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

Abstract

We develop further the theory of q-deformations of real numbers introduced in [MGO20] and [MGO19b] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of the modular group PSLq(2, Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the elements of PSLq(2, Z) are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q-deformed quadratic irrationals. Contents 1. Introduction 2. q-deformed numbers and q-continued fractions 2.1. q-rationals 2.2. q-irrationals 2.3. Proof of Theorem 1 2.4. Infinite continued fractions 3. q-deformations of matrices and of the modular group 3.1. Elementary matrices in SL(2, Z) 3.2. q-deformation of the modular group (proof of Proposition 1.1) 3.3. Möbius transformation of the q-reals 3.4. q-deformed matrices and continued fractions 3.5. Proof of Theorem 4 3.6. q-Continuants 3.7. Traces of q-deformed matrices (proof of Theorem 3) 3.8. Examples: Cohn matrices 3.9. Proof of Lemma 3.8 3.10. Proof of Lemma 3.10 4. Quadratic irrationals 4.1. Real quadratic irrational numbers 4.2. Proof of Theorem 2 4.3. Explicit expressions 4.4. Palindromicity of P (end of proof of Theorem 2) 4.5. Examples References

On radiuses of convergence of q-metallic numbers and related q-rational numbers

“…Remark 2.6. For general q-real quadratic irrational numbers, the palindromic of discriminant has been proved by considering a q-deformation of the elements of the modular group P SL(2, Z) (see [6]).…”

“…On the study of q-deformation of the real quadratic irrational numbers, L. Leclere and S. Morier-Genoud [6] proved some properties of q-deformation of them corresponding to classical real quadratic irrational numbers. On the other hand, since the q-real numbers are defined as power series in q, under the assumption that q is a complex number, L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov [7] study its radiuses of convergence and give the following conjecture which can be viewed as a q-deformation of Hurwitz's Irrational Number Theorem.…”