Talk details
Lectures and research talks for “Geometry and analysis of quantum groups”
Schedule

Mon

Tue

Wed

Thu

Fri

9:3010:30

Bichon 1

Bichon 2

Safronov 2

Bichon 3

Giselsson

11:0012:00

Yuncken

Vlaar

Arici

Safronov 3 
Nest

13:3014:30

Brochier

Safronov 1

Vos (12:0012:20)

De Commer

Martos (12:0012:20)

14:3015:30

Letzter 1

Letzter 2


Letzter 3


16:0017:00

Carotenuto, Norkvist (16:40)

AsadiVasfi (16:20)


Schrader


Lectures
Julien Bichon
Title: Cohomological dimension of Hopf algebras
Abstract: This minicourse 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 minicourses can be found at https://lmbp.uca.fr/~bichon/Cordobacdh.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 3manifolds
Abstract: In these series of lectures I will define skein modules associated to quantum groups. These are certain vector spaces associated to 3manifolds "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.
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 KKequivalent to the algebra of complex numbers so that the (K)Ktheory 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 KohnoDrinfeld 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 socalled 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 deformationquantization.
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 wellknown 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: quasiinvariant 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 quasiinvariant 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 2graph.
Abstract: It is wellknown 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 2graph. 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 Ωtwisted representation for some unitary measurable 2cocycle 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 Ωcocycle 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 2cocycle Ω, we then study Ωtwisted 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 torsionfree 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 qdeformation—now associated to split real quantum groups—which are eigenfunctions for the qdifference analogs of the Toda chains. These qdeformed 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 clusteralgebraic construction of the Toda chain and its eigenfunctions.
Bart Vlaar
Title: Universal Kmatrices 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 A_{2}) on the triple tensor product of a finitedimensional vector space V. In fact, there is a natural variation of this with an analytic flavour: End(V)valued meromorphic functions satisfying the "spectral" (=parameterdependent) YangBaxter 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 finitedimensional 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 B_{2} 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) qdeforming the fixedpoint subalgebra of an involution of g of the second kind. K can be evaluated in (integrable) category𝒪 modules, thus generalizing a result by Balagović & Kolb (2019) for finitedimensional 𝔤. If 𝔤 is an affine Lie algebra we can also let K act on (finitedimensional) loop modules V as an End(V)valued formal Laurent series K(z). Then K(z) satisfies a (Cherednikgeneralized) 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: HarishChamdra’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 BernsteinGelfandGelfand complex plays an essential role. (Joint work with Christian Voigt.)
Contributed talks
Ali AsadiVasfi
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 torsionfreeness for discrete quantum groups and survey some of its properties. Namely, I will report on a recent result about the persistence of torsionfreeness by divisible discrete quantum subgroups.
Axel Tiger Norkvist
Title: Projective real calculi and the Levi–Civita connection
Abstract: Real calculi is a derivationbased 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 sigmafinite von Neumann algebras, taking time to introduce the necessary concepts such as L2implementations 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 finitedimensional subalgebras. This result is new even for the original Haagerup property. Finally, we illustrate some examples coming from Heckevon Neumann algebras and free orthogonal quantum groups. This is joint work with Martijn Caspers, Mario Klisse, Adam Skalski and Mateusz Wasilewski.