Quantum principal bundles over quantum real projective spaces

Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U(1)U(1) as a structure group, the other has the quantum group SUq(2)SUq(2) as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf–Galois extension.