Talk details

Schedule

Lectures

Julien Bichon

Title: Cohomological dimension of Hopf algebras

Abstract: This mini-course will introduce the necessary homological material to define the cohomological dimension of Hopf algebras, prove some of its basic properties, and compute it in a number of key examples. Lecture notes for similar mini-courses can be found at https://lmbp.uca.fr/~bichon/Cordoba-cdh.pdf or https://lmbp.uca.fr/~bichon/Seoul.pdf.

Gail Letzter

Title: A Guide to Quantum Symmetric Pairs: Foundations and Applications

Abstract: These lectures serve as a guide to the study of coideal subalgebras used to form quantum symmetric pairs (QSP algebras).   The series begins with an introduction that discusses both origins and applications. Next, the classical case  is reviewed with an emphasis on essential structures used in the quantum setting. QSP algebras are then defined in terms of generators and relations;  connections to the matrix formulation via reflection equations is explained.  Realizations of quantum symmetric spaces are presented and the role of various subalgebras, such as the center, are highlighted. Throughout the talk,  concepts and constructions are illustrated with examples.

Pavel Safronov

Title: Integrating quantum groups over 3-manifolds

Abstract: In these series of lectures I will define skein modules associated to quantum groups. These are certain vector spaces associated to 3-manifolds "globalizing" the category of representations of the quantum group. I will describe applications of skein modules to topology (link invariants) and explain the associated algebraic structures (skein algebras, reflection equation algebras, ...). I will go over the definition of the ribbon structure on the category of representations of the quantum group and will be fairly explicit with the algebra without assuming much knowledge from category theory.

lecture note

Research talks

Francesca Arici

Title: From SU(2)-representations to subproduct system and their C*-algebras

Abstract: Motivated by the study of (quantum) group symmetries, in this talk we will introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We will describe their Toeplitz and Cuntz–Pimsner algebras and provide results about their topological invariants through K(K)- theory. In particular, we will show that the Toeplitz algebra of the subproduct system of an irreducible SU(2) representation is equivariantly KK-equivalent to the algebra of complex numbers so that the (K)K-theory groups of the Cuntz–Pimsner algebra can be effectively computed using an exact sequence involving an analogue of the Euler class.

Based on joint work with Jens Kaad (SDU).

Adrien Brochier

Title: Towards a higher genus Kohno-Drinfeld theorem

Abstract: The Kohno–Drinfeld theorem is a remarkable equivalence between two ribbon categories associated with a complex reductive group G. The first one is the category of representation of the quantum group associated with G. The second one is constructed using a so-called Drinfeld associator. Existence of associators follows from analytic techniques, by computing the monodromy of the KZ equation in conformal field theories. this construction is one of the cornerstones of the fascinating relation between higher algebra and topological operads, and deformation-quantization.

In this talk, I'll sketch a combinatorial construction of higher genus analogs of associators. in genus one this recovers Calaque–Enriquez–Etingof's formula for an elliptic associator encoding the monodromy of the KZB equation on an elliptic curve. I'll then explain how those are related to certain well-known quantum algebras. This should be thought of as a quantization of a certain combinatorial version of the Riemann–Hilbrt isomorphism between character varieties of surfaces, and their moduli spaces of flat connections.
 

Kenny De Commer

Title: Coideals for quantum groups: quasi-invariant integrals and Drinfeld double 

Abstract: Let G be a compact quantum group with associated *-algebra A of regular functions. A coideal *-subalgebra B of A corresponds precisely to a quantum homogeneous space X for G with a distinguished (classical) point. As such, one can make sense of the stabilizer of this point as a coideal *-subalgebra associated to the dual of G. In this talk, we will discuss conditions guaranteeing the existence of a quasi-invariant functional on this stabilizer coideal, and we introduce the notion of Drinfeld double of the original coideal with its stabilizer coideal. We briefly explain how the representation theory of the latter forms (somewhat surprisingly) a monoidal category. We illustrate the abstract theory with the example of quantum SL(2,R). This is joint work with Joel Dozkou Talla.

Olof Giselsson

Title:SU(3) as a C*-algebra generated by a 2-graph.

Abstract: It is well-known that the compact quantum group SU_q(2), considered as a C*-algebra, is isomorphic to a Cuntz–Kreiger algebra of a directed graph. Around 2000 Kumjian and Pask introduced the concept of C*-algebras generated by a higher rank directed graph. As it was shown by Nagy in the 90’s that the compact quantum group SU_q(3) was independent, as C*-algebra, of the parameter q,  it is a natural question if it is a C*-algebra coming from an apropriate higher rank graph. Based on recent work, I show that SU_q(3) is indeed isomorphic to a C*-algebra generated by a 2-graph. Moreover, I will try to give som idea how this can be generalized to SU_q(N).

Ryszard Nest

Title: Projective representation theory for compact quantum groups and the quantum Baum–Connes assembly map

Abstract: We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators K(H), if and only if the associated Ω-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of Ĝ in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C*-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj–Skandalis duality and a quantum version of the Packer–Raeburn's trick. As an application, we provide a twisted version of the Green–Julg isomorphism and obtain the quantum Baum–Connes assembly map for permutation torsion-free discrete quantum groups.

Gus Schrader

Title: Whittaker functions for quantum groups

Abstract: In one of the first applications of representation theory to quantum integrability, Kostant showed in the 1970's that the classical Whittaker functions for split real Lie groups are eigenfunctions for quantum Toda chains. These Whittaker functions admit a q-deformation—now associated to split real quantum groups—which are eigenfunctions for the q-difference analogs of the Toda chains. These q-deformed Whittaker functions have turned out to have many important applications across different areas of mathematics and physics: they govern the decomposition of a tensor product of principal series representations of the split real quantum group into irreducibles, provide the key to the proof of the modular functor conjecture in quantum higher Teichmuller theory, and they encode the BPS spectrum of 4d N=2 supersymmetric quiver gauge theories at strong coupling. I will survey these results, and explain how they arise from  the cluster-algebraic construction of the Toda chain and its eigenfunctions.

Bart Vlaar

Title: Universal K-matrices for quantum symmetric Kac–Moody pairs

Abstract: An original motivation of quantum group theory is the study of the Yang–Baxter equation. A solution of this corresponds to a representation of the Artin braid group with 2 generators (i.e. type A₂) on the triple tensor product of a finite-dimensional vector space V. In fact, there is a natural variation of this with an analytic flavour: End(V)-valued meromorphic functions satisfying the "spectral" (=parameter-dependent) Yang-Baxter equation. A range of quantum groups is provided by quantized enveloping algebras U_q(𝔤) associated to symmetrizable Kac–Moody algebras 𝔤 (defined over C), such as finite-dimensional simple Lie algebras or their, suitably extended, loop algebras: affine Lie algebras.

The reflection equation is the ambitious younger sister of the Yang–Baxter equation, namely the type B₂ analogue, describing braids in the presence of a cylinder, or particles interacting with a boundary. In work with A. Appel (arXiv:2007.09218 and ongoing) we construct a universal solution K. It lies in the twisted centralizer of a given Letzter–Kolb quantum symmetric pair, which is a coideal subalgebra of U_q(𝔤) q-deforming the fixed-point subalgebra of an involution of g of the second kind. K can be evaluated in (integrable) category-𝒪 modules, thus generalizing a result by Balagović & Kolb (2019) for finite-dimensional 𝔤. If 𝔤 is an affine Lie algebra we can also let K act on (finite-dimensional) loop modules V as an End(V)-valued formal Laurent series K(z). Then K(z) satisfies a (Cherednik-generalized) spectral reflection equation. Subject to mild conditions on V we can show that, up to a scalar, K(z) is an End(V)-valued rational function.

Robert Yuncken

Title: The Plancherel formula for a complex semisimple quantum group.

Abstract: Harish-Chamdra’s Plancherel formula gives an explicit decomposition of the regular representation of a semimsimple Lie group G, such as SL(n,C), into irreducible components.  I will describe the quantum anallogue of this formula for a complex semisimple quantum group, such as SL_q(n,C), and give an account of the representation theoretic techniques necessary for the proof.  In particular I will show how the quantized Bernstein-Gelfand-Gelfand complex plays an essential role.  (Joint work with Christian Voigt.)

Contributed talks

Ali Asadi-Vasfi

Title: Weak tracial approximate representability and its application

Abstract: We describe a weak tracial analog of approximate representability under the name “weak tracial approximate representability” for finite group actions on simple unital C*-algebras. We then investigate the dual actions on the crossed products by this class of group actions. This generalizes the results of Izumi in 2004 and Phillips in 2011 on the dual actions of finite abelian groups on simple unital C*-algebras. Time permitting, we give further information on some important applications of this notion in the radii of comparison of crossed products by this class of group actions, the dynamical Cuntz semigroups, and the dynamical radii of comparison.

Alessandro Carotenuto

Title: Principal pair of quantum homogeneous spaces

Abstract: I will review the recently introduced notion of Principal Pairs. These objects were firstly introduced to study quantum flag manifolds and their associated quantum Poisson homogeneous spaces and they provide a simple but effective framework for producing examples of faithfully flat Hopf–Galois extensions from a nested pair of quantum homogeneous spaces. This construction is modelled on the classical situation of a homogeneous fibration G/N → G/M, for G a group, and N ⊆ M ⊆ G subgroups. Moreover, I will present a large collection of noncommutative fibrations in the spirit of Brzeziński and Szymański.

Ruben Martos

Title: On the notion of torsion for quantum groups.

Abstract: I will present the notion of torsion-freeness for discrete quantum groups and survey some of its properties. Namely, I will report on a recent result about the persistence of torsion-freeness by divisible discrete quantum subgroups.

Axel Tiger Norkvist

Title: Projective real calculi and the Levi–Civita connection

Abstract: Real calculi is a derivation-based approach to noncommutative geometry which makes it possible to generalize several notions from classical differential geometry to a noncommutative setting. One such notion is that of affine connections, and in this talk we shall go over some of the current research that I am involved in regarding real calculi over projective modules and what can be said about the Levi–Civita connection in this case. As this talk is based on an ongoing research project, the focus will be on exploring some questions I have at the moment, rather than presenting finished results and conclusions.

Gerrit Vos

Title: Relative Haagerup property for arbitrary von Neumann algebras

Abstract: We introduce the relative Haagerup approximation property (rHAP) for a unital, expected inclusion of sigma-finite von Neumann algebras, taking time to introduce the necessary concepts such as L2-implementations and relative compactness. We explore how the definition depends on the choice of state or conditional expectation, and look at some variations of the definition. It turns out that the rHAP always holds if the subalgebra is finite dimensional, which implies that the rHAP is stable under taking free products with amalgamation over finite-dimensional subalgebras. This result is new even for the original Haagerup property. Finally, we illustrate some examples coming from Hecke-von Neumann algebras and free orthogonal quantum groups. This is joint work with Martijn Caspers, Mario Klisse, Adam Skalski and Mateusz Wasilewski.