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

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arxiv 2101.02953 v1 pith:FSNOHQME submitted 2021-01-08 math.NT math.COmath.QA

Ludivine Leclere , Sophie Morier-Genoud This is my paper · ORCID
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keywords deformationsgroupmodularquadraticrealirrationalsmathbbnumbers
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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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  1. Nuancing the unicity of $q$-rationals

    math.QA 2026-07 conditional novelty 6.0 of 10

    Exactly two modular-group-equivariant deformations of rationals reproduce the standard q-integers, and the newly identified one has positive coefficients and directly yields Jones polynomials of rational knots.

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