The Chern-Weil homomorphism for deformed Hopf-Galois extensions

In this article, we study the Chern-Weil theory for Hopf-Galois extensions originally introduced by Hajac and Maszczyk in the context of coalgebra extensions. We show that the cyclic homology Chern-Weil homomorphism defines natural transformations between Hopf-Galois extensions with a strong connection (principal comodule algebras) and cyclic homology, thereby generalizing the concept of characteristic classes to the noncommutative setting. In the second part, we study the effect of $2$-cocycle deformations of Hopf-Galois extensions on the aforementioned homomorphism. We consider the $2$-cocycle coming from the structure Hopf algebra of the extension, an external symmetry, and finally the combined case.