This meeting aims bringing together French researchers on the theory of compact quantum groups, and to share approaches from noncommutative geometry, operator algeras, harmonic analysis, topological dynamics, noncommutative probability, quantum information.
It will take place from June 22nd to 26th on the campus Bouloie of the University Marie et Louis Pasteur in Besançon. It will start at 2pm on Monday, June 22, and end at 12pm on Friday, June 26, 2026.
Registration is free but mandatory to benefit from the meals offered by the conference.
Organizer: Uwe Franz
Local Organizers: Christian Le Merdy, Alexandre Nou
Programme
Introductory lectures: Ivo Dell'Ambrogio and Robert Yunken will give a series of lectures on topics related to Task 2 "New approaches to K-theory computations" and Task 3 "Locally compact quantum groups" of our work programme.
Lecture by Robert Yuncken
Title: Quantised semisimple Lie groups and their representations
Abstract: In the early 1980s Russian mathematical physicists (Kulish & Reshetikhin, Sklyanin, …) discovered the Hopf algebra $U_q(\mathfrak{sl}_2 \mathbb{C})$ whose representation theory closely resembles that of the Lie group $SL_2(\mathbb{C})$, or more precisely its universal enveloping algebra. A few years later, Woronowicz and Vaksman & Soibelman discovered the Hopf algebra $\mathbb{O}_q(SL_2(\mathbb})$ whose corepresentation theory resembles the representation theory of $SL_2(\mathbb{C})$. These discoveries opened the door to a new world of quantised semisimple Lie groups, which have no geometrical existence but manifest themselves in a huge collection of related Hopf algebras.
In these introductory talks, we will describe the definitions and representation theory of compact and complex quantised semisimple Lie groups and their representation theory. If time permits, we may also discuss the phenomenon of crystallisation, due to Kashiwara and Lusztig, which uses the extreme limit of these quantisations to reduce representation theory to combinatorics.
Lecture by Ivo Dell'Ambrogio
Title: Homological algebra for Kasparov theory
Abstract: KK-theory, or bivariant K-theory, is a powerful invariant of topological spaces and C*-algebras due to Gennadi Kasparov in the early 80's, with countless applications. Like classical topological K-theory, which it generalizes, it remains relatively computable thanks to Bott periodicity and the long exact sequences associated to nice extensions. In the mid 2000's, Ralf Meyer and Ryszard Nest realized that most flavours of KK-theory organise themselves into triangulated categories and even tensor-triangulated categories; this observation enabled a more systematic approach to homological computations, as well as to deep structural issues such as the Baum-Connes conjecture.
In these introductory talks I will first recall the axiomatic approach to KK-theory, its (tensor) triangulated structure, and its equivariant versions for locally compact groups and quantum groups, hinting at a couple of the known constructions. I will then present a choice of recent applications afforded by the triangular point of view, focussing on the case of equivariant KK-theory for compact and finite groups.
International Invited Speaker: Christian Voigt (University of Glasgow) on "Quantum symmetries of locally compact spaces"
Abstract: Starting from Wang’s discovery of the quantum permutation group of a finite set, the notion of quantum symmetry has been studied intensively over the past three decades, most notably for finite graphs. In this talk I’ll discuss how the notion of quantum symmetry can be defined in the setting of locally compact spaces. The corresponding (discrete) quantum groups are not very tractable in general, but I’ll present some concrete problems where quantum permutations (of finite sets) and topology come together in an interesting way.
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