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

TL;DR

This paper classifies generalized kinematical Lie algebras by showing they inherently possess a symplectic involutive structure, leading to symplectic symmetric spaces with invariant connections and structures, enriching the understanding of relativity algebra classifications.

Contribution

It introduces a canonical symplectic involutive Lie algebra structure for generalized kinematical Lie algebras and classifies these structures comprehensively.

Findings

Every generalized kinematical Lie algebra has a canonical siLa structure.

Associated homogeneous spaces are symplectic symmetric spaces with invariant connections.

Complete classification of the fine structure of these siLas.

Abstract

We generalize the notion of kinematical Lie algebra introduced in physics for the classification of the various possible relativity algebras an isotropic spacetime can accommodate. We first give an elementary proof of the fact that such a generalized kinematical Lie algebra $\mathfrak{g}$ always carries a canonical structure of symplectic involutive Lie algebra (shortly ``siLa''). In other words, if $G$ is a connected Lie group admitting $\mathfrak{g}$ as Lie algebra, there always exists a Lie subgroup $H$ of $G$ constituted by the elements of $G$ that are fixed under an involutive automorphism of $G$ and such that the homogenenous space $M=G/H$ is a symplectic symmetric space. In particular, the manifold $M$ canonically carries a $G$-invariant linear torsionfree connection $\nabla$ whose geodesic symmetries centered at all points extend as global $\nabla$-affine transformations of $M$. The manifold $M$ is also canonically equipped with a symplectic structure $\omega$ which is invariant under every geodesic symmetry, implying in particular that it is parallel w.r.t the linear connection: $\nabla\omega=0$. In a second part, we give a complete description of the fine structure of our generalized siLa's. Our discussion yields a complete classification of such sila's.