  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.
