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We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, thesimple Lie algebra $\mathfrak{sl}_2$ and the Witt algebra. They are constructedthrough a realization as differential operators. These operators are related tothe modular group and $q$-deformed rational numbers defined by Morier-Genoudand Ovsienko and lead to $q$-deformed M\"obius transformations acting on thehyperbolic plane.

Infinitesimal modular group: $q$-deformed $\mathfrak{sl}_2$ and Witt algebra