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 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 i.e. affine symmetric spaces equipped with parallel symplectic structures (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 currently fastly developing field of geometric actions.

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( paper | slides | video )

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Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

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