Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 053, 16 pages      arXiv:2308.06158      https://doi.org/10.3842/SIGMA.2024.053

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

Alexander Thomas
Universität Heidelberg, Berliner Str. 41-49, 69120 Heidelberg, Germany

Received December 01, 2023, in final form June 03, 2024; Published online June 20, 2024

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 Möbius transformations acting on the hyperbolic plane.

Key words: quantum algebra; Lie algebra deformations; $q$-Virasoro; Burau representation.

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