Curved momentum spaces associated to the k-deformation of the (3+1) de Sitter and Anti-de Sitter algebras are constructed as orbits of suitable actions of the dual Poisson-Lie group associated to the k-deformation with non-vanishing cosmological constant. The k-de Sitter and k-Anti-de Sitter curved momentum spaces are separately analysed, and they turn out to be, respectively, half of the (6+1)-dimensional de Sitter space and half of a space with SO(4, 4) invariance. Such spaces are made of the momenta associated to spacetime translations and the ‘hyperbolic’ momenta associated to boost transformations. The known k-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.
