Kinematical Lie algebras via deformation theory

TL;DR

This paper uses deformation theory to classify all possible kinematical Lie algebras in 3+1 dimensions, including their central extensions and invariant inner products, providing foundational results for higher dimensions.

Contribution

It introduces a systematic deformation theory approach to classify kinematical Lie algebras and their extensions in 3+1 dimensions, including new deformations and invariant structures.

Findings

Classified all deformations of the static kinematical Lie algebra

Identified which Lie algebras admit invariant symmetric inner products

Discovered new non-central extensions of deformations

Abstract

We present a deformation theory approach to the classification of kinematical Lie algebras in 3+1 dimensions and present calculations leading to the classifications of all deformations of the static kinematical Lie algebra and of its universal central extension, up to isomorphism. In addition we determine which of these Lie algebras admit an invariant symmetric inner product. Among the new results, we find some deformations of the centrally extended static kinematical Lie algebra which are extensions (but not central) of deformations of the static kinematical Lie algebra. This paper lays the groundwork for two companion papers which present similar classifications in dimension D + 1 for all D>3 and in dimension 2+1.