Curved momentum spaces associated to the $\kappa$-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 $\kappa$-deformation with nonvanishing cosmological constant. The $\kappa$-de Sitter and $\kappa$-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 $\mathrm{SO}(4,4)$ invariance. Such spaces are made of the momenta associated to spacetime translations and the “hyperbolic” momenta associated to boost transformations. The known $\kappa$-Poincaré curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.

• Received 5 February 2018

DOI:https://doi.org/10.1103/PhysRevD.97.106024

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Published by the American Physical Society