The de Rham cohomology of a Lie group modulo a dense subgroup

TL;DR

This paper establishes that the de Rham cohomology of a Lie group modulo a dense subgroup is isomorphic to the Lie algebra cohomology of a related quotient algebra.

Contribution

It proves a new isomorphism between the diffeological de Rham cohomology of G/H and the Lie algebra cohomology of g/h, where H is dense in G.

Findings

De Rham cohomology of G/H equals Lie algebra cohomology of g/h.

The result applies to dense subgroups of Lie groups.

Provides a cohomological characterization of quotient spaces by dense subgroups.

Abstract

Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak g$. We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of $\mathfrak g/\mathfrak h$, where $\mathfrak h$ is the ideal $\{Z\in\mathfrak g:\exp(tZ)\in H \text{ for all } t\in\mathbf R\}$.