Curved momentum spaces from quantum (anti-)de Sitter groups in (3+1) dimensions

Curved momentum spaces associated to the κ-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 κ-deformation with nonvanishing cosmological constant. The κ-de Sitter and κ-anti-de Sitter curved momentum spaces are separately analyzed, 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 κ-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.