Curved momentum spaces from quantum groups with cosmological constant

We bring the concept that quantum symmetries describe theories with nontrivial momentum space properties one step further, looking at quantum symmetries of spacetime in presence of a nonvanishing cosmological constant $\Lambda$. In particular, the momentum space associated to the $\kappa$-deformation of the de Sitter algebra in (1+1) and (2+1) dimensions is explicitly constructed as a dual Poisson-Lie group manifold parametrized by $\Lambda$. Such momentum space includes both the momenta associated to spacetime translations and the `hyperbolic' momenta associated to boost transformations, and has the geometry of (half of) a de Sitter manifold. Known results for the momentum space of the $\kappa$-Poincar\'e algebra are smoothly recovered in the limit $\Lambda\to 0$, where hyperbolic momenta decouple from translational momenta. The approach here presented is general and can be applied to other quantum deformations of kinematical symmetries, including (3+1)-dimensional ones.