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 $\mathrm{\Lambda }\ne 0$. In this article, we introduce by hand the quantum group ${\mathcal{U}}_q(\mathfrak{s}u(2))$ into the loop quantum gravity (LQG) framework; that is, we deal with ${\mathcal{U}}_q(\mathfrak{s}u(2))$-spin networks. We explore some of the consequences, focusing in particular on the structure of the observables. Our fundamental tools are tensor operators for ${\mathcal{U}}_q(\mathfrak{s}u(2))$. We review their properties and give an explicit realization of the spinorial and vectorial ones. We construct the generalization of the $\mathrm{U}(N)$ 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, 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.

• Received 30 May 2014

DOI:https://doi.org/10.1103/PhysRevD.90.104037