On 3+1 anti-de Sitter and de Sitter Lie bialgebras with dimensionful deformation parameters

We analyze among all possible quantum deformations of the 3+1 (anti)de Sitter algebras, so(3,2) and so(4,1), which have two specific non-deformed or primitive commuting operators: the time translation/energy generator and a rotation. We prove that under these conditions there are only two families of two-parametric (anti)de Sitter Lie bialgebras. All the deformation parameters appearing in the bialgebras are dimensionful ones and they may be related to the Planck length. Some properties conveyed by the corresponding quantum deformations (zero-curvature and non-relativistic limits, space isotropy,...) are studied and their dual (first-order) non-commutative spacetimes are also presented.Comment: 7 pages. Communication presented in the XIII Int.Colloq. Integrable Systems and Quantum Groups, June 17-19, 2004, Prague, Czech Republic. Some misprints and dimensions of parameters have been fitte

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

The $\kappa$-deformation of the (2+1)D anti-de Sitter, Poincar\'e 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.Comment: 25 pages. Updated version with more content, comments and reference