Alexander Thomas, “Infinitesimal Modular Group: <nobr>$q$</nobr>-Deformed <nobr>$\mathfrak{sl}_2$</nobr> and Witt Algebra”, SIGMA, 2024, Volume 20,053, 16 pp.

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

Alexander Thomas

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

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.

Keywords: quantum algebra, Lie algebra deformations, $q$-Virasoro, Burau representation.

MSC: 35A01, 65L10, 65L12, 65L20, 65L70

Received: December 1, 2023; in final form June 3, 2024; Published online June 20, 2024

Language: English

DOI: 10.3842/SIGMA.2024.053