Maxwell kinematical algebras and 3D gravities

TL;DR

This paper develops a systematic expansion framework to derive Maxwell extensions of kinematical Lie algebras, enabling the construction of 3D Chern-Simons gravity theories with new algebraic structures.

Contribution

It introduces a semigroup expansion method to generate Maxwell kinematical algebras from parent algebras, unifying various cases and extending to an infinite hierarchy.

Findings

Systematic derivation of Maxwell algebras via semigroup expansion.

Construction of invariant tensors for these algebras.

Formulation of 3D Chern-Simons gravity theories based on the new algebras.

Abstract

In this paper, we present a Maxwell extension of kinematical Lie algebras by promoting the contraction method underlying the Bacry and L\'evy-Leblond cube to a semigroup expansion framework. Within this approach, we show that both non- and ultra-relativistic Maxwell algebras admitting non-degenerate invariant bilinear forms can be systematically obtained from different parent algebras through a unified expansion scheme, leading to a Maxwellian kinematical cube. This construction is further generalized to an infinite hierarchy of kinematical algebras. The expansion method naturally provides the corresponding invariant tensors, allowing for the systematic construction of three-dimensional Chern-Simons gravity theories.