Conference 62. Seminar Sophus Lie

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.

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.

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,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,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,R)-limit set.

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

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 $Z_2$-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