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

TL;DR

This paper extends the theory of q-deformations of real numbers, especially quadratic irrationals, using a q-deformed modular group, revealing palindromic polynomial traces with positive coefficients.

Contribution

It introduces a q-deformation of the modular group and demonstrates its action on quadratic irrationals, providing explicit polynomial expressions for these deformations.

Findings

Traces of PSL_q(2,Z) elements are palindromic polynomials with positive coefficients.

Explicit formulas for q-deformed quadratic irrationals are derived.

The q-deformation commutes with the modular group action.

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.