Quantum Symmetries

Workshop January - March 2024


The aim of this workshop is to stimulate research concerning quantum symmetries. Our goal is to gather critical mass for synergic exchange of ideas and actual research progress. The idea is to bring together mathematicians who would like to work with each other but reside in different areas of the world. For instance, one can convince her/his remote collaborators to come to Warsaw at the same time to work together. We can provide at least partial funding for all qualified participants.

The workshop begins with the arrival of its first participant (6 January 2024) and ends with the departure of its last participant (9 March 2024). All participants attend on site the core part of the workshop that spans the 2023/24 semester break  at the University of Warsaw. The core part is devoted mostly to research collaboration and discussions rather than formal lectures, dissemination of results, and teaching activities, so its schedule included below consists of only 24 short research talks throughout the four weeks. 

Organizers will schedule talks on a weekly basis. There will be no recordings of lectures, and we strongly discourage slides in favor of traditional chalk talks. The talks will take place in Rooms 321, 403 and 405 in IMPAN, Mondays 14:00-16:00, Wednesdays 16:00-18:00, Fridays 14:00-16:00.


Monday 14:00 - 14:45, Room 403: Piotr M. Hajac (IMPAN, Poland)

The functoriality of graph algebras

Motivated by natural examples in noncommutative topology, we study pullbacks of graph C*-algebras. While some pullbacks can be easily understood as coming from pushouts of directed graphs via a contravariant functor assigning C*-algebras to graphs, other pullbacks require both covariant and contravariant functors assigning C*-algebras to graphs being objects in two different categories. In the latter case, one cannot explain the pullback structure of a graph C*-algebra in terms of a commuting diagram of underlying graphs in one category of graphs. We solve this problem by introducing a new category of directed graphs, where morphisms are relations rather than maps, and define a covariant functor from this category to the category of C*-algebras. Now, the new functor generalizes both the covariant and contravariant functors used before. Our main result is a pullback theorem stating under which assumptions the new functor maps a commutative diagram of graphs into the pullback diagram of their graph C*-algebras. (Based on recent joint works with Mariusz Tobolski and Gilles Gonçalves de Castro.)

Monday 15:00 - 15:45, Room 403: Gaston Nieuviarts (Università di Genova, Italy)

Skew torsion from twisted spectral triples

Non-Commutative Geometry (NCG) is a framework that extends Riemannian geometry by permitting noncommutative algebras of functions on "spaces". A significant achievement of NCG has been the re-expression of gravitation and Yang-Mills theories within a unified geometric framework in which gauge and Higgs fields emerge as fluctuations of the Dirac operator. However, this approach faces two fundamental problems: it is Riemannian and requires the artificial introduction of a scalar field to obtain the experimental Higgs mass. A new approach to address these deficiencies is provided by the framework of twisted spectral triples. The starting point of this discussion lies in the observation that given a twisted manifold, twisted fluctuations of the Dirac operator generate a 1-form field that corresponds precisely to a metric and geodesic-preserving torsion. The objective of this presentation is to to characterize this torsion field, and to show the corresponding Einstein-Hilbert action. 

Wednesday 16:00 - 16:45, Room 403: Alistair Miller (Syddansk Universitet, Denmark)

Functoriality of the homology and operator K-theory of a groupoid

Ample groupoids provide a framework for studying all sorts of discrete time totally disconnected topological dynamical systems which we might study through a C*-algebra. The homology of an ample groupoid is a useful approximant to the K-theory of the associated groupoid C*-algebra. I will describe functoriality of both the homology and K-theory with respect to a broad class of groupoid morphisms known as proper étale correspondences, which encompass for example étale homomorphisms and Morita equivalences. By obtaining functoriality of the approximation too, I will describe how to deduce K-theoretic results from computations in homology.

Wednesday 17:00 - 17:45, Room 403: Emil V. Prodan (Yeshiva University, USA)

Operator product states on tensor powers of C*-algebras

The program of matrix product states on (infinite) tensor powers of finite algebras, initiated by Fannes, Nachtergaele and Werner in Commun. Math. Phys. 144, 443-490 (1992), supplies a constructive algorithm to generate ergodic states relative to the natural action of the group of integer numbers. In this seminar, I re-assess this program for the generic context of tensors of nuclear C*-algebra. Key to our approach is a rigorous definition and characterization of what we call entanglement kernel. The quotient of the half-space tensor algebra by this kernel supplies an operator system and a reduced state on it, and we call this stage of analysis the reduction process. The next stages of the analysis are the factorization and reconstruction processes that supply positive maps that can be iterated to reconstruct the initial state on the tensor power. Stinespring representations of the mentioned positive maps, together with their iterates, lead to a presentation of the original state in terms of operator products. New examples of states generated with the algorithm will be presented.

Friday 14:00 - 14:45, Room 321: Valerio Proietti (Universitetet i Oslo, Norway)

Elliott invariant in a geometric context

Given a class of topological dynamical systems, we study the associated mapping torus from the point of view of foliated spaces. By studying the interaction between the leafwise Dirac operator and the invariant transverse measures, we reframe in a geometric fashion the Elliott invariant for the crossed product of the dynamical system, and prove a rigidity result for the mapping torus, lifting leafwise homotopy equivalences to isomorphism of the noncommutative leaf space. Joint work with Hao Guo and Hang Wang.

Friday 15:00 - 15:45, Room 321: Lucas Hall (University of Haifa, Israel)

Topological classification of some coactions on C*-algebras of topological quivers

Topological quivers are the broadest topological analogue of directed graphs, and may be used to construct C*-algebras. I will present a topological version of the combinatorial skew product and show how this designs a natural coaction on the associated C*-algebra. With classical intuition at hand, this develops a class of coactions which one can “see” in a noncommutative framework. I’ll gesture toward some work in progress generalizing these coactions to a broader setting.


Monday 14:00 - 14:45, Room 403: Francesco D'Andrea (Università di Napoli, Italy)

From graph to groupoid C*-algebras: the example of quantum spheres

Odd-dimensional quantum spheres were introduced by Vaksman and Soibelman in the 1990s as quantum homogeneous spaces of Woronowicz's quantum unitary groups. Since then, they have been a main example in testing ideas about noncommutative spaces. In 1997, Sheu proved that the C*-algebra of a quantum sphere is isomorphic to a groupoid C*-algebra. Five years later, Hong and Szymański proved that it is a graph C*-algebras. In this talk I will explain how the groupoid introduced by Sheu is related to the graph groupoid of Hong-Szymański's graph.

Monday 15:00 - 15:45, Room 403: Elmar Wagner (Universidad Michoacana, Mexico)

Spectral triples on quantum flag manifolds from the Bernstein-Gelfand-Gelfand resolution

It is a general observation that the most studied spectral triples on quantum homogeneous spaces make extensive use of the representation theory of quantum groups. One way to construct a Dirac operator is by starting with a covariant differential calculus. However, in the case of irreducible quantum flag manifolds, there is a direct way to describe the differential calculus explicitly by using a quantum version of the Bernstein-Gelfand-Gelfand resolution found by Heckenberger and Kolb. For reasons explained in the talk, we will find enough first order derivations for a description of the Dolbeault operator by left invariant vector fields. An inner product will be determined by the basic requirement of unitary representations. Then the main problem for determining the spectrum of the Dobeault-Dirac operator resides in determining the branching laws and computing the spectrum of the associated Laplacians. We will discuss how this can be reduced to computations on highest weights.

Wednesday 16:00 - 16:45, Room 403: Petr Somberg (Univerzita Karlova, Czechia)

Dirac operator and its cohomology for the quantum group Uq(sl2)

I will introduce a Dirac operator D for the quantum group Uq(sl2), as an element of the tensor product of Uq(sl2) and the Clifford algebra on two generators. I will demonstrate some basic properties of D, including an analogue of Vogan's conjecture. I compute the cohomology of D acting on various Uq(sl2)-modules.

Wednesday 17:00 - 17:45, Room 403: Stefan Wagner (Blekinge Tekniska Högskola, Sweden)

The noncommutative geometry of frame bundles

Vector bundles in classical geometry typically arise as objects associated with something more profound, a principal bundle. In particular, each vector bundle E with fibre V is naturally associated with a principal GL(V)-bundle, the frame bundle of E. Frame bundles thus constitute a key tool for studying vector bundles. Indeed, a connection on a frame bundle induces covariant derivatives on all associated bundles in a coherent way, leading to many important geometric constructions. This is the situation in Riemannian geometry where, for a Riemannian manifold M, the Levi-Civita connection on the frame bundle of M induces a covariant derivative on the tensor fields, leading, for instance, to the Riemannian curvature of M. The noncommutative geometry of frame bundles, however, has not been studied conclusively, although the notion of a noncommutative principal bundle is certainly available. In this talk we present our approach to the subject. Our study is part of a larger program with the purpose to give a novel bundle-theoretic perspective on noncommutative Riemannian spin geometry.

Friday 14:00 - 14:45, Room 403: Natã Machado (Universidade Federal de Santa Catarina, Brazil)

Non-self-adjoint operator algebras associated with étale categories

Étale categories naturally arise in non-commutative frame theory and in the study of non-involutive combinatorial objects. In this presentation, I will introduce a definition for the non-self-adjoint operator algebra of an étale category and highlight some properties that connect with the established case of C*-algebras of étale groupoids.

Friday 15:00 - 15:45, Room 403: Philipp Schmitt (Universität Hannover, Germany)

Deformation quantization of polynomial Poisson structures

Deformation quantization is a general framework for quantizing classical mechanical systems by deforming the classical observable algebra into a non-commutative algebra, with the deformation parameter playing the role of Planck's constant. In a formal setting, Kontsevich's Formality Theorem provides a formula to quantize any Poisson structure on R^n. However, the weights appearing in this formula make it hard to understand its convergence properties, and the existence of "strict deformation quantizations", where the deformation parameter can be evaluated to the actual physical value of Planck's constant, is a widely open problem. In this talk, I will present a combinatorial approach to the formal deformation quantization of polynomial Poisson structures on R^n, due to Barmeier-Wang, which does not involve weights. This makes the convergence properties more accessible, and yields strict quantizations for several polynomial Poisson structures on R^n. A particularly striking example is the quantization of the constant Poisson structure on R^2 perturbed by a quadratic term, which results in a quantum Weyl algebra that is, in some sense, much "bigger" than the usual Weyl algebra. This talk is based on joint work with S. Barmeier.


Monday 14:00 - 14:45, Room 405: Ludwik Dąbrowski (SISSA, Italy)

Metric and Einstein tensors from spectral functionals

Given a Laplace operator, using the noncommutative residue we define two bilinear functionals of vector fields, the values of which yield metric and Einstein  tensors, respectively. Alternatively, given a Dirac operator on a spin manifold, we define two bilinear functionals of differential 1-forms, the values of which yield the dual metric and Einstein tensors, respectively. We generalise these concepts to Laplace-type and Dirac-type operators, including Hodge-de Rham operator, and in non-commutative geometry. In particular we show that for the conformally rescaled noncommutative torus the Einstein functionals vanish. Based on Adv. Math. 427 (2023) 1091286;  J. Noncommut. Geom. and Commun. Math. Phys. (in press), with A. Sitarz and P. Zalecki.

Monday 15:00 - 15:45, Room 405: Mariusz Tobolski (Uniwersytet Wrocławski, Poland)

Local-triviality dimensions

Local-triviality dimensions are invariants of actions of compact (quantum) groups on unital C*-algebras. When both the group and the algebra are classical, the finiteness of these dimensions is equivalent to having a (locally trivial) principal bundle. In my talk, I will discuss the most important results concerning the local-triviality dimensions and many natural examples coming from gauge actions on graph C*-algebras.

Wednesday 16:00 - 16:45, Room 405: Andrzej Sitarz (Uniwersytet Jagielloński, Poland)

Spectral torsion

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We provide examples of this functional over several spectral triples, including Hodge-de Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum SU(2) group. Based on a joint work with Ludwik Dąbrowski and Paweł Zalecki.

Wednesday 17:00 - 17:45, Room 405:  Joseph C. Várilly (Universidad de Costa Rica)

How do symmetries arise in quantum theory?

Many writings on quantum fields come with preassigned symmetries in the form of a local gauge group (or Lie algebra) or its possible q-deformations. However, such local symmetries need not be taken as inputs. We briefly explain how the structure of interaction polynomials automatically imposes a Lie algebra of compact type at second order in perturbation theory.

Friday 14:00 - 14:45, Room 405: Roberto Conti (Sapienza Università di Roma, Italy)

Modular algebraic quantum theory (part I)

Part I is dedicated to some general issues. Motivated by algebraic quantum field theory (suitably freed from its usual background space-time geometry), we exploit a somehow hidden ingredient (modular theory) in order to uncover and restore a link between geometry and quantum theory. This further supports the proposal for a modular theory of quantum gravity (arXiv:1007.4094v1), where quantum space-time is spectrally reconstructed a posteriori by covariant states, via Tomita-Takesaki modular theory as dynamical constraint, and covariance is described via categorical principles.

Friday 15:00 - 15:45, Room 405: Paolo Bertozzini (Thammasat University, Thailand)

Modular algebraic quantum theory (part II)

Part II is dedicated to modular spectral geometry (some heretic views on quantization). Motivated by algebraic quantum field theory (suitably freed from its usual background space-time geometry), we exploit a somehow hidden ingredient (modular theory) in order to uncover and restore a link between geometry and quantum theory. This further supports the proposal for a modular theory of quantum gravity (arXiv:1007.4094v1), where quantum space-time is spectrally reconstructed a posteriori by covariant states, via Tomita-Takesaki modular theory as dynamical constraint, and covariance is described via categorical principles.


Monday 14:00 - 14:45, Room 403: Tomasz Maszczyk (Uniwersytet Warszawski, Poland)

Dirac propagator as a functor from 4D-cobordism to Hilbert spaces

We construct a functor from the category of oriented Riemannian 3-folds, with the causal collar Spin(1, 3)-structure with ``from-space-like-to-space-like’’ oriented causal Spin(1, 3)-cobordisms as morphisms, to the groupoid of Hilbert spaces with unitary isomorphisms. In defining such a functor, we identify the positive definiteness of the hermitian scalar product as the space-likeness of the boundary. We show that this functor can be extended via the fermionic Fock space construction to a Quantum Field Theory.  

Monday 15:00 - 15:45, Room 403: Eusebio Gardella (Chalmers tekniska högskola, Sweden)

Classifiability of crossed products

To every action of a discrete group on a compact Hausdorff space one can canonically associate a C*-algebra, called the crossed product. The crossed product construction is an extremely popular one, and there are numerous results in the literature that describe the structure of this C*-algebra in terms of the dynamical system. In this talk we will focus on one of the central notions in the realm of the classification of simple, nuclear C*-algebras, namely Jiang-Su stability. We will review the existing results and report on the most recent progress in this direction, going beyond the case of free actions both for amenable and nonamenable groups. Parts of this talk are joint works with Geffen, Kranz and Naryshkin, and with Geffen, Gesing, Kopsacheilis and Naryshkin.

Wednesday 16:00 - 16:45, Room 403: Arkadiusz Bochniak (Max Planck Institut für Quantenoptik, Garching, Germany)

Quantum version of Mycielski transformation: construction and consequences

Mimicking constructions known from classical graph theory, we propose an analog of Mycielski transformation for quantum graphs and study its consequences. In particular, we analyze its impact on (quantum) chromatic and clique numbers of quantum graphs. We also briefly discuss the role of (quantum) symmetries. Based on joint work with P. Kasprzak and work in progress with P. Kasprzak, P. M. Sołtan, and I. Chełstowski.

Wednesday 17:00 - 17:45, Room 403: Cristian Ivanescu (MacEwan University, Edmonton, Canada)

Notes on Villadsen algebras

The study of C*-algebras is often thought to be the study of noncommutative topological spaces. M. Rieffel introduced stable rank as a noncommutative version of the covering dimension of topological spaces. All simple C*-algebras were thought to be either stable rank one or infinity. J. Villadsen constructed simple C*-algebras with a stable rank equal to 2, 3, 4…. In my talk, I will discuss Villadesn's construction and present some recent results on Villadsen algebras obtained in joint work with  Dan Kucerovsky from UNB.

Friday 14:00 - 14:45, Room 403: Mahesh Krishna (Indian Statistical Institute, Bangalore Centre)

Unitary group actions: exploring frames and their links to noncommutative geometry

The intersection of frames generated through unitary group actions represents a captivating and contemporary research domain, establishing intriguing ties with diverse fields, notably noncommutative geometry. This presentation will elucidate key findings within this realm. Additionally, time permitting, I will delve into the evolution of frame theory concerning the Hilbert C*-modulus.

Friday 15:00 - 15:45, Room 403: Jan Gundelach (Chalmers tekniska högskola, Sweden)

Amenability of Banach algebras

An important family of Banach algebras are the L^1(G) algebras for a locally compact group. In some sense, all information on the group can be recovered from L^1(G). Johnson proved that amenability of a group is equivalent to a completely Banach algebraic property of L^1(G), which therefore was baptized to be the definition of amenability of Banach algebras. Concretely, a Banach algebra is amenable if it admits a bounded approximate diagonal. An approximate diagonal of a Banach algebra B is an asymptotically central net in B tensor B so that the image net in B, when we collapse tensors to multiplication signs, is a right approximate identity for B. There are two Banach algebra constructions that generalize the construction G -> L^1(G): For groupoids with a Haar system, the analogue is the completion of compactly supported functions in the I-norm L^1(groupoid) and for C*-dymanical systems G -> A the analogue is the twisted convolution algebra L^1(G,A). In this talk, I will discuss ongoing research with Eusebio Gardella and Eduard Ortega on the generalizations of Johnson’s theorem in those cases. Ultimately, the question is how amenability for those Banach algebras can be characterized in terms of the underlying groupoid or the C*-dynamical system, respectively.