Topological Hopf algebras and their Hopf-cyclic cohomology

TL;DR

This paper extends Hopf-cyclic cohomology to topological Hopf algebras, enabling the study of infinite-dimensional Lie algebras and establishing new isomorphisms and connections with relative Lie algebra cohomology.

Contribution

It introduces a topological version of Hopf-cyclic cohomology, identifies the coefficient categories with representation categories of topological algebras, and establishes a van Est type isomorphism.

Findings

Extended Hopf-cyclic cohomology to topological Hopf algebras.

Identified coefficient categories with representation categories of anti-Drinfeld doubles.

Established a van Est type isomorphism linking cohomologies.

Abstract

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the Hopf-cyclic cohomology to the relative Lie algebra cohomology with respect to a maximal compact subalgebra.