q-deformed sl2 and Witt algebra

I'll present new q-deformations of Lie algebras linked to the modular group and the q-rationnal numbers by Morier-Genoud and Ovsienko. In particular we describe deformations of sl2 and the Witt algebra. These deformations are realized as differential operators acting on the hyperbolic plane, giving new insights into q-rationnals.