Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra

TL;DR

This paper introduces new $q$-deformations of key algebraic structures like the Heisenberg, $rak{sl}_2$, and Witt algebras, connecting them to the modular group and $q$-deformed rational numbers through differential operator realizations.

Contribution

It presents novel $q$-deformations of important Lie and related algebras using differential operators linked to the modular group and $q$-rational numbers.

Findings

New $q$-deformations of the Heisenberg, $rak{sl}_2$, and Witt algebras.

Construction of these deformations via differential operators.

Connection to $q$-deformed M"obius transformations on the hyperbolic plane.

Abstract

We describe new $q$-deformations of the 3-dimensional Heisenberg algebra, the simple Lie algebra $\mathfrak{sl}_2$ 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 M\"obius transformations acting on the hyperbolic plane.