QGS: Session 2023-2024

• Title: Quantum Cayley graphs.

• Abstract: I will talk about a method of associating a quantum graph to a discrete quantum group together with a projection in its function algebra. These quantum graphs are analogues of Cayley graphs and they do not depend on the choice of a generating projection in the sense of metric geometry. Later I will show how they can help in finding examples of finite quantum groups having Frucht property, i.e. arising as quantum automorphism groups of quantum graphs.

Part of the talk will be based on an on-going joint work with Michael Brannan and Adam Skalski.

• Slides.

• Title: Classical actions of quantum permutations.

• Abstract: Quantum permutation groups can act non-trivially, and even ergodically, on finite spaces. This is, in view of many quantum rigidity results, an exception and it is natural to wonder whether there are other classical spaces on which quantum permutations can act. H. Huang constructed a family of such spaces, and we will show that these are the only possibilities. This is a joint work with F. Taipe and S. Wang.

• Title: Topological Boundaries of Representations and Coideals.

• Abstract: We will introduce and study quantum analogues of Furstenberg-Hamana boundaries of representations of discrete quantum groups, where the Furstenberg boundary is the Furstenberg-Hamana boundary of the left regular representation. Our focus is on the GNS representations of idempotent states, or to put it differently, the quasi-regular representations coming from coideals associated to compact quasi-subgroups. We use their Furstenberg-Hamana boundaries to study (co)amenability properties of such coideals. Then, we combine our work with recent work of Hataishi and De Ro to settle open problems of Kalantar, Kasprzak, Skalski, and Vergnioux for wide classes of quantum groups, including unimodular discrete quantum groups and C*-exact discrete quantum groups. For example, we prove that a unimodular discrete quantum group has the unique trace property iff it acts faithfully on its Furstenberg boundary.

This is joint work with Fatemeh Khosravi.

• 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.

• Slides.

• Title: Cohomology of free unitary quantum groups.

• Abstract: In the talk, we will discuss the free unitary quantum groups (or "universal quantum groups") of Wang and van Daele from a (co)homological perspective. We find a /free resolution of the counit/, a versatile tool which helps to compute cohomological data such as Hochschild cohomology or bialgebra cohomology of the associated Hopf algebras. For free orthogonal quantum groups, such resolutions have been found by Collins, Härtel, and Thom (in the Kac-case) and Bichon (in the general case), and they will serve as our starting point for finding resolutions for free unitary quantum groups.

Based on joint work with I. Baraquin, U. Franz, A. Kula and M. Tobolski [arXiv:2309.07767]

• Slides.

• Title: Proper actions, fixed-point algebras, and deformation via coactions.

• Abstract: The notion of proper actions of groups on spaces has various generalizations for group actions of noncommutative C*-algebras A, which all allow the construction of generalized fixed-point algebras A^G which are Morita equivalent to ideals in the reduced crossed products A⋉G. The weakest version was introduced by  Rieffel in 1990 and it played an important role in his theory of deformations via actions of ℝ^d.

In this talk we want to report on some joint work with Alcides Buss on a version of proper actions which allows the construction of maximal (or exotic) generalized fixed-point algebras which are Morita equivalent to ideals in the maximal (resp. exotic) crossed products. We will report on several applications including Landstad duality for coactions and deformation of C*-algebras via coactions in the sense of Kasprzak and Bowmick, Neshveyev, and Sangha.

• Slides.

• Title: Gaussian Parts of Compact Quantum Groups.

• Abstract: We introduce the Gaussian part of a compact quantum group G, namely the largest quantum subgroup of G supporting all the Gaussian functionals of G. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and q-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups. Joint work with Amaury Freslon and Adam Skalski.

• Slides.

• Title: Equivariant injectivity of crossed products.

• Abstract: We introduce the notion of a G-operator space, which consists of an action of a locally compact quantum group G on an operator space X, and we study the notion of G-equivariant injectivity for such an operator space. We define a natural associated crossed product operator space X ⋊ G, on which both the locally compact quantum group G and its dual act. We completely characterise when these crossed products are equivariantly injective with respect to these actions. We discuss how these results generalise and unify several recent results from the literature and we give some new applications.

• Slides.

• Title: Intertwining techniques for actions of C*-tensor categories.

• Abstract: Intertwining techniques, first used in the realm of C*-algebras in Elliott’s classification of AF-algebras, have been essential in the classification theory of C*-algebras and their group actions. In this talk I will discuss intertwining and how it appears in C*-classification, I will then outline an adaptation of these techniques to the tensor category equivariant setting. Time permitting I will discuss an application of these techniques to the study of tensor category equivariant Jiang-Su stability.

• Slides.

• Title: Quantum-symmetric equivalence via Manin's universal quantum groups.

• Abstract: We study 2-cocycle (and more generally quantum-symmetric equivalences between) twists of graded algebras via their associated universal quantum groups, in the sense of Manin. We prove that Zhang twists arise as a special case of 2-cocycle twist, and that 2-cocyle twisting preserves many fundamental homological invariants of graded algebras. As a consequence, we give a characterization of Artin--Schelter regular algebras using the language of 2-cocycle twists.

• Slides.

• Title: Induced functors on Drinfeld centers.

• Abstract: I will explain how the right/left adjoint of a monoidal functor induced a braided lax/oplax monoidal functors between the corresponding Drinfeld centers. This requires some mild technical assumptions, namely that the projection formulas hold for the adjoint functor. This holds, for example, when the monoidal categories are rigid. As the induced functors on the Drinfeld centers are (op)lax and compatible with braiding, they preserve commutative (co)algebra objects. As classes of examples, we consider monoidal restriction functors along extensions of Hopf algebras leading to (co)induction functors on Yetter-Drinfeld module categories. This is joint work in progress with Johannes Flake (Bonn) and Sebastian Posur (Münster).

• Slides.

• Title: No-signalling values of cooperative quantum games.

• Abstract: Finding values, the optimal winning probability, of various non-local games over different strategies has been an important task in Quantum Information Theory and also for resolving the Connes Embedding Problem. In this talk I will discuss values of quantum games (games with quantum inputs and outputs), arising from the type hierarchy of quantum no-signalling correlations, establishing operator space tensor norm expressions for each of the correlation types. This is a joint work with Jason Crann, Rupert Levene and Ivan Todorov.

• Slides.

• Title: Growth in tensor powers.

• Abstract: This talk is based on joint work with K.Coulembier, P.Etingof, D.Tubbenhauer. Let G be any group and let V be a finite dimensional representation of G over some field. We consider tensor powers of V and their decompositions into indecomposable summands. The main question which will be addressed in this talk: what can we say about count (e.g. total number) of these indecomposable summands? It turns out that there are reasonable partial answers to this question asymptotically, i.e. when the tensor power is large.

• Slides.

• Title: Equivariant formality and reduction.

• Abstract: In this talk, we discuss the reduction-quantization diagram in terms of formality. First, we propose a reduction scheme for multivector fields and multidifferential operators, phrased in terms of L-infinity morphisms. This requires the introduction of equivariant multivector fields and equivariant multidifferential operator complexes, which encode the information of the Hamiltonian action, i.e., a G-invariant Poisson structure allowing for a momentum map. As a second step, we discuss an equivariant version of the formality theorem, conjectured by Tsygan and recently solved in a joint work with Nest, Schnitzer, and Tsygan. This result has immediate consequences in deformation quantization, since it allows for obtaining a quantum moment map from a classical momentum map with respect to a G-invariant Poisson structure.

• Title: Compact Matrix Quantum Group Equivariant Neural Networks.

• Abstract: In deep learning, we would like to develop principled approaches for constructing neural networks. One important approach involves identifying symmetries that are inherent in data and then encoding them into neural network architectures using representations of groups. However, there exist so-called “quantum symmetries” that cannot be understood formally by groups. In this talk, we show how to construct neural networks that are equivariant to compact matrix quantum groups using Woronowicz’s version of Tannaka-Krein duality. We go on to characterise the linear weight matrices that appear in these neural networks for a class of compact matrix quantum groups known as “easy”.  In particular, we show that every compact matrix group equivariant neural network is a compact matrix quantum group equivariant neural network.

• Slides.

• Title: Modular invariants of quantum groups.

• Abstract: A very interesting feature of compact quantum groups is that their Haar integral, which is a normal state on L^inf(G), can be non-tracial. Via Tomita-Takesaki theory, this gives rise to two groups of automorphisms: modular automorphisms and scaling automorphisms. One can use them to define a number of invariants, related to whether these automorphisms are trivial, inner or approximately inner. During the talk I'll introduce such invariants (also in the general locally compact case), discuss a conjecture related to one of them, and present their calculation in the case of q-deformed compact, simply connected, semisimple Lie group G_q. The talk is based on a joint work with Piotr Sołtan.

• Slides.

• Title: Maximal amenability of the radial subalgebra in free quantum groups factors.

• Abstract: The free orthogonal quantum groups O^+(N), introduced by Shuzhou Wang, are monoidally equivalent to the SU_q(2) 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 the maximal amenability of the associated radial MASA, obtained in recent joint work with Xumin Wang.

• Slides.

• Title: Quantum automorphism groups of discrete structures.

• Abstract: Given any mathematical structure, it is a natural question to ask which quantum symmetries it admits. One can in general not hope to find a quantum automorphism group for any structure in the framework of Kustermans-Vaes, as a necessary condition for its existence is local compactness of the classical automorphism group. In recent work, a wide range of discrete structures, those which are connected and locally finite in a suitable sense, were shown to admit an algebraic quantum automorphism group. The main tool for their construction is a generalization of the Tannaka-Krein-Woronowicz reconstruction theorem. In particular, this allows to construct quantum automorphism groups of connected locally finite quantum graphs, such as Wasilewski's quantum Cayley graphs, generalizing joint results with Stefaan Vaes.

• Title: Spectra of quantum algebras.

• Abstract: The talk will survey what is known and/or conjectured about the prime and primitive spectra of quantum algebras, particularly quantized coordinate rings and related algebras such as quantized Weyl algebras. The topological structure of these spectra, their relations with classical algebraic varieties, and their relations with each other will be discussed.

• Slides.

• Title: Equivariant quantizations of the positive nilradical and covariant differential calculi.

• Abstract: We consider the problem of quantizing the positive nilradical of a complex semisimple Lie algebra of finite rank, together with a certain fixed direct sum decomposition. The decompositions we consider are in one-to-one correspondence with total orders on the simple roots, and exhibit the nilradical as a direct sum of graded modules for appropriate Levi factors. We show that this situation can be quantized equivariantly as a finite-dimensional subspace within the positive part of the corresponding quantized enveloping algebra. Furthermore, we show that such subspaces give rise to left coideals, with the possible exception of components corresponding to some exceptional Lie algebras, and this property singles them out uniquely. Finally, we discuss how to use these quantizations to construct covariant first-order differential calculi on quantum flag manifolds, which coincide with those introduced by Heckenberger-Kolb in the irreducible case.