The Hopf-algebra symmetry, or more generally coalgebra symmetry, is also a key concept in studying Galois type extensions of noncommutative algebras. The Galois condition is an algebraic incarnation of the principal bundle condition. The structure of such Galois extensions can be studied through modules associated via quantum-group representations. The C*-algebraic tool of the Noncommutative Index Theorem has been used to compute the Chern numbers (Chern-Connes pairing) for algebraic quantum vector bundles (projective modules) over the standard Podle\'s quantum sphere. This example lead to some general conclusions on how to compute projector matrices of modules associated with Hopf-Galois extensions. Thus the Hopf symmetry of an algebra extension can aid the computation of the Chern-Connes pairing from the K-theory side. On the other hand, it is also the Hopf symmetry principle from which A.Connes and H.Moscovici constructed cyclic cocycles for the pairing. These two approaches should be more interrelated than they are today.

Syllabus: Galois type extensions as a natural generalization of the classical Galois theory and affine group-scheme torsors of algebraic geometry, faithfully flat principal homogeneous spaces and the Galois type correspondence between coideals right ideals and coinvariant subalgebras (non-standard quotients, e.g., Podles quantum spheres), entwining structures, C*-algebraic approach to principal actions of locally compact quantum groups and gauge actions for graph C*-algebras, connections on faithfully flat Hopf-Galois extensions, strong connections and Chern-Connes pairing in the Hopf-Galois theory, application of the Noncommutative Index Theorem to line bundles over standard Podles quantum sphere, further examples of quantum fibrations (e.g., construction of the quantum real projective space).