Principal pairs of quantum homogeneous spaces

We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf–Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation of a homogeneous fibration G/N → G/M , for G a group, and N ⊆ M ⊆ G subgroups. Variations on Takeuchi’s equivalence and Schneider’s descent theorem are presented in this context. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples. Moreover, a large collection of noncommutative fibrations (in the spirit of Brzeziński and Szymański) are constructed.