Cyclic Cohomology.pdf

The main result is a natural isomorphism between the equivariant periodic cyclic homology of a bornological algebra and its crossed product with a bornological quantum group. This isomorphism is compatible with composition products and allows for the derivation of the Green-Julg theorem and its dual in periodic cyclic homology. The construction relies on the notion of a modular pair for a bornological quantum group, which is closely related to the concept introduced by Connes and Moscovici in their work on cyclic cohomology for Hopf algebras. The paper works within the framework of bornological quantum groups, incorporating actions of classical Lie groups, compact or discrete quantum groups, and provides a more conceptual and general approach to the dual Green-Julg theorem. Key points include:

Construction of a duality isomorphism in equivariant periodic cyclic homology
Introduction of modular pairs for bornological quantum groups
Derivation of the Green-Julg theorem and its dual in periodic cyclic homology
Compatibility with composition products
Application to actions of bornological quantum groups on bornological algebras
Relation to Connes and Moscovici's work on cyclic cohomology for Hopf algebras.