# Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that

• Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$

• de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and

• anti-de Sitter spacetime has $$SO(d-1,2)/SO(d-1,1).$$

One then is interested in unitary irreducible representations of the space-time symmetry group.

Question: Is this correct that the above is the symmetry of Minkowski, de Sitter spacetime, and anti-de Sitter spacetime? It this the same as the isometry of these spacetimes? How to show this is the complete symmetry?