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 $SO(p,q+1)\mathrm{/}SO(p,q)$ where $p+q=\mathcal{N}$, $\mathcal{N}\in \mathbb{N}$. 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 $\mathcal{R}(\mathcal{N}+1)$. The spectrum obtained from ${\cal R}({\cal N}+1)$ via the Daskaloyannis method depends on undetermined signs that can be associated to the signatures. Two examples are worked out explicitly for the cases $SO(2,1)\mathrm{/}SO(2)$ and $SO(3)\mathrm{/}SO(2)$ 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 $\mathcal{R}(3)$.