Classification of kinematical Lie algebras

TL;DR

This note surveys the landscape of kinematical Lie algebras across dimensions, detailing their static baseline and all possible deformations, with a focus on when such algebras admit an invariant inner product. It highlights dimension-specific phenomena: the Bianchi classification governs $D=1$, a rich $D=2$ sector arising from a symplectic structure, and three-to-four families in $D=3$ enabled by the vector product, plus higher-$D$ results showing that only simple algebras (and their trivial central extensions) are metric for $D>3$. The classification draws on deformation theory and central extensions, consolidating outcomes from Bassry–Nuyts, Bacry–Lévy-Leblond, and subsequent high-dimensional analyses. The work provides a reference taxonomy for the metric and non-metric kinematical algebras across all $D eq 2$, including explicit central-extension-induced families like $rak{so}(D+1,1)igoplusrak{R}$, $rak{so}(D+2)igoplusrak{R}$, and $rak{so}(D,2)igoplusrak{R}$.

Abstract

We summarise the classification of kinematical Lie algebras in arbitrary dimension and indicate which of the kinematical Lie algebras admit an invariant inner product.