Abstract
Theκ-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 theκ-deformation are explicitly given, and the three quantum kinematical groups are obtained as quantizations of such Poisson–Lie algebras. As a consequence, the noncommutative (2 + 1)D spacetimes that generalize theκ-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 noncommutative worldlines. Furthermore, quantum (anti-)de Sitter algebras are presented both in the known basis related to 2 + 1 quantum gravity and in a new one which generalizes the bicrossproduct one. In this framework, the quantum deformation parameter is related to the Planck length, and the existence of a kind of “duality” between the cosmological constant and the Planck scale is also envisaged.
Highlights
The connection between quantum groups and Planck scale physics was early suggested in [1]
Quantum deformations of Lie algebras and Lie groups [2,3,4,5,6,7,8] have been broadly applied in the construction of deformed symmetries of spacetimes [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23], especially for the Poincareand Galilei cases, for which the deformation parameter is known to play the role of a fundamental scale
Among all these quantum kinematical algebras the well known κ-Poincarealgebra [9, 13, 14, 16, 18] has been frequently considered. These deformed Poincaresymmetries were later applied in the context of the so-called doubly special relativity (DSR) theories [24,25,26,27,28,29,30,31,32] which introduced two fundamental scales: the usual observer-independent velocity scale c as well as an observer-independent length scale lp, which was related to the deformation parameter in the algebra
Summary
The connection between quantum groups and Planck scale physics was early suggested in [1]. As an intermediate stage in the search of the quantum (A)dS groups, we compute in Section 4 the invariant (A)dS vector fields and the Poisson–Lie structures coming from the classical r-matrix generating the κ-deformation Since the first-order structure of the complete quantum deformation of soω(2, 2) is described by the corresponding Drinfel’d-double, some preliminary information concerning the physical properties of the associated noncommutative spaces can be extracted from it Notice that, in this firstorder approach, all the expressions will be linear both on the generators and on the dual quantum group coordinates
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