Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

TL;DR

This paper extends quantum deformations related to (2+1) gravity to (3+1) dimensions using Drinfel'd double structures, introducing a new quantum spacetime that differs from κ-Minkowski and incorporates the cosmological constant explicitly.

Contribution

It generalizes Drinfel'd double-based quantum deformations to (3+1) gravity, providing a new twisted r-matrix and analyzing its noncommutative spacetime properties.

Findings

Introduces a new twisted r-matrix for (3+1) Lorentzian algebras.

Shows the resulting quantum spacetime differs from κ-Minkowski.

Demonstrates the cosmological constant as an explicit parameter.

Abstract

We present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincar\'e Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding principle towards the description of (3+1) gravity. To this aim, a canonical classical $r$-matrix arising from a Drinfel'd double structure for the three (3+1) Lorentzian algebras is obtained. This $r$-matrix turns out to be a twisted version of the one corresponding to the (3+1) $\kappa$-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the $\kappa$-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincar\'e limit to be straightforwardly obtained.