Hopf-cyclic versus Kadison-relative cyclic homology of instantons

For noncommutative principal bundles corresponding to Hopf-Galois extensions Jara and Stefan established an isomorphism between the relative cyclic homology and a cyclic dual Hopf-cyclic homology with appropriate stable anti-Yetter-Drinfeld coefficients. However, known noncommutative deformations of principal bundles, such as the quantum circle bundles over Podles spheres and the quantum instanton bundles, go beyond the Hopf-Galois context.

The case of quantum circle bundles over Podles spheres were discussed by the authors in a previous work, and an analogical isomorphism for homogeneous quotient coalgebra-Galois extensions were constructed. In the present work, the isomorphism for arbitrary module coalgebra-Galois comodule algebra extensions, covering the case of quantum instanton bundles, is established.

In the classical (commutative) case the corresponding anti-Yetter-Drinfeld coefficients specialize to the algebra of functions on the Brylinski space, playing a role in orbifold and stringy cohomology (Chen-Ruan, Brylinski-Nistor) and local Langlands duality (Aubert-Baum-Plyman-Solleveld).