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 $\mathrm{PSL}(2,\mathbb{Z})$. 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.

中文翻译：

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q-模群中的变形和实二次无理数

我们将进一步发展[10]和[9]中引入的实数q形变的理论，并特别关注实数二次无理数的类别。我们的关键工具是模群矩阵的q变形$\mathrm{PSL}（\mathrm{2个}，\mathbb{ž}）$。莫比乌斯（Möbius）变换的模块化组的作用与q变形相对应。我们证明q变形矩阵的踪迹是具有正系数的回文多项式。这些迹线出现在q变形的二次无理数的显式表达式中。