Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration and its associated twistor geometry. Our quantum sphere arises as the unit sphere inside a q-deformed quaternion space . The resulting four-sphere is a quantum analogue of the quaternionic projective space . The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space and use it to study a system of anti-self-duality equations on , for which we find an 'instanton' solution coming from the natural projection defining the tautological bundle over .