Noncommutative homogeneous spaces: The matrix case

Given a quantum subgroup G⊂Un and a number k≤n we can form the homogeneous space X=G/(G∩Uk), and it follows from the Stone-Weierstrass theorem that C(X) is the algebra generated by the last n−k rows of coordinates on G. In the quantum group case the analogue of this basic result does not necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the “easy quantum group” case, with the construction and study of several algebras associated to the noncommutative spaces of type X=G/(G∩Uk+).