We describe new q-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra sl₂ and the Witt algebra. They are constructed through a realization as differential operators. These operators are related to the modular group and q-deformed rational numbers defined by Morier-Genoud and Ovsienko and lead to q-deformed Möbius transformations acting on the hyperbolic plane.
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Alexander Thomas (2024) studied this question.
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