  A basic introduction to the $su(1,1)$ algebra is presented, in which we
discuss the relation with canonical transformations, the realization in terms
of quantized radiation field modes and coherent states. Instead of going into
details of these topics, we rather emphasize the existing connections between
them. We discuss two parametrizations of the coherent states manifold
$SU(1,1)/U(1)$: as the Poincar{\'e} disk in the complex plane and as the
pseudosphere (a sphere in a Minkowskian space), and show that it is a natural
phase space for quantum systems with SU(1,1) symmetry.
