Abstract
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 non-vanishing cosmological constant. The $\kappa$-de Sitter and $\kappa$-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 $\kappa$-Poincar\'e curved momentum space is smoothly recovered as the vanishing cosmological constant limit from both of the constructions.
Highlights
It is understood that in models of Planck-scale deformed special relativity (DSR), where the Planck energy plays the role of a second relativistic invariant besides the speed of light, momentum space has a nontrivial geometry [1,2,3,4,5]
Curved momentum spaces associated to the κ-deformation of the (3 þ 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
We focused on the κ-de Sitter (dS) algebra, which can be seen as a deformation of the standard Poincarealgebra governed by two deformation parameters: the cosmological constant Λ 1⁄4 −ω is a classical deformation parameter, controlling the effects of spacetime curvature, while z 1⁄4 1=κ is the quantum deformation parameter, controlling the Planck-scale effects
Summary
It is understood that in models of Planck-scale deformed special relativity (DSR), where the Planck energy plays the role of a second relativistic invariant besides the speed of light, momentum space has a nontrivial geometry [1,2,3,4,5]. The nontrivial features that arise in the (3 þ 1) construction with respect to the (2 þ 1) case for both quantum dS and AdS symmetries are emphasized The fact that a momentum space relying only on the spacetime translations cannot be constructed is made apparent by the fact that the coalgebra of translations does not close on its own, but it contains generators of the Lorentz sector These two features are analogous to what we had observed for the lower-dimensional models studied in our previous paper [the mixing between the translation sector and the Lorentz sector being somewhat less invasive in the (1 þ 1) case, since it only concerns the algebra sector]. VI summarizes our findings and additional insights on the nontrivial properties of rotations that emerge in (3 þ 1) dimensions are included in an appendix
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