Symmetries of the q-deformed real projective line

We generalize in two steps the quantized action of the modular group on $q$-deformed real numbers introduced by Morier-Genoud and Ovsienko. First, we let the projective general linear group $PGL_2(\mathbb{Z})$ act on $q$-real numbers via a $q$-deformed action. The quantized matrices we get have combinatorial interpretations. Then we consider an extension of the group $PGL_2(\mathbb{Z})$ by the $2$-elements cyclic group, and define a quantized action of this extension on $q$-real numbers. We deduce from these actions some underlying relations between $q$-real numbers, and between left and right versions of $q$-deformed rational numbers. In particular we investigate the case of some algebraic numbers of degree $4$ and $6$. We also prove that the way of quantizing real numbers defined by Morier-Genoud and Ovsienko is an injective process.