Kinematical Lie algebras and symplectic symmetric spaces I: Lie algebraic aspects

Bieliavsky, Pierre;BOULANGER, Nicolas
(2025) Letters in Mathematical Physics : a journal for the rapid dissemination of short contributions in the field of mathematical physics — Vol. 116, n° 1, p. null-null (2025)

Files

KINE-NICO-LMP.pdf
  • Open Access
  • Adobe PDF
  • 489.27 KB
Download

Details

Authors
Abstract
The aim of this note is to present a close relation between kinematical Lie algebras and symmetric spaces in a symplectic context: to every kinematical Lie algebra is canonically associated a symplectic symmetric space. For non-flat symmetric spaces, this correspondence is one-to-one onto a specific class of symplectic symmetric spaces whose structure we describe in details. In particular, the transvection Lie algebra of such a symmetric space is either three-graded or of the Poincar{\'e} type. The denomination “Poincar{\'e} type” refers to symplectic symmetric spaces characterized by a property that generalizes the fact that the classical Poincar{\'e} group $SO_o(1, D) \ltimes \Bbb{R}^{D+1}$ turns out to be the transvection group of an unexpected purely symplectic symmetric space structure on the cotangent bundle of the hyperbolic space $SO_o(1, D)/SO(D)$ (i.e., the mass-shell orbit). The Lie triple system associated with every such symplectic symmetric space is of Jordan type (in the sense of W. Bertram), i.e., it is a homotope of the Lie triple system associated with a Hermitian symmetric space. However, the class of those JordanLie triple systems associated with a classical kinematical Lie algebra is not stable under the natural operations of homotopies and dualities defined on Jordan-Lie triple systems. In order to restore stability, we need to introduce a natural generalization of the notion of kinematical Lie algebras, which is the framework where the present work is formulated. The last section of this work presents some remarks on the coadjoint orbits that are naturally associated with symplectic symmetric spaces.
Affiliations

Citations

  • APA
  • Chicago
  • FWB

Bieliavsky, P., & BOULANGER, N. (2025). Kinematical Lie algebras and symplectic symmetric spaces I: Lie algebraic aspects. Letters in Mathematical Physics : a journal for the rapid dissemination of short contributions in the field of mathematical physics, 116(1), null-null. https://doi.org/10.1007/s11005-025-02041-9 (Original work published 2025)