On 3+1 Anti-de Sitter and de Sitter Lie Bialgebras with Dimensionful Deformation Parameters

Abstract

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.

Curved momentum spaces from quantum (anti–)de Sitter groups in ( 3+1 ) dimensions

“…At this point it is natural to wonder whether there could exist another quantum AdS ω algebra in (3 þ 1) dimensions that generalizes the κ-Poincaré algebra and has a nondeformed rotation subalgebra, δðJ 1 Þ ¼ δðJ 2 Þ ¼ δðJ 3 Þ ¼ 0. This question can be addressed by recalling that in [43,55] it was proven that all AdS ω deformations in (3 þ 1) dimensions with primitive coproducts for P 0 and J 3 [i.e., with δðP 0 Þ ¼ δðJ 3 Þ ¼ 0] are generated by one of the two (disjoint) families of two-parametric classical r-matrices,…”

Noncommutative Relativistic Spacetimes and Worldlines from 2 + 1 Quantum (Anti-)de Sitter Groups

“…This term does not appear either in the (3 + 1)D -Poincaré algebra (with = 0) or in the (2 + 1)D -(A)dS algebras. A twisted version of (66) with a second deformation parameter was considered in [97] by imposing some physical requirements, and the very same classicalmatrix has been derived in [98] from a Drinfel' d-double approach; namely,…”

Hamilton geometry: Phase space geometry from modified dispersion relations

“…The model we consider is inspired by the q-de Sitter Hopf algebra [35][36][37][38], which is the only known fully consistent example of Planck-scale deformations of the de Sitter relativistic symmetries 6 . This example is particularly interesting since the nontrivial interaction between spacetime curvature and Planck-scale effects are fully apparent.…”

“…As we already mentioned, the most prominent advantage to study modified dispersion relations as Hamiltonians with Hamiltonian geometry is that the framework naturally incorporates a nontrivial curved geometry of position and momentum space consistently at the same time, a feature that is cumbersome in other approaches that study modified dispersion relations. To demonstrate the features of the general framework we will in particular derive the Hamilton geometry of the cotangent bundle induced by dispersion relations inspired form the q-de Sitter and κ-Poincaré quantum groups [22,[35][36][37][38][39][40]. The κ-Poincaré quantum group is one of the most studied models in quantum gravity phenomenology encoding departures from standard relativistic kinematics without spoiling the relativity principle, thanks to modified laws of transformations between inertial observers.…”