*Dr. Maïté Dupuis (University of Waterloo)*

In the quantum gravity regime, the introduction of a nonzero cosmological constant is usually done using a quantum group. From the phase space construction of loop quantum gravity, it is not clear how such a quantum group appears. Using a Poisson-Lie group, I will generalize the standard holonomy-flux algebra to introduce the cosmological constant and describe (2d) hyperbolic discrete geometries. I will show how the closure constraint imposed on a vertex can be related to the hyperbolic cosine law for the dual triangle. At the quantum level, this formulation will give rise to U_q(su(2)) spin networks or equivalently in the dual picture to hyperbolic quantum triangles glued together. If time permits, I will also discuss the flatness constraint that appears in this deformed context.