Kinematical Lie algebras via deformation theory

TL;DR

The paper develops a deformation-theoretic framework for classifying kinematical Lie algebras in 3+1 dimensions by computing the cohomology $H^2(rak g;rak g)$ and its obstructions via the Nijenhuis–Richardson bracket and the Maurer–Cartan equation $\partial\varphi = \tfrac12 [\![\varphi,\varphi]\!]$, then modding out by $\text{GL}(V)$. Applying Hochschild–Serre, the authors recover Bacry–Nuyts’ classification of kinematical algebras and extend it to deformations of the static algebra and its universal central extension, identifying Carroll, Lorentzian/Euclidean Newton, Galilean/Bargmann, conformal-type, and other Newton-type algebras, and determining which admit invariant inner products. The work provides a unified, constructive deformation-theory approach, clarifying how central extensions interact with deformations and laying groundwork for higher dimensions ($D>3$) and the 2+1D case in companion papers. This advances the geometric understanding of homogeneous spacetimes associated with kinematical Lie algebras and offers a rigorous toolkit for future classifications.

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.