# Global symmetries of spacetime For a fixed spacetime, one can study its symmetry. This can be done by finding Killing vector fields. The symmetry of a spacetime is very different from the symmetry of a gravitational theory, where only [[0060 Asymptotic symmetry|asymptotic symmetries]] are true symmetries: bulk diffeomorphisms are gauge symmetries. ## 4d Minkowski - translation - Lorentz - full $O(1,3)$ - proper $SO(1,3)$ - proper orthochronous $SO^+(1,3)$ - all have the same Lie algebra ==$so(1,3)\cong sl(2,\mathbb{C})$== - $sl(2,\mathbb{C})$ is 3 dimensional when viewed as complex algebra but 6 dimensional when viewed as real algebra ## AdS$_3$ - take only the $m=-1,0,1$ generators of each of the two Witt algebras (which is [[0085 Asymptotic symmetry of AdS3|the asymptotic symmetry algebra of AdS3]]) - each isomorphic to $sl(2,\mathbb{R}) \simeq so(2,1)$ - two of these make ==$sl(2,\mathbb{R})\oplus sl(2,\mathbb{R}) \simeq so(2,2)$== i.e. symmetry algebra of global AdS3