Cyclic cohomology and Baaj-Skandalis duality

TL;DR

This paper develops a duality isomorphism in equivariant periodic cyclic homology for bornological quantum groups, extending classical theorems and unifying actions of Lie groups and quantum groups.

Contribution

It introduces a duality isomorphism in equivariant cyclic homology for bornological quantum groups, generalizing Baaj-Skandalis duality.

Findings

Establishment of a duality isomorphism in equivariant cyclic homology.

Generalized Green-Julg and dual Green-Julg theorems.

Incorporation of actions of classical Lie groups and quantum groups.

Abstract

We construct a duality isomorphism in equivariant periodic cyclic homology analogous to Baaj-Skandalis duality in equivariant Kasparov theory. As a consequence we obtain general versions of the Green-Julg theorem and the dual Green-Julg theorem in periodic cyclic theory. Throughout we work within the framework of bornological quantum groups, thus in particular incorporating at the same time actions of arbitrary classical Lie groups as well as actions of compact or discrete quantum groups. An important ingredient in the construction of our duality isomorphism is the notion of a modular pair for a bornological quantum group, closely related to the concept introduced by Connes and Moscovici in their work on cyclic cohomology for Hopf algebras.