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 $G\mathrm{/}N\to G\mathrm{/}M$, for $G$ a group, and $N\subseteq M\subseteq G$ 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 Brzezi'{n}ski and Szyma'{n}ski) are constructed.