The general Racah algebra as the symmetry algebra of generic systems on pseudo-spheres

We characterize the symmetry algebra of the generic superintegrable system on a pseudo-sphere corresponding to the homogeneous space SO(p, q + 1)/SO(p, q) where p+q=N, N∈N. These symmetries occur both in quantum as well as in classical systems in various contexts, so they are quite important in physics. We show that this algebra is independent of the signature (p, q + 1) of the metric and that it is the same as the Racah algebra R(N+1). The spectrum obtained from R(N+1) via the Daskaloyannis method depends on undetermined signs that can be associated to the signatures. Two examples are worked out explicitly for the cases SO(2, 1)/SO(2) and SO(3)/SO(2) where it is shown that their spectrum obtained by means of separation of variables coincide with particular choices of the signs, corresponding to the specific signatures, of the spectrum for the symmetry algebra R(3).