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Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra
Symmetry, integrability and geometry: methods and applications, Tome 20 (2024) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Alexander Thomas},
     title = {Infinitesimal {Modular} {Group:} $q${-Deformed} $\mathfrak{sl}_2$ and {Witt} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2024},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a52/}
}
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Alexander Thomas. Infinitesimal Modular Group: $q$-Deformed $\mathfrak{sl}_2$ and Witt Algebra. Symmetry, integrability and geometry: methods and applications, Tome 20 (2024). http://geodesic.mathdoc.fr/item/SIGMA_2024_20_a52/

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