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

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