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

Towards quantized complex numbers: $q$-deformed Gaussian integers and the Picard group

“…We already have the q-deformed operators T q , S q , U q . The remaining generator L and its q-deformation also appeared in the context of q-deformed rational numbers [11,13,9]. The operator of linear-fractional transformations associated with the matrix L is the "negation operator": L(x) = −x.…”

“…The operator of linear-fractional transformations associated with the matrix L is the "negation operator": L(x) = −x. It was observed in [11,13,9] that, besides the invariance under the modular group action, q-deformed rational numbers satisfy one more invariance property:…”

“…Proof. All of the relations, except for (U q L q ) 2 = Id, have already been checked in [11,9]. Let us give here the details of the computation for the latter relation.…”

“…It was further extended to arbitrary real numbers in [12]. Several properties of q-numbers were studied in [13,9,10]. The unimodality conjecture formulated in [11] was tackled in [1].…”

“…The notion of q-rationals was defined in [11] in a combinatorial way. It became clear later (see [9,13,10]) that the simplest, and perhaps most conceptual, way to define q-deformed rationals is to assume the modular invariance.…”