Q-Deformations in the modular group and of the real quadratic irrational numbers

Abstract

We develop further the theory of q-deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of matrices of the modular group PSL(2,Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the q-deformed matrices are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q-deformed quadratic irrationals.