QGS: Session 2021-2022

• Title: Crossed products of representable localization algebras.

• Abstract: Let X be a locally compact, Hausdorff space. The representable localization algebra for X was introduced and studied by Willett and Yu. The K-theory of the algebra serves as the representable K-homology of the space X.

Now let G be a second countable, locally compact group and suppose that X is a proper G-space. It turns out that the K-theory of the crossed product by G of the representable localization algebra for X serves as the representable G-equivariant K-homology of the proper G-space X.

The goal of this talk is to describe these facts and roles of the representable localization algebras in the study of the Baum—Connes conjecture.

• Slides.

• Title: A covariant Stinespring theorem.

• Abstract: We will introduce a finite-dimensional covariant Stinespring theorem for compact quantum groups. Let G be a compact quantum group, and let T:= Rep(G) be the rigid C*-tensor category of finite-dimensional continuous unitary representations of G. Let Mod(T) be the rigid C*-2-category of cofinite semisimple finitely decomposable T-module categories. We show that finite-dimensional G-C*-algebras (a.k.a C*-dynamical systems) can be identified with equivalence classes of 1-morphisms out of the object T in Mod(T). For 1-morphisms X: T -> M1, Y: T -> M2, we show that covariant channels between the corresponding G-C*-algebras can be 'dilated' to isometries t: X -> Y \otimes E, where E: M2 -> M1 is some 'environment' 1-morphism. Dilations are unique up to partial isometry on the environment; in particular, the dilation minimising the quantum dimension of the environment is unique up to a unitary. When G is a compact group this implies and generalises previous covariant Stinespring-type theorems.

We will also discuss some results relating to rigid C*-2-categories, including that any connected semisimple rigid C*-2-category is equivalent to Mod(T) for some rigid C*-tensor category T. (Here semisimple means not just semisimplicity of Hom-categories but also idempotent splitting for 1-morphisms, direct sums for objects, etc.)

This talk is based on the paper arXiv:2108.09872.

• Slides.

• Title: Gaussian states and Gaussian parts of compact quantum groups.

• Abstract: I will motivate and explain the notion of a Gaussian state on a compact quantum group G, as introduced by Michael Schürmann. This concept leads to the idea of the Gaussian part of G, understood as the smallest quantum subgroup of G which supports all the Gaussian states of G. I will discuss properties of Gaussian states and compute Gaussian parts for several examples. This turns out to be related to quantum connectedness and certain topological generation questions for quantum subgroups. The talk will be based on joint work with Uwe Franz and Amaury Freslon.

• Slides.

• Title: Amenability and weak containment for étale groupoids.

• Abstract: A famous theorem of Hulanicki says that a locally compact group is amenable if and only if its full and reduced C*-algebras coincide. For groupoids, the situation is more delicate: While amenabiltiy implies equatility of the full and reduced C*-algebra, the converse fails according to examples by Willett. The behavior of Willett's groupoids can be explained by their non-exactness. We show that if an étale groupoid satisfies a certain exactness condition, then equality of its full and reduced C*-algebra is equivalent to amenability of the groupoid.

• Slides.

• Title: An introduction to diagram algebras.

• Abstract: In this talk, I will introduce the notion of a diagram algebra and explain their connection to the representation theory of compact quantum groups. I will also describe the role that they play for loop models in statistical physics as well as the correspondence between their traces and random walks on graphs.

• Slides.

• Title: Cluster quantization from factorization homology.

• Abstract: The character variety of a manifold is its moduli space of flat G-bundles.  These moduli spaces and their quantizations appear in a number of places in mathematics, representation theory, and quantum field theory.  Famously, Fock and Goncharov showed that a certain "decorated" variant of character varieties carries the structure of a cluster variety -- that is, the moduli space contains a distinguished set of toric charts, with combinatorially defined transitions functions (called mutations).  This led them to a now-famous quantization of their decorated character varieties.

In this talk I'll explain that the by-hands construction of these charts by Fock and Goncharov can in fact be extracted from a more general framework called stratified factorization homology, and I'll outline how this allows us to extend the Fock-Goncharov story from surfaces to 3-manifolds.

• Title: C(X) -Algebras and their K-Stability.

• Abstract: Non-stable K-theory is the study of the homotopy groups of the group of (quasi-) unitaries of a C*-algebra. We will give an overview of the theory, and discuss a special class of C*-algebras, termed as K-stable C*-algebras along with its rational analogue. We shall give a permanence property related to K-stability (rational Kstability) concerning continuous C(X)-algebras. We will end with an application of the aforementioned result to crossed product C*-algebras.

• Slides.

• Title: Strongly 1-Bounded Quantum Group von Neumann Algebras.

• Abstract: Strong 1-boundedness is a property for a tracial von Neumann algebra M that was introduced by Jung that allows one to distinguish M from the (interpolated) free group factors. Many examples came from group von Neumann algebras, such as those from certain groups having property (T). For quantum group von Neumann algebras, Brannan and Vergnioux showed in a landmark paper that those coming from the orthogonal free quantum groups are strongly 1-bounded, despite sharing many structural properties with the free group factors. We first review these developments, and then report on recent progress concerning permanence of strong 1-boundedness under finite index subfactors and applications to quantum automorphism groups such as the quantum permutation group S_{N^2}^+. This last part is based on ongoing joint work with Brannan, Harris, and Yamashita.

• Slides.

• Title: Dynamical quantum graphical calculus.

• Abstract: Graphical calculus provides a diagrammatic framework for performing topological computations with morphisms in strict tensor categories. The key idea is to identify such morphisms with oriented diagrams labeled by their in- and output objects. This was formalized by Reshetikhin and Turaev, by constructing for every strict tensor category C a strict tensor functor that assigns isotopy classes of C-colored ribbon graphs to morphisms in C. This can be applied to the tensor category of finite-dimensional representations of a quantum group U_q(g).

In this talk, I will first outline the fundamentals of this finite-dimensional quantum graphical calculus. Then I will explain how it can be extended to a larger category of quantum group representations, encompassing the quantum group analog of the BGG category O. In particular, this extended framework allows to visualize U_q(g)-intertwiners on Verma modules, as well as morphisms depending on a dynamical parameter, such as dynamical R-matrices. Finally, I will describe how this dynamical quantum graphical calculus can be used to obtain q-difference equations for quantum spherical functions.

This talk is based on joint work with Nicolai Reshetikhin (UC Berkeley) and Jasper Stokman (University of Amsterdam).

• Title: On the quantum symplectic sphere.

• Abstract: The algebra of the quantum symplectic (4n-1)-sphere O(S_q^{4n-1}) is defined as a subalgebra of the quantum symplectic group by Faddeev, Reshetikhin and Takhtajan. Recently D'Andrea and Landi investigated faithfull irreducible *-representations of O(S_q^{4n-1}). They proved that the first n-1 generators of its enveloping C^*-algebra C(S_q^{4n-1}) are all zero. The result is a generalisation of the case where n=2 which was shown by Mikkelsen and Szymański.

In this talk, I will first present how C(S_q^{4n-1}) can be described as a graph C^*-algebra, from which it follows that C(S_q^{4n-1}) is isomorphic to the quantum (2n+1)-sphere by Vaksman and Soibelman. Then, I present a candidate of a vector space basis for O(S_q^{4n-1}) which is constructed by a nontrivial application of the Diamond lemma. The conjecture is supported by computer experiments for n=1,...,8.  By finding a vector space basis we can moreover conclude that the n-1 generators are non-zero inside the algebra O(S_q^{4n-1}).

• Slides.

• Title: The Frucht property in the quantum group setting.

• Abstract: A classical theorem of Frucht states that every finite group is the automorphism group of a finite graph. Is every quantum permutation group the quantum automorphism group of a finite graph? In this talk we will answer this question with the help of orbits and orbitals.

This talk is based on joint work with Teo Banica.

• Slides.

• Title: A metric characterization of freeness.

• Abstract: Freeness of random variables has many characterizations, with free cumulants, free entropy, Schwinger-Dyson equations, etc. Here, we discuss a new metric characterization with the norm of the sum of generators tensored by their adjoint, and explain the relation and applications to other problems in operator algebras and von Neumann algebras. Time permitting, we will also discuss some ingredients of the proof. This is based on joint work with Leonard Cadilhac.

• Title: A graph with quantum symmetry and finite quantum automorphism group.

• Abstract: This talk concerns quantum automorphism groups of graphs, a generalization of automorphism groups of graphs in the framework of compact matrix quantum groups. We will focus on certain colored graphs constructed from linear constraint systems. In particular, we will give an explicit connection of the solution group of the linear constraint system and the quantum automorphism group of the corresponding colored graph. Using this connection and a decoloring procedure, we will present an example of a graph with quantum symmetry and finite quantum automorphism group. This talk is based on joint work with David Roberson.

• Slides.

• Title: Lp-Lq Fourier multipliers on locally compact quantum groups.

• Abstract: Hörmander proved that the Fourier multiplier is Lp-Lq bounded if the symbol lies in the weak Lr space, for certain p,q,r. In recent years, this result was generalized to more general groups and quantum groups. Here we presented an extension to certain locally compact quantum groups. It covers the known results and the proof is simpler. It also yields a family of Lp-Fourier multipliers over compact quantum groups of Kac type. The talk is based on arXiv:2201.08346.

• Slides.

• Title: Partitions, quantum group actions and rigidity.

• Abstract: In this talk, I will present a new combinatorial approach to the study of ergodic actions of compact quantum groups. The connection between compact quantum groups and the combinatorics of partitions goes back to Banica's founding work on the representation theory of free orthogonal quantum groups, and was later formalized in the seminal paper of Banica and Speicher under the theory of "easy quantum groups". Based on some new alternative version of the Tannaka-Krein reconstruction procedure for ergodic actions, we extend Banica and Speicher's combinatorial approach to the setting of ergodic actions of compact quantum groups. Our examples in particular recovers actions on finite spaces, on embedded homogeneous spaces and on quotient spaces. Moreover, we use this categorical point of view to study the quantum rigidity of ergodic actions on classical spaces, and show that the free quantum groups cannot act ergodically on a classical connected compact space, thereby answering a question of D. Goswami and H. Huang.

The talk is based on the recent preprint arXiv:2112.07506 jointly with Amaury Freslon and Frank Taipe.

• Slides.

• Title: Quantization of the 2-sphere.

• Abstract: The quantization problem is the problem of associating non-commutative quantum algebras to a classical Poisson algebra in such a way that the commutator is related to the Poisson bracket. In a formal setting, this problem and its equivariant counterpart are well-understood and can always be solved (under a mild assumption in the equivariant case). However, in a C*-algebraic setting, there exist obstructions to equivariant quantization, for example for the 2-sphere. In this talk, we will give a brief introduction to the quantization problem, and propose a way to obtain an equivariant quantization of the 2-sphere in a Fréchet algebraic setting.

• Slides.

• Title: Braided quantum symmetries of graph C*-algebras.

• Abstract: A braided compact quantum group (over T) is, roughly speaking, a “compact quantum group” object in the category of T-C*-algebras equipped with a twisted monoidal structure. In this talk, we shall explain the existence of a universal braided compact quantum group acting on a graph C*-algebra in the category mentioned above. Time permitting, we shall sketch the proof, constructing along the way a braided analogue of the free unitary quantum group. Finally, as an example, we shall compute this universal braided compact quantum group for the Cuntz algebra.

• Slides.

• Title: An introduction to crossed products by group actions on C*-algebras.

• Abstract: We give a survey of some results on crossed products by discrete group actions and discuss their basic properties. Further, we restrict our attention to finite group actions with the Rokhlin property, approximate representability, and their weakened versions. Time permitting, we outline some structure results for the crossed products by these classes of group actions and their contributions to finite-dimensional quantum groups.

• Title: On the differential geometry of Lie groups of Fröbenius type.

• Abstract: The talk will be based on the papers:

(1) In the first one, joint with V. Gayral (Memoirs AMS 2015), we construct universal deformation formulae for actions on topological algebras (C* or Fréchet) of the Lie groups which carries a negatively curved left-invariant Kähler structure.

(2) A second one, joint with V. Gayral, S. Neshveyev and L. Tuset, where we construct locally compact quantum groups from star products on a class of Lie groups.

The Lie groups on which these deformations are performed (in both (1) and (2)) are of "Frobenius type''. This means that their Lie algebras carry an exact non-degenerate two-cocycle or, equivalently, that they admit an open co-adjoint orbit. In both cases, the star products, say at the formal level, are of Fedosov type i.e. associated with a left-invariant symplectic torsion free affine connection on the group manifold at hand. In particular, they are obtained from differential theoretical considerations.

However, there is a dichotomy: the orderings of the star products considered in (1) and (2) are different. In (1), we deal with Weyl ordered star products, while in (2) with normal (or anti-normal) ones. This has, apparently, a strong effect on the regularity of the categories those constructions live in: smooth versus measurable or topological. 

More precisely: In (1), we definitely deal with a ``smooth object'', e.g. the universal deformation formula (i.e. the twist) allows to deform smooth vectors of the group action, e.g. they are relevant in differential noncommutative geometry in the sense of A. Connes. But, no locally compact quantum group is present there. And until now, I haven't be able to define a reasonable notion of ``smooth quantum group'' attached to the construction. In (2), the quantum group is present, but the deformation procedure apparently breaks smoothness: smooth vectors of strongly continuous actions (i.e. smooth module-algebras) of the group are not stable under twisting.

In the talk, I will discuss  differential geometrical aspects of Frobenius Lie groups within this deformation quantization context. I will end with a suggestion based on the possible use of a Lie group theoretical version of a microlocal analytical tool: Hörmander's smooth wave front set.

• Title: Yetter-Drinfeld algebras, module categories and injectivity.

• Abstract: Many examples of quantum group actions carry a Yetter-Drinfeld structure. Among them, you find C*-algebras coming from the boundary theory of Drinfeld doubles, which is closely related to the theory of ucp maps and injective envelopes of Hamana. Exploring Tannaka-Krein duality for quantum group actions, it is possible to extend many concepts and results of boundary theory to the categorical setting, but the lack of a categorification of non-braided-commutative Yetter-Drinfeld algebras impose an obstruction to a full analogy. In this talk, I will sketch how to perform such a categorification and relate it to the study of injectivity for module categories. Based on joint works with E. Habbestad, S. Neshveyev and M. Yamashita.

• Title: Pivotality, twisted centres and the anti-double of a Hopf monad.

• Abstract: Pairs in involution are an algebraic structure whose systematic study is motivated by their applications in knot theory, representation theory and cyclic homology theories. In this talk, we will explore a categorical view for these objects from the perspective of representation theory of monoidal categories. A focus will lie on illustrating how their existence is linked to a particular well-behaved notion of duality called pivotality. In particular, we will show how the language of monads allows us to combine the algebraic with the categorical perspective of these pairs. 

This talk is based on the article arXiv:2201.05361.

• Title: Partial algebraic quantum groups and their Drinfeld doubles.

• Abstract: In this talk, we will define partial algebraic quantum groups, which are special cases of weak multiplier Hopf algebras, as introduced by Van Daele and Wang. At the same time, they provide a generalization to the notion of a partial compact quantum group, as introduced by De Commer and Timmermann. The main aim of the talk will be to realize the Drinfeld double of a partial compact quantum group as a partial algebraic quantum group.

This talk is based on joint work with K. De Commer.

• Slides.

• Title: Hecke algebras and the Schlichting completion for discrete quantum groups.

• Abstract: In recent joint work with Skalski and Voigt we construct and study the  Hecke algebra and Hecke operators associated with an almost normal subgroup in a discrete quantum group. We also give in this framework a quantum version of the Schlichting completion, which yields an algebraic quantum group with a compact-open subgroup. We describe a class of examples arising from HNN extensions.

• Slides.

• Title: Q-systems and higher unitary idempotent completion for C*-algebras.

• Abstract: Q-systems were introduced by Longo to study finite index inclusions of infinite von Neumann factors. A Q-system is a unitary version of a Frobenius algebra object in a tensor category or a C* 2-category. By the work of Müger, Q-systems give an axiomatization of the standard invariant of a finite index subfactor.

Following work of Douglass-Reutter, a Q-system is also a unitary version of a higher idempotent. In this talk, we will describe a higher unitary idempotent completion for C* 2-categories called Q-system completion.

Our main goal is to show that C*Alg, the C* 2-category of right correspondences of unital C*-algebras is Q-system complete. To do so, we will use the graphical calculus for C* 2-categories, and adapt a subfactor reconstruction technique called realization, which is inverse to Q-system completion. This result allows for the straightforward adaptation of subfactor results to C*-algebras, characterizing finite index extensions of unital C*-algebras equipped with a faithful conditional expectation in terms of the Q-systems in C*Alg. If time allows, we will discuss an application to induce new symmetries of C*-algebras from old via Q-system completion.

This is joint work with Chen, C. Jones and Penneys (arXiv: 2105.12010).

• Title: Filtered Frobenius algebras in monoidal categories.

• Abstract: We develop filtered-graded techniques for algebras in monoidal categories with the goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. We then produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form. These two results of independent interest are used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category is represented by a Frobenius algebra in it. This is joint work with Dr. Chelsea Walton (Rice University).

• Slides.