Kinematical Lie algebras and symplectic symmetric spaces I: Lie algebraic aspects
Pierre Bieliavsky, Nicolas Boulanger
TL;DR
The paper develops a unified framework linking generalized kinematical Lie algebras to symplectic geometry by proving that any such algebra $\mathfrak{g}$ carries a canonical symplectic involutive Lie algebra (siLa) structure. This yields a simply connected symplectic symmetric space $M=G/H$ with a $G$-invariant torsionfree connection $\nabla$ and a $\omega$-form parallel under $\nabla$, unifying spacetime symmetry classifications with symplectic geometry. It provides a complete description of the fine structure of these sila’s, including complex, real, semisimple, and Poincaré-type nilpotent regimes, and shows that in many cases the associated spaces are cotangent bundles of symmetric spaces or Hermitian/para-Hermitian/cayley-type duals, thereby linking kinematical algebras to Jordan theory and structure varieties. The results have implications for the geometric and Hamiltonian aspects of relativity algebras, offering a robust algebraic-geometric classification that persists under Jordan-Lie dualities and contractions, with potential applications to generalized spacetime models and quantization. Overall, the work bridges physical relativity principles with a rigorous symplectic-involutive Lie theory, providing a comprehensive classification and a pathway to further harmonic-analytic and geometric investigations.
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 $ω$ which is invariant under every geodesic symmetry, implying in particular that it is parallel w.r.t the linear connection: $\nablaω=0$. 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.