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arXiv · 2101.02953

$q$-deformations of the modular group and of the real quadratic irrational numbers

Ludivine Leclere·Sophie Morier-Genoud
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Abstract

We develop further the theory of $q$-deformations of real numbers introduced by Morier-Genoud and Ovsienko, and focus in particular on the class of real quadratic irrationals. Our key tool is a $q$-deformation of the modular group $PSL_q(2,\mathbb{Z})$. The action of the modular group by M\"obius transformations commutes with the $q$-deformations. We prove that the traces of the elements of $PSL_q(2,\mathbb{Z})$ are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the $q$-deformed quadratic irrationals.

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Ludivine Leclere, Sophie Morier-Genoud. 2021-01-08. $q$-deformations of the modular group and of the real quadratic irrational numbers. https://arxiv.org/abs/2101.02953

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