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

Understand

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.