QGS: Session 2020-2021

• Title: Furstenberg boundary of a discrete quantum group.

• Abstract: The notion of topological boundary actions has recently found striking applications in the study of operator algebras associated to discrete groups. We will discuss the analogue concept for discrete quantum groups, show that in this generalization there still always exists a maximal boundary action - the so-called Furstenberg boundary. We discuss applications in problems of C*-simplicity and uniqueness of the Haar state of the dual.

This is joint work with Pawel Kasprzak, Adam Skalski and Roland Vergnioux.

• Slides.

• Title: On the Baum-Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture.

• Abstract: We give a decomposition of the equivariant Kasparov category for a discrete quantum group with torsions. This formulates the Baum-Connes assembly map for general discrete quantum groups possibly with torsion. As an application, we show that the group C*-algebra of a discrete quantum group in a certain class satisfies the UCT.

• Slides.

• Title: A quantization of Sylvester's law of inertia.

• Abstract: Sylvester's law of inertia states that two self-adjoint matrices A and B are related as A = X*BX for some invertible complex matrix X if and only if A and B have the same signature (N_+,N_-,N_0), i.e. the same number of positive, negative and zero eigenvalues. In this talk, we will discuss a quantized version of this law: we consider the reflection equation *-algebra (REA), which is a quantization of the *-algebra of polynomial functions on self-adjoint matrices, together with a natural adjoint action by quantum GL(N,C). We then show that to each irreducible bounded *-representation of the REA can be associated an extended signature (N_+,N_-,N_0,[r]) with [r] in R/Z, and we will explain in what way this is a complete invariant of the orbits under the action by quantum GL(N,C). This is part of a work in progress jointly with Stephen Moore.

• Slides.

• Title: The spectrum of equivariant Kasparov theory for cyclic groups of prime order.

• Abstract: In 2006, Ralf Meyer and Ryszard Nest proved that the G-equivariant Kasparov category of a locally compact group G carries the structure of a tensor-triangulated category. This structure conveniently handles the usual homological algebra, bootstrap constructions and assembly maps involved in many KK-theoretical calculations, e.g. in connection with the Baum-Connes conjecture.  As with any tensor triangulated category, we can also associate to the G-equivariant Kasparov category its spectrum in the sense of Paul Balmer. This is a topological space (similar to the Zariski spectrum of a commutative ring) which allows us, as it were, to re-inject some genuinely geometric ideas in non-commutative geometry. It turns out that the spectrum contains enough information to prove the Baum-Connes conjecture for G, hence we should expect the question of its computation to be very hard.  In this talk, after discussing such preliminaries and motivation, I will present joint work with Ralf Meyer providing the state of the art on this subject. Although more general partial results are known, a complete answer is only known so far for finite groups of prime order and for algebras in the bootstrap category.

• Slides.

• Title: Riesz transforms on compact quantum groups and strong solidity.

• Abstract: The Riesz transform is one of the most important and classical examples of a Fourier multiplier on the real numbers. It may be described as the operator "nabla_j Delta^{-1/2}" where nabla_j = d/dx_j is the derivative and Delta is the Laplace operator. In a more general context the Riesz transform may always be defined for any diffusion semigroup on the reals. In case the generator of this semi-group is the Laplace operator the classical Riesz transform is retrieved. In quantum probability the quantum Markov semi-groups play the role of the diffusion semi-groups and again a suitable notion of Riesz transform can be described.

In this paper we show that the Riesz transform may be used to prove rigidity properties of von Neumann algebras. We focus in particular on examples from compact quantum groups. Using these tools we show that a class of quantum groups admits rigidity properties. The class has the following properties:

1. SUq(2) is contained in it.

2. The class is stable under monoidal equivalence, free products, dual quantum subgroups and wreath products with S_N+.

The rigidity properties include the Akemann-Ostrand property and strong solidity. Part of this talk is based on joint work with Mateusz Wasilewski and Yusuke Isono.

• Slides. 

• Title: Dynamics of compact quantum metric spaces.

• Abstract: The classical Gelfand correspondence justifies the slogan that  C*-algebras are to be thought of as "non-commutative Hausdorff spaces", and  Rieffel's theory of compact quantum metric spaces provides, in the same vein, a non-commutative counterpart to the theory of compact metric spaces. The aim of my talk is to introduce the basics of this theory, and explain some new results on dynamical systems of compact quantum metric spaces.   If time permits, I will also touch upon another recent result,  which shows  how quantized intervals approximate a classical interval in the quantum version of the Gromov-Hausdorff distance. This is based on joint works with Jens Kaad and Thomas Gotfredsen.

• Slides.

• Title: Discrete subfactors, realization of algebra objects, and Q-system completion.

• Abstract: In recent decades, we have seen that quantum symmetries of quantum mathematical objects, like non-commutative spaces and quantum field theories, are best described by quantum groups, subfactors, and unitary tensor categories. Subfactor classification has led to discovery of interesting "exotic" quantum symmetries and to important constructions for unitary tensor categories. For example, Q-systems (special C* Frobnius algebra objects) were introduced by Longo to characterize the canonical endomorphism for type III subfactors, which is the analog of Jones' basic construction for type II_1 and Kosaki's version for type III. We will use this perspective to discuss some subfactor results which go beyond small index classification, making connections to quantum groups along the way. We'll then discuss a version of a unitary higher idempotent completion for C*/W* 2-categories based on Gaiotto--Johnson-Freyd's theory of condensations in higher categories.

• Slides.

• Title: About the monoidal invariance of cohomological dimension of Hopf algebras.

• Abstract: I will discuss the question whether Hopf algebras having monoidally equivalent category of comodules have the same cohomological dimension, and I will present a new positive answer.

• Slides.

• Title: Quantum Cuntz-Krieger algebras.

• Abstract: The notion of a quantum graph, a concept going back to work of Erdos-Katavolos-Shulman and Weaver, provides a noncommutative generalisation of finite graphs. Quantum graphs play an intriguing role in the analysis of quantum symmetries of graphs via monoidal equivalences, and naturally appear also in quantum information theory. In this talk, I will discuss the construction of certain C*-algebras associated with directed quantum graphs, in analogy to the definition of Cuntz-Krieger algebras, and illustrate this with some examples. (Joint work with M. Brannan, K. Eifler, M. Weber).

• Slides.

• Title: How to (badly) quantum shuffle cards.

• Abstract: Card shuffles can be thought of as random walks on the symmetric group, and the study of these random walks has been a subject of interest to probabilists for more than forty years. Even for one of the simplest examples, the random transposition walk, precise results concerning the convergence to equilibrium were only very recently obtained. After briefly describing that setting, I will report on a joint work with L. Teyssier and S. Wang where we study an analogue of the random transposition walk on the quantum symmetric group, therefore a kind of "quantum card shuffle". In particular, we obtain a similar asymptotic description of the convergence to equilibrium, called the "limit profile", involving the free Poisson distribution while the classical case involved the usual Poisson distribution.

• Slides.

• Title: Noncommutative Tensor Triangular Geometry.

• Abstract: In this talk, I will show how to develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$C). Insights from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, $\mathbf K$, and then to associate to $\mathbf K$ a topological space--the Balmer spectrum $\Spc {\mathbf K}$. 

We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\Spc \bK$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of $\mathbf K$, which in turn can be applied to classify the thick two-sided ideals and $\Spc {\mathbf K}$.

If time permits applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon. 

This is joint and ongoing work with Milen Yakimov and Kent Vashaw.

• Slides.

• Title: From subfactors to actions of the Thompson group.

• Abstract: In his quest in constructing conformal field theories from subfactors Vaughan Jones found an efficient machine to construct actions of groups like the Thompson groups. I will briefly explain the story of this discovery. I will then present a general overview of those Jones actions providing explicit examples. Some of the results presented come from joint works with Vaughan Jones and with Valeriano Aiello and Roberto Conti.

• Title: Quantum Root Vectors and a Dolbeault Double Complex for the A-Series Quantum Flag Manifolds.

• Abstract: In the 2000s a series of seminal papers by Heckenberger and Kolb introduced an essentially unique covariant $q$-deformed de Rham complex for the irreducible quantum flag manifolds. In the years since, it has become increasingly clear that these differential graded algebras have a central role to play in the noncommutative geometry of Drinfeld--Jimbo quantum groups. Until now, however, the question of how to extend Heckenberger and Kolb's construction beyond the irreducible case has not been examined. Here we address this question for the $A$-series Drinfeld--Jimbo quantum groups $U_q(\frak{sl}_{n+1})$, and show that for precisely two reduced decompositions of the longest element of the Weyl group, Lusztig's associated space of quantum root vectors gives a quantum tangent space for the full quantum flag manifold $\mathcal{O}_q(F_{n+1})$ with associated differential graded algebra of classical dimension. Moreover, its restriction to the quantum Grassmannians recovers the $q$-deformed complex of Heckenberger and Kolb, giving a conceptual explanation for their origin. Time permitting, we will discuss the noncommutative K\"ahler geometry of thesespaces and the proposed extension of the root space construction to the other series. (Joint work with P. Somberg).

• Title: Quantum affine algebras and spectral k-matrices.

• Abstract: The Yang-Baxter equation (YBE) and the reflection equation (RE) are two fundamental symmetries in mathematics arising from particles moving along a line or a half-line. The quest for constant solutions of YBE (R-matrices) is at the very origin of the Drinfeld-Jimbo quantum groups and their universal R-matrix. Similarly, constant solutions of RE (k-matrices) naturally appear in the context of quantum symmetric pairs (QSP). 

In joint work with Bart Vlaar, we construct a discrete family of universal k-matrices associated to an arbitrary quantum symmetric Kac-Moody pair as operators on category O integrable representations. This generalises previous results by Balagovic-Kolb and Bao-Wang valid for finite-type QSP. In this talk, I will explain how, in affine type, this construction gives rise to parameter-dependent operators (spectral k-matrices) on finite-dimensional representations of quantum loop algebras solving the same RE introduced by Cherednik and Sklyanin in the 1980s in the context of quantum integrability near a boundary.

• Slides.

• Title: Quantum Galois Group of Subfactors.

• Abstract: In this talk, I prove the existence of a universal (terminal) object in a number of categories of Hopf algebras acting on a given subfactor N \subset M (finite index, type II_1) such that N is in the fixed point subalgebra of the action. These universal Hopf algebras can be interpreted as a quantum group version of Galois group of the subfactor. We compute such universal quantum groups for certain class of subfactors, notably those coming  outer actions of finite dimensional Hopf * algebras (joint work with Suvrajit Bhattacharjee and Alex Chirvsitu).

• Slides.

• Title: On fusion 2-categories.

• Abstract: I will revisit and categorify concepts from the theory of fusion categories — including idempotent completeness and semi-simplicity, ultimately leading to a notion of `fusion 2-category’. I will highlight structural similarities and differences between fusion 1- and 2-categories and discuss several concrete examples. If time permits, I will discuss the role of fusion 2-categories as a natural building block for 4-dimensional topological field theories.

• Title: Non-commutative balls and quantum group structures.

• Abstract: The Toeplitz algebra attached to the unit disk is the universal C*-algebra generated by an isometry, and is a non-commutative analogue of the unit disk. Similarly, one can attach algebras to non-commutative counterparts of non-compact Hermitian symmetric spaces. I will discuss results to the effect that such quantum spaces cannot admit quantum group structures, i.e. their attached non-commutative “function algebras” do not admit reasonable Hopf algebra structures.

(joint w/ Jacek Krajczok and Piotr Soltan)

• Title: Cluster realization of spherical DAHA.

• Abstract: Spherical subalgebra of Cherednik's double affine Hecke algebra of type A admits a polynomial representation in which its generators act via elementary symmetric functions and Macdonald operators. Recognizing the elementary symmetric functions as eigenfunctions of quantum Toda Hamiltonians, and applying (the inverse of) the Toda spectral transform, one obtains a new representation of spherical DAHA. In this talk, I will discuss how this new representation gives rise to an injective homomorphism from the spherical DAHA into a quantum cluster algebra in such a way that the action of the modular group on the former is realized via cluster transformations. The talk is based on a joint work in progress with Philippe Di Francesco, Rinat Kedem, and Gus Schrader.

• Slides.

• Title: Actions of fusion categories on topological spaces.

• Abstract: Fusion categories are algebraic objects which generalize the representation categories of finite quantum groups. We define an action of a (unitary) fusion category C on a compact Hausdorff space X to be a C module category structure on Hilb(X), the category of finite dimensional Hilbert bundles over a compact Hausdorff space X. When X is connected, we discuss obstructions to the existence of such actions and describe techniques for building examples.

• Slides.

• Title: Non-commutative ergodic theory of semi-simple lattices.

• Abstract: In the late 90's, Nevo and Zimmer wrote a series of papers describing the general structure of stationnary actions of higher rank semi-simple Lie groups G on probability spaces. With Cyril Houdayer we extended this result in two ways: first we upgraded it to actions on non-commutative spaces (von Neumann algebras), and we also managed to study actions of lattices in G. I will explain this non-commutative ergodic theorem and the main ingredients of proof, and give striking consequences on the unitary representations of these lattices and their characters.

• Title: Quantum groups and nonlocal games.

• Abstract: In this talk I will explain how quantum groups arise in quantum information theory via a class of graph based nonlocal games. Our point of departure will be an interactive protocol (nonlocal game) where two provers try to convince a verifier that two graphs are isomorphic. Allowing provers to take advantage of shared quantum mechanical resources will then allow us to define quantum isomorphism of graphs as the ability of quantum players to win the corresponding game with certainty. We will see that quantum isomorphism can be naturally reformulated in the language of quantum groups.

• Slides.

• Title: Globalization for Geometric Partial Comodules.

• Abstract: The study of partial symmetries (partial actions and coactions, partial representations and corepresentations, partial comodule algebras) is a relatively recent field in continuous expansion and, therein, one of the relevant questions is the existence and uniqueness of a so-called globalization (or enveloping action). For instance, in the framework of partial actions of groups any global action of a group G on a set induces a partial action of the group on any subset by restriction. The idea behind the concept of globalization of a given partial action is to find a (universal) G-set such that the initial partial action can be realized as the restriction of this global one. 

We propose here a categorical approach to partial symmetries and the globalization question, explaining several of the existing results and, at the same time, providing a procedure to construct globalizations in concrete contexts of interest. Our approach relies on the notion of geometric partial comodules, recently introduced by Hu and Vercruysse in order to describe partial actions of algebraic groups from a Hopf-algebraic point of view. 

Unlike classical partial actions, which exist only for (topological) groups and Hopf algebras, geometric partial comodules can be defined over any coalgebra in a monoidal category with pullbacks and they allow to describe phenomena that are out of the reach of the theory of partial (co)actions, even in the Hopf algebra framework. At the same time, geometric partial comodules allow to approach in a unified way partial actions of groups on sets, partial coactions of Hopf algebras on algebras and partial (co)actions of Hopf algebras on vector spaces. Thus, the question of studying the existence (and uniqueness) of globalization for geometric partial comodules naturally arises as a unifying way to address the issue.

Based on a joint work with Joost Vercruysse.

• Slides.

• Title: Hopf algebras in SupLat and set-theoretical YBE solutions.

• Abstract: Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. In this talk, we will present a new approach for studying these solutions, by looking at Hopf algebras in the category of complete lattices and join-preserving morphisms, denoted by SupLat. Any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its remnant and a co-quasitriangular structure on H induces a brading operator on R(H), which induces a skew brace structure on R(H). From this correspondence, we will recover several aspects of the theory of skew braces. In particular, we will construct the universal skew brace of a set-theoretical YBE solution, as the remnant of an FRT-type reconstruction in SupLat.

• Slides.

• Title: Equivariant spectral triple for the compact quantum group U_q(2) for complex deformation parameters.

• Abstract: Let q=|q|e^{i𝜋𝜃} be a nonzero complex number such that |q|≠1, and consider the compact quantum group U_q(2). In this talk, we discuss a complete list of inequivalent irreducible representations of U_q(2) and its Peter-Weyl decomposition. Then, for 𝜃∉ℚ\{0,1}, we discuss the K-theory of the underlying C*-algebra C(U_q(2)), and a spectral triple which is equivariant under its own comultiplication action. The spectral triple obtained here is even, 4^+-summable, non-degenerate, and the Dirac operator acts on two copies of the L^2-space of U_q(2). The Chern character of the associated Fredholm module is nontrivial.

• Slides.

• Title: Orthogonal vs unitary in the case of "easy" quantum groups.

• Abstract: We consider quantum subgroups of Wang’s free orthogonal quantum group on the one hand and of his free unitary quantum group on the other. In the first case, the generators of the underlying C*-algebras are selfadjoint which is dropped in the latter case. We compare these two cases along the lines of so called "easy" quantum groups and we observe that the step from the orthogonal to the unitary case is huge. This is a survey talk on the landscape of "easy" quantum groups with a particular emphasis on the differences between the orthogonal and the unitary case.

• Slides.