61st Seminar Sophus Lie 

The Seminar Sophus Lie is an itinerant workshop on the theory of Lie groups and their wider horizon. It was founded around 1989-90 in the framework of the activities to reconnect scientists from former West and East Germany, which had been separated for more than forty years. Since then, the Seminar Sophus Lie became an international event and was organised once or twice per year in several European universities (see the list of the previous editions). For further details on the history of the Seminar, see A. Fialowski and A. Szilard, Seminar Sophus Lie, Newsletter of the EMS 69 (Sept. 2008), pp. 14-16.

The 61st edition will take place at the Department of Mathematics, Aula dal Passo, Tor Vergata University of Rome (Italy). 

Organizing committee: Martina Lanini, Vincenzo Morinelli

We are planning to have a social aperitif (early light dinner) on Thursday 11th.
It will take place on the rooftop of Hotel Nord Nuova Roma, Via Giovanni Amendola, 3/Roma 00185, 00185 Roma (RM)

https://maps.app.goo.gl/KKoSvzh1NXTwguBF8

Schedule

• Francesca Arici (Leiden University)
  
  Title: SU(2)-Symmetries and Gysin Sequences for C*-algebras

Abstract:This talk explores the interplay between representation theory and noncommutative geometry, motivated by the classical Gysin sequence in topological K-theory. We introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces—structures that naturally encode group symmetries and arise from tensor powers of finite-dimensional representations.

Associated to these systems are Toeplitz and Cuntz–Pimsner C*-algebras, whose topological invariants we study using equivariant KK-theory. A key result is that the Toeplitz algebra built from the subproduct system of an irreducible SU(2) representation is equivariantly KK-equivalent to the complex numbers. This allows for a computation of the KK-theory of the corresponding Cuntz–Pimsner algebra via a Gysin-type exact sequence, featuring a noncommutative analogue of the Euler class.

• Pierre Bieliavsky (Université Catholique de Louvain)

Title: Kinematical Lie algebras and symplectic symmetric spaces

Abstract: The notion of kinematical Lie algebra was introduced in physics for the classification of the various possible relativity Lie algebras an isotropic spacetime can accommodate. I will explain how such a (generalized) kinematical Lie algebra underlies a symplectic symmetric space which completely governs its structure. 

I will illustrate the construction by the example of the Poincaré group (and generalizations) which turns out to be the transvection group of a (non-metric) symplectic symmetric space. This is a joint work with Nicolas Boulanger.

• Francesco D'Andrea (Università degli Studi di Napoli Federico II)
  
  Title: Relation morphisms of directed graphs
  
  Abstract: In the past decades, algebras associated with combinatorial data have been the basis for a fruitful exchange of ideas among pure algebraists, analysts, and geometers. These include Leavitt path algebras, Cuntz-Krieger algebras, graph C*-algebras, and their generalisations (ultragraph C*-algebras, higher rank graph C*-algebras, etc.), as well as convolution algebras of étale groupoids. In each of these examples, one has both a covariant and a contravariant functor, from a suitable source category of combinatorial objects (e.g., graphs or groupoids) to the category of associative (*-)algebras. The covariant and contravariant pictures can be unified by considering morphisms defined by relations rather than maps. I will discuss this problem for algebras associated to directed graphs. (Based on a joint work with G.G. de Castro & P.M. Hajac.)
  
  

• Thomas Creutzig  (FAU Erlangen-Nürnberg)
  
  Title: The status quo of representation theory of affine vertex superalgebras
  
  Abstract: Abstract: To any Lie algebra g togehter with a bilinear form B on g one can associate a vertex algebra, the universal affine vertex algebra of g associated to the "level" B. Representations of this vertex algebra are special representations of the affinization of g. One advantage of vertex algebras is that they often can give the representation categoies the structure of a ribbon ternsor category. If g is a simple Lie algebra and B a positive integer multiple of the Killing form, then one can for example obtain modular tensor categories. However for almost all levels and in particular for Lie superalgebras representation categories are not semisimple. I plan to give an overview talk about the state of the art of the theory.
  
  

• Tiziano Gaudio  (Tor Vergata University of Rome)
  
  Title: Conformal nets from minimal W-algebras
  
  Abstract: Minimal W-algebras are a remarkable family of conformal vertex superalgebras. They are obtained from any couple (g,f) by a procedure of quantum Hamiltonian reduction, where g is a basic simple Lie superalgebra and f is one of its minimal nilpotent even elements. In 2023,  Kac, Möseneder Frajria and Papi classified minimal W-algebras admitting a compatible positive-definite invariant Hermitian form. In all these cases, the minimal W-algebra is a unitary vertex operator superalgebra in the usual sense.
  From the algebraic quantum field theory point of view, it is conjectured that every unitary vertex operator superalgebra satisfies the analytic condition of strong graded locality, assuring the existence of a conformal net of von Neumann algebras naturally arising from it. In this talk, we show that all unitary minimal W-algebras are strongly graded-local. For some model in this family, we also test the conjecture that a strongly graded-local unitary vertex operator algebra is strongly rational if and only if its corresponding conformal net is completely rational. This talk is based on the joint work arXiv:2506.04270 with S. Carpi. 
  
  

• Bas Janssens  (Institute of Applied Mathematics Delft University of Technology)
  
  Title: Convergence of LSU(k) to LSDiff(S^2) on the level of central extensions

Abstract: By work of Hoppe, Bordemann, Schaller, Schlichenmaier and Meinrenken in the 1980's, the structure constants of su(k) converge to those of Svec(S^2) for large k, where Svec(S^2) is the Lie algebra of divergence-free vector fields on the sphere. As an immediate consequence, the structure constants of the loop algebra Lsu(k) converge to those of LSvec(S^2), the loop algebra with values in Svec(S^2). Using results of Neeb-Wagemann, we classify the integrable central extensions of LSvec(S^2), and show that these are the limit of affine Kac-Moody cocycles for Lsu(k) under an appropriate scaling. This raises the interesting question if there exists a projective unitary positive energy representation of LSDiff(S^2) which is -- at least in some sense -- the limit of a sequence of highest weight representations of \widehat{LSU(k)} for large k. If the answer is affirmative, one could hope to construct from this a (conformal?) field theory on S^1 x S^2, and study it in terms of limits of loop group conformal nets.

Joint work with Zhenghan Wang.

• Anne Moreau (Paris Orsay)
  
  Title: W-algebras as conformal extensions of affine vertex algebras 

Abstract: In this talk, I will present several criteria for a vertex algebra morphism from an affine vertex algebra to be conformal and surjective. The key application is W-algebras, where we can use these to obtain new examples of W-algebras that collapse to affine vertex algebras or are conformal extensions. This is based on a recent joint work with Dražen Adamović, Tomoyuki Arakawa, Thomas Creutzig, Andrew Linshaw, Pierluigi Möseneder Frajria, and Paolo Papi.

• Karl-Hermann Neeb (FAU Erlangen-Nürnberg)
  
  Title: A group theoretic perspective on modular theory: Abstract Euler wedges and their duality

Abstract: In the modular theory of operator algebras on encounters pairs (U,J) of unitary one-parameter groups U(t) with a commuting conjugation J. Taking symmetry groups G into account, one would like to understand the G-orbit of such pairs and how they relate to the causal structure of spacetimes in Algebraic Quantum Field Theory.
In this talk we develop a perspective on this problem starting with a Lie group G, an Euler element h in its Lie algebra (this means that ad h is diagonalizable with eigenvalues 1,0,-1) and a corresponding involution τh on G. The orbit W+ of the pair (h,τh) is the corresponding abstract wedge space. As these spaces are coverings of the adjoint orbit of h, they require fine information on its fundamental group, which we determine explicitely. We also explain how it can be generated by loops coming from 2-dimensional subspaces that are orbits of SL(2)-subgroups. The Lie algebras of these subgroups are generated by "orthogonal'' pairs of Euler elements, which are also classified.

This is based on recent work with V. Morinelli and G. Ólafsson.

• Daniele Valeri (Sapienza University of Rome)
  
  Title: Integrability of classical affine W -algebras

Abstract: Classical affine W -algebras W (g, O) are algebraic structures associated to a simple Lie algebra g and a nilpotent orbit O. In this talk we will describe how to associate to W (g, O) an integrable hierarchy of PDEs.

When O is the principal nilpotent orbit one gets the Drinfeld–Sokolov hierarchy, which gives the famous Korteweg–de Vries hierarchy for g = sl2.