The notion of kinematical Lie algebra was introduced in physics for the classification of the various possible relativity algebras an isotropic spacetime can accommodate (H. Bacry and J. Levy-Leblond. Possible kinematics. J. Math. Phys., 9; 1968). Kinematical Lie algebras were classified in spacetime dimension four by brute force in the middle of the eighties. (H. Bacry and J. Nuyts. Classification of Ten-dimensional Kinematical Groups With Space Isotropy. J. Math. Phys., 27; 1986). More recently, those were reconsidered in a much wider context within the mathematical framework of Cartan geometry (José Figueroa-O’Farrill. Non-lorentzian spacetimes. Differ. Geom. Appl., 82; 2022).

In a joint work with N. Boulanger (UMons), we recently gave an elementary proof of the fact that such a kinematical Lie algebra (and natural generalizations) always carries a canonical structure of symplectic involutive Lie algebra i.e. consists in the tangent version of a very specific class of symplectic symmetric spaces (Bieliavsky, P., Boulanger, N.; Kinematical Lie algebras and symplectic symmetric spaces I. Lie algebraic aspects. Letters in Mathematical Physics, 116″(1), 2026). This geometrical result yields in particular an alternative classification of (generalized) kinematical Lie algebras of arbitrary dimension in purely symplectic geometric terms. It also establishes an unexpected strong relation between these spacetimes and contact sub-Riemannian symmetric spaces. In the talk, after having introduced the basic notions, I will explain these results. If time permits, I will show the implications within the field of geometric action functionals.

The compactified Ruijsenaars–Schneider systems are completely integrable Hamiltonian systems on smooth moduli spaces of flat SU(n) connections on the one-holed torus. The monodromy around the hole is constrained to the smallest non-central conjugacy classes of SU(n), which can be labelled by a parameter $0<x<1$. Depending on the value of x, the systems arise in two drastically different forms: in type (i) these are toric systems, while in the type (ii) cases they possess globally continuous action variables that generate a Hamiltonian torus action (only) on a dense open subset of the phase space of dimension 2(n − 1). After reviewing background material, we report our recent study of the momentum polytopes and the fibers of the action map (alias the momentum map) that are contained in the complement of the domain of the densely defined torus action in the type (ii) cases. We have shown that these fibers are smooth connected isotropic submanifolds, diffeomorphic to the 3-dimensional sphere in the simplest cases, similarly to the fibers occurring, e.g., in the classical Gelfand-Cetlin systems.

The talk is based on joint work with Holger Dullin (https://arxiv.org/abs/2604.18023) and earlier publications.

Total positivity à la Lusztig and causal (homogenous) structures share a lot of similarities and can be brought together under the notion of “positivity with respect to a parabolic subgroup”. We will give a number of characterizations of positivity (Lie theoretic and more geometric), new examples and a number of specific properties.

Joint work with Anna Wienhard.

Nichols algebras emerged first in various studies of Hopf algebras, but are used in the meanwhile also elsewhere. In the talk I will first discuss established structural results connecting Nichols algebras and Lie theory. Then I will explain some recent advances regarding the classification of Nichols algebras over groups, as well as some characterizations of noetherianity and finite Gelfand-Kirillov dimension.

We discuss a functorial framework for the convergence of Drinfeld’s Universal Deformation Formula on spaces of analytic or entire vectors. Algebraically, this type of deformation is based on a Drinfeld twist

$F = \sum_{n=0}^{\infty} \hbar^n \cdot F_n \in \bigl( \operatorname{U} (\mathfrak{g}) \otimes \operatorname{U} (\mathfrak{g}) \bigr)[[\hbar]]$

It constitutes a distinguished, and rather elusive, formal power series with coefficients in two copies of the universal enveloping algebra $\operatorname{U}(\mathfrak{g})$ of some Lie algebra $\mathfrak{g}$. Drinfeld’s principal idea is then that the defining axioms of a twist induce formal deformations of any associative algebra our Lie algebra $\mathfrak{g}$ acts on by derivations. Equipping the representation space with a locally convex topology, we overcome the formal character of this deformation, i.e. pass from a formal parameter $\hbar$ to a complex number $\hbar\in \mathbb{C}$. This is achieved by matching an equicontinuity condition on the action of the components F_n of the twist with the order of analytic vectors of the representation. Finally, we demonstrate the effectiveness of our machinery by applying it to the explicit non-abelian Drinfeld twists constructed by Giaquinto and Zhang.

This is joint work with Chiara Esposito and Stefan Waldmann.

The Lie algebra of divergence-free vector fields is one of the four classical infinite-dimensional Lie algebras studied by Élie Cartan, the other three being the Lie algebras of symplectic vector fields, of contact vector fields, and the Lie algebra of all vector fields. In much the same way as Hamiltonian vector fields are the commutator subalgebra of the Lie algebra of symplectic vector fields, the exact divergence-free vector fields are the commutator subalgebra of the Lie algebra of divergence-free vector fields. We show that the Lie algebra of exact divergence-free vector fields has a universal central extension: the Lie algebra of (n-2)-forms modulo exact forms — this is joint work with Leonid Ryvkin and Cornelia Vizman. If time permits I will also comment on central extensions of the other three classical Lie algebras of Cartan.

The smooth deformations of an ideal $\mathfrak{i}$ in a Lie algebra $\mathfrak{g}$ differentiate to cohomology classes in the cohomology of $\mathfrak{g}$ with values in its adjoint representation on $\operatorname{Hom}(\mathfrak{i}, \mathfrak{g}/\mathfrak{i})$. This cohomology associated with the ideal $\mathfrak{i}\trianglelefteq\mathfrak{g}$ is compared with other Lie algebra cohomologies defined by $\mathfrak{i}$, such as the cohomology defined by $\mathfrak{i}$ as a Lie subalgebra of $\mathfrak{g}$ in (Richardson 1969), and the cohomology defined by the Lie algebra epimorphism $\mathfrak{g} \to \mathfrak{g}/\mathfrak{i}$.

After a choice of complement of the ideal $\mathfrak{i}$ in the Lie algebra $\mathfrak{g}$, its deformation complex is enriched to the differential graded Lie algebra that controls its deformations, in the sense that its Maurer-Cartan elements are in one-to-one correspondence with the (small) deformations of the ideal. Furthermore, the $L_{\infty}$-algebra that simultaneously controls the deformations of $\mathfrak{i}$ and of the ambient Lie bracket is identified. Under appropriate assumptions on the low degrees of the deformation cohomology of a given Lie ideal, the (topological) rigidity and  stability of ideals are  described, as well as obstructions to deformations of ideals of Lie algebras.

This is a joint work with Ilias Ermeidis.

A theorem of Monod-Py states that unlike finite dimensional settings, there are continuous irreducible representations of PO(1,n) in PO(1,∞). These representations are called exotic. In this talk, we will use the Debin-Fillastre hyperbolic model for convex bodies to give a different construction of non-totally geodesic quasi-isometric embedding of the hyperbolic plane in the infinite dimensional hyperbolic space, which will induce an exotic representation of PSL(2,$\mathbb{R}$). With the classical results from convex geometry, we show that this exotic representation is convex-cocompact, and that the quotient of the minimal PSL(2,$\mathbb{R}$)-invariant closed convex set by its action is homeomorphic to the oriented Banach-Mazur compactum. Moreover, we also compute the dimension of this convex set and the Hausdorff dimension of the PSL(2,$\mathbb{R}$)-limit set.

This is a joint work with François Fillastre and David Xu.

Hardy spaces arise naturally as normal forms for unitary one-parameter groups and, more generally, for representations of involutive semigroups. Motivated by this representation-theoretic viewpoint, we introduce an abstract notion of Hankel operators associated with unitary representations of involutive semigroups. For suitable choices of semigroups, this framework recovers the classical Hankel operators on Hardy spaces. This perspective also leads to an investigation of the unit group of the algebra of bounded holomorphic functions on a simply connected domain. Although the corresponding Hardy spaces are all isomorphic as Banach algebras by the Riemann Mapping Theorem, different domains naturally single out different one-parameter semigroups, leading to different group-theoretic realizations of the same Banach algebra.

In the classical theory of Lie groups, Kirillov’s coadjoint orbit method establishes a profound link between algebra and geometry, equipping orbits in the dual of a Lie algebra with a canonical symplectic structure. In this talk, we explore a parallel construction developed for Jordan algebras and their $\mathbb{Z}_2$-graded extensions, Jordan superalgebras.

We demonstrate how the Jordan product can be used to define a generalized distribution on the dual space, inducing canonical pseudo-Riemannian metrics on the resulting orbits. For formally real (Euclidean) Jordan algebras, we show that these orbits possess a natural Riemannian structure that recovers fundamental objects of information geometry: specifically, the Fisher–Rao metric in the classical setting and the Bures–Helstrom metric in the quantum setting.

This is based on joint work with Florio Ciaglia, Shuhan Jiang and Jürgen Jost