Higher-dimensional kinematical Lie algebras via deformation theory

TL;DR

This work extends deformation-theoretic classification to kinematical Lie algebras in dimension $D+1$ with $D\ge 4$ by analyzing deformations of the static kinematical Lie algebra and its universal central extension. Using Chevalley--Eilenberg cohomology and Hochschild--Serre factorization, the authors compute infinitesimal deformations, identify obstructions, and derive explicit normal forms, revealing branches parameterized by a central parameter $t_5$ and, for $t_5\neq 0$, orbit types (zero, spacelike, timelike, lightlike) corresponding to well-known algebras such as Carroll, Galilei, Newton, Poincaré, and (A)dS variants. They show that up to isomorphism these deformations yield algebras including $\mathfrak{so}(D+1,1)$, $\mathfrak{so}(D+2)$, $\mathfrak{so}(D,2)$, as well as their trivial or non-central central extensions, with invariant inner products existing only for the simple algebras (via the Killing form) or their trivial central extensions. The results for $D=4$ align with $D\ge 5$ after detailed calculation, while the $D=3$ case remains distinct due to special low-dimensional phenomena. Overall, the paper provides a comprehensive, deformation-theoretic map of higher-dimensional kinematical Lie algebras and their metric properties, organized into explicit tables and normal forms.

Abstract

We classify kinematical Lie algebras in dimension $D \geq 4$. This is approached via the classification of deformations of the relevant static kinematical Lie algebra. We also classify the deformations of the universal central extension of the static kinematical Lie algebra in dimension $D\geq 4$. In addition we determine which of these Lie algebras admit an invariant inner product.