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 detail. In particular, the transvection Lie algebra of such a symmetric space is either three-graded or of the Poincaré type. The denomination “Poincaré type” refers to symplectic symmetric spaces characterized by a property that generalizes the fact that the classical Poincaré group \(SO_o(1,D)\ltimes \mathbb {R}^{D+1}\) 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 Jordan-Lie 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.
