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

TL;DR

This paper constructs curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions, revealing their geometric structure and relation to known $ ext{kappa}$-Poincaré momentum space, with implications for quantum gravity models.

Contribution

It introduces a novel geometric construction of $ ext{kappa}$-deformed momentum spaces using dual Poisson-Lie groups for (3+1)D (A)dS algebras, extending previous flat-space results.

Findings

$ ext{kappa}$-de Sitter and Anti-de Sitter momentum spaces are half of (6+1)-dimensional de Sitter space and a space with $SO(4,4)$ invariance.

The constructed spaces include momenta from spacetime translations and boost transformations.

The flat $ ext{kappa}$-Poincaré momentum space is recovered in the zero cosmological constant limit.

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.