The general solution of the real $Z_2^{\otimes N}$ graded contractions of $so(N+1)$

TL;DR

This paper provides an explicit general solution for graded contractions of the real Lie algebra so(N+1), revealing how contraction parameters relate to geometric curvatures and kinematical algebras.

Contribution

It explicitly solves the graded contraction equations for so(N+1) with a $bz_2^{ imes N}$ grading, detailing the structure and geometric interpretation of contraction parameters.

Findings

Solution depends on 2^N - 1 parameters.

Low-dimensional cases are explicitly analyzed.

Contraction parameters relate to geometric curvatures and space-time properties.

Abstract

The general solution of the graded contraction equations for a $\zz_2^{\otimes N}$ grading of the real compact simple Lie algebra $so(N+1)$ is presented in an explicit way. It turns out to depend on $2^N-1$ independent real parameters. The structure of the general graded contractions is displayed for the low dimensional cases, and kinematical algebras are shown to appear straightforwardly. The geometrical (or physical) meaning of the contraction parameters as curvatures is also analysed; in particular, for kinematical algebras these curvatures are directly linked to geometrical properties of possible homogeneous space-times.