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

TL;DR

This paper classifies specific quantum deformations of 3+1 (anti)de Sitter algebras with dimensionful parameters, exploring their properties and associated non-commutative spacetimes, relevant for quantum gravity models.

Contribution

It identifies only two families of two-parametric (anti)de Sitter Lie bialgebras with specific primitive operators, characterized by dimensionful parameters possibly linked to the Planck length.

Findings

Two families of (anti)de Sitter Lie bialgebras identified

Deformation parameters are dimensionful, related to Planck length

Properties like zero-curvature limits and space isotropy analyzed

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.