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 non-vanishing cosmological constant. The $κ$-de Sitter and $κ$-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 $κ$-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.