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)$U\left(1\right)$ as a structure group, the other has the quantum group S_Uq(2)$S{U}_q\left(2\right)$ 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.