Observables in loop quantum gravity with a cosmological constant
An open issue in loop quantum gravity (LQG) is the introduction of a non-vanishing cosmological constant $\Lambda$. In 3d, Chern-Simons theory provides some guiding lines: $\Lambda$ appears in the quantum deformation of the gauge group. The Turaev-Viro model, which is an example of spin foam model is also defined in terms of a quantum group. By extension, it is believed that in 4d, a quantum group structure could encode the presence of $\Lambda\neq0$. In this article, we introduce by hand the quantum group $\mathcal{U}_{q}(\mathfrak{su}(2))$ into the LQG framework, that is we deal with $\mathcal{U}_{q}(\mathfrak{su}(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for $\mathcal{U}_{q}(\mathfrak{su}(2))$. 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 $\mathcal{U}_{q}(\mathfrak{u}(n))$. We are then able to build geometrical observables, such as the length, area or angle operators ... We show that these operators characterize a quantum discrete hyperbolic geometry in the 3d LQG case. Our results confirm that the use of quantum group in LQG can be a tool to introduce a non-zero cosmological constant into the theory.