2+1 Kinematical expansions: from Galilei to de Sitter algebras

TL;DR

This paper introduces a new method for expanding Lie algebras using Casimir invariants and curvature parameters, applied to kinematical algebras, revealing two main expansion types related to space-time and worldline curvatures.

Contribution

It proposes a novel expansion technique for Lie algebras based on Casimir invariants and curvature parameters, applied to 3d and (2+1)d kinematical algebras, classifying expansions into two types.

Findings

Expansions from Galilei to Newton--Hooke or Poincaré algebras.

Further expansions to de Sitter algebras.

Identification of two main expansion types based on curvature interpretations.

Abstract

Expansions of Lie algebras are the opposite process of contractions. Starting from a Lie algebra, the expansion process goes to another one, non-isomorphic and less abelian. We propose an expansion method based in the Casimir invariants of the initial and expanded algebras and where the free parameters involved in the expansion are the curvatures of their associated homogeneous spaces. This method is applied for expansions within the family of Lie algebras of 3d spaces and (2+1)d kinematical algebras. We show that these expansions are classed in two types. The first type makes different from zero the curvature of space or space-time (i.e., it introduces a space or universe radius), while the other has a similar interpretation for the curvature of the space of worldlines, which is non-positive and equal to $-1/c^2$ in the kinematical algebras. We get expansions which go from Galilei to either Newton--Hooke or Poincar\'e algebras, and from these ones to de Sitter algebras, as well as some other examples.