Hopf cyclic cohomology and Hodge theory for proper actions

TL;DR

This paper constructs a Hopf algebroid for proper Lie group actions, linking its cyclic cohomology to invariant de Rham cohomology, and develops a generalized Hodge theory showing finite-dimensionality of cyclic cohomology classes.

Contribution

It introduces a new Hopf algebroid framework for proper actions and establishes a generalized Hodge theory connecting cyclic cohomology with invariant differential forms.

Findings

Cyclic cohomology equals invariant de Rham cohomology.

Every cyclic cohomology class has a generalized harmonic form.

Cyclic cohomology space is finite dimensional.

Abstract

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of invariant differential forms. We prove that every cyclic cohomology class of the Hopf algebroid is represented by a generalized harmonic form. This implies that the space of cyclic cohomology of the Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we discuss properties of the Euler characteristic for a proper cocompact action.