Non-commutative relativistic spacetimes and worldlines from 2+1 quantum (anti-)de Sitter groups

Abstract

The $\kappa$-deformation of the (2+1)D anti-de Sitter, Poincaré and de Sitter groups is presented through a unified approach in which the curvature of the spacetime (or the cosmological constant) is considered as an explicit parameter. The Drinfel'd-double and the Poisson-Lie structure underlying the $\kappa$-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson-Lie algebras. As a consequence, the non-commutative (2+1)D spacetimes that generalize the $\kappa$-Minkowski space to the (anti-)de Sitter ones are obtained. Moreover, noncommutative 4D spaces of (time-like) geodesics can be defined, and they can be interpreted as a novel possibility to introduce non-commutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related with 2+1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related with the Planck length, and the existence of a kind of "duality" between the cosmological constant and the Planck scale is also envisaged.

25 pages. Updated version with more content, comments and references

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Cited by in corpus (20)

• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
• Curved momentum spaces from quantum (Anti-)de Sitter groups in (3+1) dimensions
• The $\kappa$-Newtonian and $\kappa$-Carrollian algebras and their noncommutative spacetimes
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• Noncommutative spaces of worldlines
• Physical Constraints on Quantum Deformations of Spacetime Symmetries
• Curved momentum spaces from quantum groups with cosmological constant
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