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

Abstract

We present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincaré 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) κ-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 κ-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) Poincaré limit to be straightforwardly obtained. Keywords: (3+1) gravity, Anti-de Sitter, Cosmological constant, Quantum groups, Classical r-matrices, Drinfel'd doubles, Noncommutative spacetime