Projective module description of the q-monopole

The Dirac q-monopole connection is used to compute projector matrices of qu antum Hopf line bundles for arbitrary winding number. The Chern-Connes pair ing of cyclic cohomology and K-theory is computed for the winding number -1 . The non triviality of this pairing is used to conclude that the quantum p rincipal Hopf fibration is non-cleft. Among general results, we provide a l eft-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also p rovide for arbitrary strong connections on algebraic quantum principal bund les (Hopf-Galois extensions) their associated covariant derivatives on proj ective modules.