Program: Geometry and analysis of quantum groups 2023

Rooms

• Monday morning: Vilhelm Bjerkenes' house, Auditorium 4
• Monday afternoon: Vilhelm Bjerkenes' house, Auditorium 5
• Tuesday morning: Georg Sverdrup's house, Auditorium 2
• Tuesday afternoon: Vilhelm Bjerkenes' house, Auditorium 5
• Wednesday morning: Physics Building, Lille Physical Auditorium
• Thursday morning: Vilhelm Bjerkenes' house, Auditorium 4
• Thursday afternoon: Vilhelm Bjerkenes' house, Auditorium 5
• Friday morning: Niels Henrik Abels hus, UE26

Invited talks

Francesca Arici

Title: Quantum group symmetries, subproduct systems and \(KK\)-theory

Abstract: In this talk, we shall discuss subproduct systems of Hilbert spaces and their Toeplitz and Cuntz–Pimsner algebras. We will provide results about their topological invariants through \(K(K)\)- theory. More specifically, I will recall how the Toeplitz algebras of the subproduct system of an \(SU(2)\)-representation is equivariantly \(KK\)-equivalent to the algebra of complex numbers and show that the \((K)K\)- theory groups of the Cuntz–Pimsner algebra can be effectively computed. Further, I will discuss Temperley-Lieb subproduct systems as defined by Habbestad-Neshveyev, and explain how C*-algebras in this class satisfy Poincaré duality.

Pierre Bieliavsky

Title: Geodesic hyperbolic triangles and the \(\Gamma\)-equivariant form of Berezin quantization
 
Abstract: A family of Type \(\textrm{II}_1\) factors were constructed by Florin Radulescu in the context of Berezin quantization of Riemann surfaces. I will explain how his construction can be obtained by means of hyperbolic geometry. Part of this work is joint with Florin. 

Arkadiusz Bochniak

Title: Mycielski transformation for quantum graphs

Abstract: Quantum graphs and their characteristics are intriguing generalizations of notions and tools known from discrete mathematics into the quantum world. Their non-trivial relations with quantum information theory provide a bridge between this branch of mathematics and quantum mechanics. In classical graph theory, there are several characteristics that one can associate with given graphs, e.g., chromatic or clique numbers. The famous problem, solved by Mycielski, was to construct a graph that contains a given graph as a subgraph and can have an arbitrarily large chromatic number, but no larger clique is produced. We propose an analog of the Mycielski transformation and its generalizations in the quantum setting and study how they affect the (quantum) characteristics of quantum graphs. Based on joint work with P. Kasprzak (arXiv:2306.09994) and work in progress with P. Kasprzak, P.M. Sołtan, and I. Chełstowski.

Francesco Bonechi

Title: Groupoid quantization for Poisson compact hermitian symmetric spaces

Abstract: I will review an approach to quantization of Poisson manifolds via symplectic groupoid. The basic fact is that a Poisson Nijenhuis (PN) structure gives a canonical real polarization of the symplectic groupoid integrating any of the infinite Poisson structures defined by the PN geometry. This polarization is very singular but still, under favorable circumstances, can produce a meaningful groupoid of lagrangian leaves. The basic example is given by a class of PN structures defined on compact hermitian symmetric spaces. I will describe the construction in the case of complex projective spaces and discuss the present understanding for Grassmannians.

Tomasz Brzeziński

Title: Lie algebras on affine spaces

Abstract: We first explore the definition of an affine space which makes no reference to the underlying vector space and then formulate the notion of a Lie bracket and hence a Lie algebra on an affine space in this framework. Since an affine space has neither distinguished elements nor additive structure, the concepts of anti-symmetry and Jacobi identity need to be modified. We provide suitable modifications and illustrate them by a number of examples. The talk is based on a joint work with James Papworth.

Chiara Esposito

Title: Equivariant Formality

Abstract: In this seminar we discuss the compatibility between deformation quantization and phase space reduction. More explicitly, on the classical side one can
reduce (formal) Poisson structures with formal momentum maps, and on the quantum
side one can reduce equivariant star products. First, we describe these reductions by
morphisms of \(L\)-infinity algebras. This allow us to rephrase the "reduction commutes
with quantization" problem, originally formulated in the setting of geometric quantization, in terms of \(L\)-infinity morphisms. Time permitting we describe the same 
procedure on the quantum side and show the open questions regarding the conjecture
of equivariant formality. This is a joint project with A. Kraft, R. Nest, J. Schnitzer and B. Tsygan.

Victor Gayral

Title: Locally compact quantum groups and essentially bijective \(1\)-cocycles

Abstract: Consider an extension \(0\to V\to G\to Q\to 1\), where \(Q\) and \(V\) are lc groups with \(V\) abelian. In this talk, I will explain how to construct a locally compact quantum group, deforming \(G\), starting from a \(1\)-cocycle \(\eta\in Z^1(Q,\hat V)\) (here \(\hat V\) is the Pontryagin dual of \(V\)) subject to an "essentially bijectivity" condition and to a compatibility condition with the class of the extension \([\beta] \in H^2(Q,V)\). This is a joint work with Pierre Bieliavsky, Lars Tuset and Sergey Neshveyev.

Jens Kaad

Title: Spectral metrics on quantum projective spaces

Abstract: A spectral metric space is a unital spectral triple satisfying that the coordinate algebra becomes a compact quantum metric space via the seminorm which measures the size of first order derivatives. In this talk we investigate the spectral metric properties of quantum projective spaces. The geometric framework for this investigation is provided by the unital spectral triples introduced by D’Andrea and Dabrowski in their CMP paper from 2010. We shall see that these unital spectral triples are in fact spectral metric spaces and, if time permits, indicate how this can be proved. This result makes it possible to investigate the spectral metric continuity properties of quantum projective spaces under variations of the deformation parameter \(q\). It can moreover be viewed as a first step for understanding the higher Vaksman-Soibelman spheres from the point of view of spectral metric spaces. The talk is based on work in progress with Max Holst Mikkelsen.

Giovanni Landi

Title: On Atiyah sequences of braided Lie algebras and their splittings

Abstract: To an equivariant noncommutative principal bundle one associates an Atiyah sequence of braided derivations whose splittings give connections on the bundle. There is an explicit action of vertical braided derivations as infinitesimal gauge transformations on connections. From the sequence one derives a Chern-Weil homomorphism and braided Chern-Simons terms.

On the principal bundle of orthonormal frames over the quantum sphere \(S^{2n}_\theta\), the splitting of the sequence leads to a Levi-Civita connection on the corresponding module of braided derivations. The connection is torsion free and compatible with the `round' metric. We work out the corresponding Riemannian geometry.

Marco Matassa

Title: Quantum Levi-Civita connections on quantum projective spaces

Abstract: I will present various results on a quantum analogue of the Levi-Civita connection corresponding to the Fubini-Study metric on a projective space. This is done within the framework of quantum Riemannian geometry à la Beggs-Majid (and other authors), where the starting point is a differential calculus on the algebra. Here we use the canonical differential calculus introduced by Heckenberger-Kolb for the class of quantum irreducible flag manifolds, of which projective spaces are examples. Starting from this data, for any quantum projective space we introduce a quantum metric and its inverse, as well as a covariant connection. This turns out to be a bimodule connection and to be appropriately compatible with the quantum metric, making it a quantum Levi-Civita connection in a precise sense. Furthermore, we prove various properties of its curvature (the Riemann tensor), such as the fact that it is a bimodule map and satisfies an appropriate antisymmetry property. Finally, we consider the Ricci tensor, defined as a contraction of the Riemann tensor with the quantum metric (this also requires the introduction of a splitting map for the differential calculus). We prove that, upon making certain natural choices, this Ricci tensor is proportional to the quantum metric, therefore giving a quantum analogue of the Einstein condition for projective spaces. In the classical limit, all these objects reduce to the familiar ones for a projective space endowed with the Fubini-Study metric.

Réamonn Ó Buachalla

Title: Quantum flag manifolds and noncommutative geometric representations

Abstract: We present recent progress on noncommutative geometric representations of quantum algebraic objects, such as finite-dimensional Drinfeld-Jimbo modules, Nichols algebras, and quantum homogeneous coordinate rings of quantum flag manifolds. The noncommutative geometry underlying these realisations is a \(q\)-deformed Dolbeault complex for the \(A\)-series quantum flag manifolds. This complex is built in a very natural way from Lusztig's quantum root vectors, and is shown to be quite sensitive to the required choice of reduced decomposition of the longest element of the Weyl group. When these constructions are restricted to the quantum Grassmannians, they coincide with earlier research on the celebrated Heckenberger-Kolb differential calculus.

Roland Vergnioux

Title: Maximal amenability of the radial subalgebra of free quantum groups

Abstract: The free orthogonal quantum groups \(O^+(N)\), introduced by Shuzhou Wang, are monoidally equivalent to \(SU_q(2)\) as compact quantum groups, but on an analytical level they behave much like the quantum duals of the classical free groups, when \(N > 2\). I will review their definition and main properties, and present a new result about maximal amenability.

Christian Voigt

Title: Self-similar quantum symmetries

Abstract: This talk is devoted to quantum symmetries of regular rooted trees, and the notion of a self-similar discrete quantum group arising this way. I will discuss how these quantum groups are related to certain finite quantum automata, and illustrate this with some examples.

(Based on joint work in progress with N. Brownlowe, D. Robertson and M. Whittaker)

Robert Yuncken

Title: Crystallizing compact semisimple Lie groups

Abstract: The theory of crystal bases is a means of simplifying the representation theory of semisimple Lie algebras by passing through quantum groups.  Varying the parameter \(q\) of the quantized enveloping algebras, we pass from the classical theory at ​\(q=1\) through the Drinfeld-Jimbo algebras at \(0<q<1\) to the crystal limit at \(q=0\). At this point, the main features of the representation theory crystallize into purely combinatorial data described by crystal graphs.  In this talk, we will describe what happens to the C*-algebra of functions on a compact semisimple Lie group under the crystallization process, yielding higher-rank graph algebras. This is joint work with Marco Matassa.

Sophie Emma Zegers 

Title: On the C*-algebraic aspects of the quantum twistor bundle

Abstract: In the study of noncommutative geometry, many important concepts from classical geometry have been imported into the noncommutative world - one such example being the notion of a compact principal bundle. However, until recently, the
concept of a noncommutative fibre bundle was far from being understood. In a series of works by Brzeziński and Szymański, quantum fiber bundles with quantum homogeneous spaces as fibers, are described in the purely algebraic framework.

We study an example that fits into this framework, namely the quantum twistor
bundle \(\mathbb{C} \mathrm{P}^{1}_{q} \to \mathbb{C} \mathrm{P}^{3,\mu}_{q} \to \Sigma_q^4\) defined from the quantum instanton bundle due to Bonechi-Ciccoli-Dąbrowski-Tarlini. Our overall aim is to to relate the \(K\)-theory (and \(K\)-homology) groups of the fiber, the total space and the base of the quantum twistor bundle. As a first step, we construct \(1\)-summable Fredholm modules in the \(q=0\) case which afterwards can be transported into the case when \(q \neq 0\). This approach relies heavily on the graph C*-algebraic description of the Vaksman-Soibelman quantum sphere \(S^{7}_{q}\) due to Hong and Szymański and of \(\mathbb{C} \mathrm{P}^{3,\mu}_{q}\) due to our previous work. Based on an ongoing work with Suvrajit Bhattacharjee.

Contributed talks

Arnab Bhattacharjee

Title: Kostant differential for Quantum Grassmannian

Abstract: In the seminal article of Kostant in 1961, the author described an equivariant
(in a particular sense) differential complex of twisted \(U(\mathfrak{g})\)-module, where \(\mathfrak{g}\) is a semi-simple lie algebra. In this talk, we shall discuss, an analogous version of this equivariant differential complex over a \(q\)-deformed Drinfeld-Jimbo quantum groups \(U_q(\mathfrak{g})\) for the case of Quantum Grassmannian and relevant results. This is an ongoing work with Petr Somberg.

Marzieh Forough   

Title: The equivariant lifting problem for completely positive maps

Abstract: In this talk, I will first discuss the lifting problem for equivariant completely positive maps where the underlying C*-algebras carry continuous actions of locally compact second countable groups.  As an application of the existence of almost equivariant lift for equivariant lifting problem, I present a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. Then I describe an extension of our results on the lifting problem in the category of C*-dynamical systems to the category of groupoid dynamical systems.

(Based on joint work with Eusebio Gardella and Klaus Thomsen and on ongoing work with Suvrajit Bhattacharjee)

Piotr M. Hajac 

Title: Cyclic-Homology Chern-Weil Theory for Principal Coactions

Abstract: Viewing the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model, we construct the cyclic-homology Chern-Weil homomorphism. It turns out that in this construction it is natural to replace the unital subalgebra of coaction invariants by its natural H-unital nilpotent extension (row extension). Although the row-extension algebra provides a drastically different model of the cyclic object, we prove that, for any row extension of any unital algebra over a commutative ring, the row-extension Hochschild complex and the usual Hochschild complex are chain homotopy equivalent. We instantiate our noncommutative Chern-Weil theory by computing the cotrace space and analyzing a dimension-drop-like effect for the quantum-deformation family of the standard quantum Hopf fibrations. (Based on joint work with Tomasz Maszczyk.)

Xiao Han 

Title: Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids.

Abstract: We construct Hopf bimodules and Yetter-Drinfeld modules of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we show that the categories of Hopf bimodules and Yetter-Drinfeld modules are equivalent (pre)braided monoidal categories.

Bhishan Jacelon

Title: Quantum metric Choquet simplices

Abstract: I will describe some work (very much) in progress on a special class of compact quantum metric spaces in the sense of Rieffel. The structures under investigation are designed for witnessing metric features of morphisms into the category of classifiable C*-algebras. I will give some examples and explain how to build new ones by forming 'quantum crossed products' associated to dynamical systems on compact, connected metric spaces.

Prem Nigam Kar

Title: Transitive Nonlocal Games and Quantum Permutation Groups

Abstract: We study transitive nonlocal games, which are nonlocal games with composable strategies. We establish several interesting correspondences of bisynchronous transitive nonlocal games with the theory of compact quantum groups and semigroups. In particular, we associate a quantum permutation group with each bisychronous transitive game and vice versa. We prove that the existence of a C*-strategy, existence of a quantum commuting strategy, and existence of classical strategy are equivalent for bisynchronous transitive games. If time permits, we see how to use some of these correspondences to establish necessary and sufficient conditions for some classes of correlations, that arise as perfect strategies of transitive nonlocal games, to be nonlocal.

Juan Carlos Morales Parra

Title: Towards a Combinatorial Quantization of Dynamical Character varieties    

Abstract: The moduli space of flat connections over a Riemann surface (of generic genus and number of punctures) could be realized as a constrained Poisson structure, with brackets defined in terms of a classical \(r\)-matrix of the underlying Lie algebra. After gauge fixing the constraints, the derived Dirac brackets resemble the original ones, but are now defined in terms of a classical dynamical r-matrix.   

Following Fock-Rosly approach, we explain how these gauge-fixed structures could be understood as "special" decorated character varieties and so, following Alekseev-Grosse-Schomerus combinatorial quantization scheme, we will describe how quantum dynamical R-matrices and Quantum Dynamical Groups appear naturally at the quantum level, using decorated \(SL(2,\mathbb{C})\) character varieties over punctured spheres as an illustrative example.

(This talk is based on joint work with Bernd Schroers).

Lukas Rollier

Title: Equivariant Tannaka-Krein Duality

Abstract: I will report on joint work with Stefaan Vaes. For a reasonably wide class of actions of locally compact quantum groups on direct sums of matrix algebras, one may show that their equivariant representation theory mimics in important ways the representation theory of compact quantum groups. In particular, enough of the structure is retained to prove an equivariant version of the well-known Tannaka-Krein duality. This new reconstruction theorem may be used to study quantum automorphism groups of various discrete structures.

Piotr Sołtan

Title: Invariants of quantum groups related to the scaling and modular groups

Abstract: I will discuss some new results concerning invariants of locally compact quantum groups defined by the scaling and modular groups. Apart from presenting many examples I will focus on a new approach to proving that for second countable compact quantum groups triviality of one of the invariants implies that the quantum group is of Kac type. This new approach is related to the i.c.c. condition for discrete groups.

Josse van Dobben de Bruyn

Title: Graphs with trivial automorphism group and non-trivial quantum automorphism group

Abstract: A current research theme in quantum graph theory is to determine which classical/quantum groups can occur as the classical/quantum automorphism group of finite graphs. One question in this direction is to determine how far the classical and quantum automorphism groups can be apart. In this talk, I will present a class of graphs which have trivial automorphism group and non-trivial quantum automorphism group, thereby answering a question asked by several authors in recent years. The construction is inspired by linear constraint system games in quantum information theory. This talk is based on joint work with David E. Roberson and Simon Schmidt.

Henrik Winther

Title: Jets and differential operators in noncommutative geometry

Abstract: We construct an infinite family of endofunctors on the category of left modules over a unital associative algebra equipped with a differential calculus. These functors generalize the jet functors on vector bundles from differential geometry. In particular, our construction coincides with the classical jet functor for vector bundles when the algebra is the smooth functions on a manifold and the calculus is generated by the classical exterior derivative. We show that our jet functors give rise to a category of linear differential operators between modules, that these satisfy many good properties one might expect, and that most maps which are expected to be differential operators (connections, differentials, partial derivatives) are. We also discuss representability, symbols and define a notion of vector fields in this setting. Joint work with K. Flood and M. Mantegazza.

Peter Zeman

Title: Quantum automorphism groups of trees

Abstract: We give a characterisation of quantum automorphism groups of trees. In particular, for every tree, we show how to iteratively construct its quantum automorphism group using free products and free wreath products. This can be considered a quantum version of Jordan's theorem for the automorphism groups of trees. This is the one of the first characterisations of quantum automorphism groups of a natural class of graphs with quantum symmetry.