Quantum groups, non-commutative Lorentzian spacetimes and curved momentum spaces

The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $\Lambda$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $\kappa$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $\kappa$-deformation, while curved momentum spaces are defined as orbits generated by the $\kappa$-dual of the Hopf algebra of quantum symmetries. Throughout this construction we use kinematical coordinates, in terms of which the physical interpretation becomes more transparent, and the cosmological constant $\Lambda$ is included as an explicit parameter whose $\Lambda \rightarrow 0$ limit provides the Minkowskian case. The generalization of these results to the physically relevant (3+1)-dimensional deformation is also commented.