q-deformation of $z\to {az+b\over cz+d}$

TL;DR

This paper constructs a quantum deformation of the Möbius transformation acting on the Podle's sphere, extending classical conformal transformations to a q-deformed quantum setting.

Contribution

It introduces a quantum double action of q on the Podle's sphere, generalizing the classical Lorentz group action on the Riemann sphere.

Findings

Defined the quantum double action on the Podle's sphere

Generalized the classical conformal transformation to the q-deformed setting

Provided a framework for quantum projective transformations

Abstract

We construct the action of the quantum double of $\uq$ on the standard Podle\'s sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.