SAYD modules over Lie-Hopf algebras

TL;DR

This paper establishes a van Est isomorphism linking Lie algebra cohomology and Hopf cyclic cohomology for Lie-Hopf algebra structures, introduces a correspondence between SAYD modules, and constructs explicit examples demonstrating the theory.

Contribution

It generalizes the van Est isomorphism to the first spectral sequence level and classifies SAYD modules over certain Hopf algebras, including the Schwarzian Hopf algebra.

Findings

Proves a one-to-one correspondence between SAYD modules over Lie algebra and Hopf algebra

Shows Connes-Moscovici Hopf algebras lack finite-dimensional SAYD modules except the known one

Constructs a nontrivial four-dimensional SAYD module over the Schwarzian Hopf algebra

Abstract

In this paper a general van Est type isomorphism is established. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and SAYD modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is found at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes- Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.