Kinematical Lie algebras in 2+1 dimensions

TL;DR

This work provides a complete classification of real kinematical Lie algebras in $2+1$ dimensions by deforming the static kinematical algebra, employing a relative deformation complex and automorphism analysis to identify integrable deformations. The authors compute the second and third cohomology of the deformation complex, solve obstruction equations up to second order, and classify integrable deformations into four branches that are then reduced to canonical Lie brackets via $G$-stabilisers. They construct a comprehensive table of algebras and determine which are metric, revealing that, in $2+1$ dimensions, non-semisimple metric algebras (such as Carroll, Euclidean, and Poincaré variants) do exist, alongside a family of time-like algebras unique to this dimension. The results sharpen the understanding of spacetime symmetry algebras in $2+1$ dimensions and set the stage for a parallel spacetime classification in a forthcoming work.

Abstract

We classify kinematical Lie algebras in dimension 2+1. This is approached via the classification of deformations of the static kinematical Lie algebra. In addition, we determine which kinematical Lie algebras admit invariant symmetric inner products.