Extended Kinematical 3D Gravity Theories

TL;DR

This paper classifies extended kinematical Lie algebras derived from $ ext{so}(2,2)$ using semigroup expansions, leading to new non- and ultra-relativistic algebras suitable for well-defined 3D Chern-Simons gravity actions.

Contribution

It introduces a systematic classification of extended kinematical algebras via algebra expansion methods, ensuring non-degenerate invariant forms for gravity theories.

Findings

Expanded algebras include non- and ultra-relativistic cases

Constructed well-defined Chern-Simons gravity actions

Avoided degeneracy without algebra extension

Abstract

In this work, we classify all extended and generalized kinematical Lie algebras that can be obtained by expanding the $\mathfrak{so}\left(2,2\right)$ algebra. We show that the Lie algebra expansion method based on semigroups reproduces not only the original kinematical algebras but also a family of non- and ultra-relativistic algebras. Remarkably, the extended kinematical algebras obtained as sequential expansions of the AdS algebra are characterized by a non-degenerate bilinear invariant form, ensuring the construction of a well-defined Chern-Simons gravity action in three spacetime dimensions. Contrary to the contraction process, the degeneracy of the non-Lorentzian theories is avoided without extending the relativistic algebra but considering a bigger semigroup. Using the properties of the expansion procedure, we show that our construction also applies at the level of the Chern-Simons action.