tsetlin and positive representations thereof. This talk will be based on joint works with Gus Schrader.

Piotr Sołtan: On some invariants of quantum groups
I will describe ongoing work with Jacek Krajczok on the invariants of quantum groups defined by the actions of the scaling and modular automorphisms. I will show how an application of these invariants leads to distinguishing between uncountably many compact quantum groups described by isomorphic von Neumann algebras of type III. Next I will introduce a more systematic approach to the invariants, present results of some calculations and discuss a conjecture relating one of the invariants to the question whether a given compact quantum group is of Kac type. Finally I will describe how to obtain proofs that the conjecture holds in many cases.

Lyudmila Turowska: No-signaling quantum bicorrelations and quantum graph isomorphisms
I will discuss quantum no-signaling correlations introduced by Duan and Winter and its different subclasses (quantum commuting, quantum and local). They will appear as strategies of non-local games with quantum inputs and quantum outputs.  I will then introduce an analogue of bisynchronous correlations and characterise  them by tracial  states on the universal C*-algebra of the projective free unitary group, showing that in the quantum input/output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. As an application, I will discuss quantum graph isomorphisms by giving their non-local game interpretation, and compare our approach with the existing algebraic notions of quantum graph isomorphisms. In the case of classical graphs our operational notion of quantum isomorphism leads to new quantum symmetries. This is a joint work with Michael Brannan, Sam Harris and Ivan Todorov.

Stefaan Vaes: Quantum automorphism groups of connected locally finite graphs
I present a joint work with Lukas Rollier. We construct the quantum automorphism group of any connected locally finite, possibly infinite, graph as a locally compact quantum group that has the classical (locally compact) automorphism group as a closed quantum subgroup. For finite graphs, we get the quantum automorphism group of Banica and Bichon. One of the key tools is the construction of a unitary tensor category associated with any connected locally finite graph. When this graph is the Cayley graph of a finitely generated group, the associated unitary tensor category has a canonical fiber functor. We thus also obtain a quantization procedure for arbitrary finitely generated groups, which turns out to be isomorphic to dual of the quantum isometry group of the corresponding spectral triple, as defined by Bhowmick and Skalski.

Roland Vergnioux: An analogue of the Radulescu basis for orthogonal free quantum groups
In past joint work with Freslon we proved that the radial subalgebra of orthogonal free quantum group algebras is maximal abelian and singular. In ongoing joint work with Xumin Wang we investigate the corresponding bimodule by constructing an analogue of the Radulescu basis from the free group case, and showing that it is a Riesz basis. This is motivated by potential applications to maximal injectivity. As an easy application this already allows to determine the Pukánsky invariant of the inclusion.

Ami Viselter: Invariant objects on quantum homogeneous spaces
We will discuss several types of invariant objects, mostly ones related to convolution semigroups, on quantum homogeneous spaces and the relations between them. We will focus on homogeneous spaces coming from idempotent states.

Hua Wang: Some observations on the reconstruction of certain quantum groups
I will present some observations of Woronowicz's version of Tannak-Krein reconstruction. Instead of reconstructing the compact quantum group, the focus shall be on the corresponding multiplicative unitary instead, aiming to go (perhaps slightly) beyond the compact/discrete case. If time permits, I shall also talk about some relevant topological Hopf algebra structures. This is a work in progress. 

Anna Wysoczańska-Kula: Hochschild cohomology of universal unitary quantum groups
My talk will be devoted to a recent result computing the Hochschild cohomology of universal unitary quantum groups U_F^+, F in GL_n(C). For that purpose we exhibit a free-glued product structure of U_F^+, and use the projective resolution of O_E^+ given by J.Bichon (2013). This is a joint result with I. Baraquin, U. Franz, M. Gerhold and M. Tobolski.

Makoto Yamashita: Tensor categories of classical Lie type
Representation theory of simple Lie groups has interesting deformations in the framework of tensor categories. What kind of categories can have the fusion rules of type A theory (corresponding to finite-dimensional representations of special linear groups) is well understood by the work of Kazhdan and Wenzl (1993). We look at the remaining classical series, namely the fusion rules of type BCD theories (corresponding to orthogonal and symplectic groups), and give a similar classification. The proof is based on Liu’s technique to find Kauffman bracket relations in planar algebras, and certain torsion freeness of the Lie type categories by Voigt and Goffeng for compact quantum groups, and by Schopieray and Gannon for fusion categories. This, combined with our previous work on the classification of dimension-preserving fiber functors and categorical Poisson boundary, leads to a classification of non-Kac type compact quantum groups of the classical fusion type. Based on ongoing joint work with P. Grossman and S. Neshveyev.

Sophie Emma Zegers: Classification of quantum lens spaces
In the study of noncommutative geometry, various of classical spaces have been given a quantum analogue. One such example is the quantum lens spaces described by Hong and Szymański as graph C*-algebras. The graph C*-algebraic description has made it possible to obtain important information about their structure and to work on classification. Moreover, every quantum lens space comes with a natural circle action, leading to an equivariant isomorphisms problem.
In this talk, I will first present results on classification of low dimensional quantum lens spaces based on joint work with Thomas Gotfredsen. Then we investigate the existence and construction of equivariant isomorphisms of low dimensional quantum lens spaces based on recent joint work with Søren Eilers. Contrary to the isomorphism problem, we can no longer use the graph C*-algebraic description to solve this problem.