Observables in loop quantum gravity with a cosmological constant

In many quantum gravity approaches, the cosmological constant is introduced by deforming the gauge group into a quantum group. In three dimensions, the quantization of the Chern-Simons formulation of gravity provided the first example of such a deformation. The Turaev-Viro model, which is an example of a spin-foam model, is also defined in terms of a quantum group. By extension, it is believed that in four dimensions, a quantum group structure could encode the presence of Λ≠0. In this article, we introduce by hand the quantum group Uq(su(2)) into the loop quantum gravity (LQG) framework; that is, we deal with Uq(su(2))-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for Uq(su(2)). We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the U(N) formalism in this deformed case, which is given by the quantum group Uq(u(N)). We are then able to build geometrical observables, such as the length, area or angle operators, etc. We show that these operators characterize a quantum discrete hyperbolic geometry in the three-dimensional LQG case. Our results confirm that a quantum group structure in LQG can be a tool to introduce a nonzero cosmological constant into the theory. Our construction is both relevant for three-dimensional Euclidian quantum gravity with a negative cosmological constant and four-dimensional Lorentzian quantum gravity with a positive cosmological constant.