Hopf images of coactions and effective symmetry of quantum principal bundles

Abstract: We introduce Hopf images of coactions of Hopf algebras and develop their role in the geometry of quantum principal bundles. Assuming cosemisimplicity of the structure Hopf algebra, we show that every quantum principal bundle equipped with a right-covariant first-order differential calculus admits a canonical and functorial reduction to one with inner-faithful quantum symmetry. This yields a classification of quantum principal bundles up to effective quantum symmetry and a rigidity result identifying the minimal effective symmetry acting on the reduced total space. Examples from quantum groups are discussed.