Determine when filtered and unfiltered Chevalley–Eilenberg cohomology agree

Determine, for a filtered Lie algebra g and a filtered g-module V, when the natural comparison map from filtered Chevalley–Eilenberg cohomology H^n_{fCE}(g,V) to ordinary Chevalley–Eilenberg cohomology H^n_{CE}(g,V) is an isomorphism, and characterize how far this map can fail to be an isomorphism.

Background

The paper introduces filtered Chevalley–Eilenberg cohomology by restricting cocycles and coboundaries to filtration-preserving maps. The forgetful functor from filtered to unfiltered Lie∞ algebras induces a natural comparison map from filtered cohomology to ordinary Chevalley–Eilenberg cohomology.

The authors note that the comparison is an isomorphism for the constant filtration, but do not establish general criteria for isomorphism or describe the possible discrepancy for arbitrary filtrations. This leaves unresolved the relationship between the filtered theory used in the integration construction and the previously defined unfiltered theory.