QGS: Session 2022-2023
Autumn session 2022-2023
Benjamin Anderson-Sackaney (Université de Caen)
October 18, 2022.
Title: Relative Amenability, Amenability, and Coamenability of Coideals.
Abstract: Amenability is a deeply studied property of groups, with many interesting reformulations and connections to the operator algebraic aspects of groups. For example, the reduced C*-algebra C*_r(G) of a discrete group has a unique tracial state if and only if there are no non-trivial amenable normal subgroups. This, among other related results, makes it apparent that the structure of the amenable subgroups of G contains important information about C*_r(G). For a quantum group 𝔾, an appropriate analogue of a subgroup is a coideal N ⊆ L∞(𝔾). We will present notions of relative amenability, amenability, and coamenability for coideals of discrete and compact quantum groups motivated by "relativizations" of amenability and coamenability of a subgroup of a group. We will discuss the known relationships between these formally distinct notions and their relevance to certain properties of the reduced C*-algebras of discrete quantum groups.
No talk
October 25, 2022.
No talk
November 1, 2022.
Magnus Landstad (Norwegian University of Science and Technology)
November 8, 2022.
Title: Exotic group algebras, crossed products, and coactions.
Abstract: If G is a locally compact group, we have the full group C*-algebra C*(G) and the reduced C*_r(G). We call a C*-algebra properly between C*(G) and C*_r(G) exotic.
Similarly, if G acts on a C*-algebra A we can form the full crossed product C*(GxA) and the reduced crossed product C*_r(GxA). An exotic crossed product is a C*-algebra properly between the two. Work by Baum, Guentner, and Willett show that these algebras are relevant to the Baum-Connes conjecture.
We think that the best way to study these algebras is by also looking at the corresponding dual theory of coactions. I will discuss some of these aspects, but there will be more questions than answers.
This is joint work with Steve Kaliszewski and John Quigg.
Alfons Van Daele (KU Leuven)
November 15, 2022.
Title: Algebraic quantum hypergroups and duality.
Abstract: Let G be a finite group and H a subgroup. The set 𝒢 of double cosets HpH, with p∊G has the structure of an hypergroup. The product of two elements HpH and HqH is the set of cosets HrH where r∊pHq. The algebra A of functions on 𝒢 is the space of functions on G that are constant on double cosets. It carries a natural coproduct, dual to the product, and given by:
∆(p,q)=1/n ∑_{h\in H} f(phq),
where n is the number of elements in H. The dual algebra is known as the Hecke algebra associated with the pair G, H.
In this talk I will discuss the notion of an algebraic quantum hypergroup, its fundamental properties and duality for algebraic quantum hypergroups. I will illustrate this with an example, coming from bicrossproduct theory, constructed from a pair of closed subgroups H and K of a group G, with the assumption that the map H ⋂ K = {e}.
This is part of more general work in progress with M. Landstad (NTNU Trondheim).
Stefaan Vaes (KU Leuven)
November 22, 2022 at 5:00 pm (CET) - Note the unusual time -
Title: Quantum automorphism groups of connected locally finite graphs and quantizations of finitely generated groups.
Abstract: 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. In the particular example of groups defined by a triangle presentation, this construction gives the property (T) discrete quantum groups from earlier joint work with Valvekens.
Fatemeh Khosravi (Seoul National University)
November 29, 2022 at 11:00 am (CET) - Note the unusual time -
Title: Co-amenable quantum homogeneous spaces of compact Kac quantum groups.
Abstract: Given a locally compact group G, Leptin's theorem states that G is amenable if and only if the Fourier algebra A(G) admits a bounded approximate identity, where the latter property is known as co-amenability of the quantum dual of G. In the quantum setting, this characterization is known as the duality between amenability and co-amenability. It is proved that a discrete quantum group is amenable if and only if its dual compact quantum group is co-amenable. The definition of co-amenability for quantum homogeneous spaces is given by Kalantar-Kasprzak-Skalski-Vergnioux. Furthermore, they ask whether the co-amenability of a quantum homogeneous space is equivalent to the (relative) amenability of its co-dual. In this talk, we will answer this question for quantum homogeneous spaces of compact Kac quantum groups under a mild assumption. Based on joint work with Mehrdad Kalantar.
No talk: Quantum Groups: Current Trends and New Perspectives (Oslo)
December 6, 2022.
Erik Habbestad (University of Oslo)
December 13, 2022.
Title: C*-algebras associated to Temperley-Lieb polynomials.
Abstract: We define Temperley-Lieb polynomials and consider the (standard) subproduct systems they generate. This subproduct system turns out to be equivariant with respect to a compact quantum group G monoidally equivalent to U_q(2). Exploiting this we are able to describe the C*-algebras associated to the subproduct system, which turn out to be closesly related to the linking algebra B(U_q(2),G). This is joint work with Sergey Neshveyev.
Kan Kitamura (University of Tokyo)
December 20, 2022.
Title: Partial Pontryagin duality for actions of quantum groups on C*-algebras.
Abstract: We compare actions on C*-algebras of two constructions of locally compact quantum groups, the bicrossed product due to Vaes-Vainerman and the double crossed product due to Baaj-Vaes. We give a one-to-one correspondence between them up to Morita equivalence, in the same spirit as Takesaki-Takai and Baaj-Skandalis dualities. This includes a duality between a quantum double and the product of the original quantum group with its opposite. We will explain its consequences for equivariant Kasparov theory in relation to the quantum analog of the Baum-Connes conjecture.
Christmas holidays!
December 13, 2021 - Januay 24, 2022.
Winter session 2022-2023
Julia Plavnik (Indiana University Bloomington)
January 24, 2023.
Title: Comparing different constructions of modular categories.
Abstract: Modular categories arise naturally in many areas of mathematics, such as conformal field theory, representations of braid groups, quantum groups, and Hopf algebras, and low dimensional topology, and they have important applications in condensed matter physics.
Despite recent progress in the classification of modular categories, we are still in the early stages of this theory and the general landscape remains largely unexplored. One important step towards deepening our understanding of modular categories is to have well-studied constructions. In this talk, we will present an overview of various of these constructions and compare their properties. We will focus on ribbon zesting and symmetry gauging, and we will comment on some constructions in the G-crossed setting.
Matthew Daws (University of Central Lancashire)
January 31, 2023.
Title: Around the Approximation Property for Quantum Groups.
Abstract: I will introduce what the "approximation property" (AP) is for (locally compact) groups, and provide a few applications. I will then talk about how one might give an analogous definition for (locally compact) quantum groups, explaining some of the need technology along the way. Time allowing, I will discuss how the AP interacts with various common constructions, and also about "central" versions and links with tensor categories.
Oleg Aristov
February 7, 2023.
Title: Complex-analytic approach to quantum groups.
Abstract: We discuss quantum analogues of complex Lie groups. Our approach is closer to classical quantum group theory than to C*-algebraic one (no multipliers and no invariant weights). I propose to consider a topological Hopf algebra with a finiteness condition (holomorphically finitely generated or HFG for short). This topic seems to offer a wide range of research opportunities.
Our focus is on examples, such as analytic forms of some classical quantum groups (a deformation of a solvable Lie group and Drinfeld-Jimbo algebras). I also present some general results: (1) the category of Stein groups is anti-equivalent to the category of commutative Hopf HFG algebras; (2) If G is a compactly generated Lie group, the associated convolution cocommutative topological Hopf algebra (introduced by Akbarov) is HFG. When, in addition, G is connected and linear, the structure of this cocommutative algebra can be described explicitly. I also plan to discuss briefly holomorphic duality (which is parallel to Pontryagin duality).
Arthur Troupel (Université de Paris)
February 14, 2023.
Title: Free wreath products as fundamental graph C*-algebras.
Abstract: The free wreath product of a compact quantum group by the quantum permutation group S_N^+ has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such groups is well-known, but some results about their operator algebras were still open, for example Haagerup property, K-amenability or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental C*-algebras of certain graphs of C*-algebras, and we deduce these properties from these constructions.
Devarshi Mukherjee (Universidad de Buenos Aires)
February 21, 2023.
Title: Noncommutative geometry in mixed characteristic.
Abstract: I will give an overview of noncommutative topological algebras and their cohomology theories in the setting of the p-adic integers.
This will entail constructions that are familiar from the complex case, such as the formation of a smooth subalgebra of a C*-algebra. The examples I will specialise these constructions to are group algebras of discrete and p-adic Lie groups. It turns out that these are also examples of bornological quantum groups (in the sense of Voigt). Finally, if time permits, I will also discuss the computations of the Hochschild homology of the completions of such algebras.
No talk
February 28, 2023.
Mao Hoshino (University of Tokyo)
March 7, 2023.
Title: Equivariant covering spaces of quantum homogeneous spaces.
Abstract: In this talk I will explain the imprimitivity theorems for equivariant correspondences in two cases: for a general compact quantum group under a finiteness condition, and for the Drinfeld-Jimbo deformation of a semisimple compact Lie group. These results involve the representation theories of function algebras and the Tannaka-Krein duality for equivariant correspondences. I also would like to give some applications if time allows.
Melody Molander (UC Santa Barbara)
March 14, 2023.
Title: Skein Theory for Affine ADE Subfactor Planar Algebras.
Abstract: Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. These planar algebras also encode two other invariants of the subfactors: the index and the principal graph. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and give some presentations of subfactor planar algebras of index 4 which have affine ADE Dynkin diagrams as their principal graphs.
No talk: Nowrooz (Persian New Year)
March 21, 2022.
Yasuyuki Kawahigashi (University of Tokyo)
March 28, 2023 at 1:00 pm (CEST) - Note the unusual time -
Title: Topological order, tensor networks and subfactors.
Abstract: I will explain interactions between two-dimensional topological order and subfactors from a viewpoint of tensor networks. The range of a certain finite dimensional projection appearing in statistical physics is identified with the higher relative commutant of the subfactor arising from such a tensor network. We then work out the machinery of alpha-induction for braided fusion categories in the setting of certain 4-tensors, called bi-unitary connections, appearing in subfactor theory.
Easter holidays!
March 28, 2023 - April 25, 2023.
Spring session 2022-2023
Michael Brannan (University of Waterloo)
April 25, 2023.
Title: No-signalling bicorrelations and generalized quantum automorphisms of graphs.
Abstract: I'll report on some recent joint work with Sam Harris, Lyudmila Turowska and Ivan Todorov (arXiv:2302.04268), where we introduce an analogue of bisynchronous correlations in the context of quantum input-quantum output non-local games. One of the main motivations of this work was to find a non-local game interpretation of the quantum automorphisms and isomorphisms of quantum graphs that have appeared recently in the literature. I'll explain how these considerations are related to tracial representations of quantum automorphism groups of matrix algebras, and in the case of ordinary graphs, lead us to a softer (and possibly more general) notion of quantum symmetry for graphs.
No talk
May 2, 2023.
No talk
May 9, 2023.
No talk
May 16, 2023.
No talk: Canadian Operator Symposium 2023
May 23, 2023.
No talk: Canadian Operator Symposium 2023
May 23, 2023.
No talk
May 30, 2023.
Joeri De Ro (Vrije Universiteit Brussel)
June 6, 2023.
Title: Actions of compact and discrete quantum groups on operator systems.
Abstract: We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. Given an action of a discrete quantum group on an operator system X, we introduce associated crossed products, and we prove that equivariant injectivity of the operator system X is equivalent with dual equivariant injectivity of the associated crossed products. As an application of this result, we prove a duality result for equivariant injective envelopes. This is joint work with Lucas Hataishi.
No talk: Noncommutative Geometry Festival 2023
June 13, 2023.
Mainak Ghosh (Indian Statistical Institute)
June 20, 2023 at 11:00 am (CEST) - Note the unusual time -
Title: Unitary connections and Q-systems.
Abstract: The standard invariant plays a major role in subfactor theory. In this talk, I will discuss a 2-categorical generalization of an axiomatization of the standard invariant and further discuss some algebraic structures associated to it. This is based on joint work with P. Das, S. Ghosh and C. Jones (arXiv:2211.03822) and on arxiv : 2302.04921.
Summer holidays!
June 20, 2023 - ?, 2023.