The cohomology of left-invariant elliptic involutive structures on compact Lie groups

TL;DR

This paper demonstrates that under specific conditions, the cohomology of left-invariant elliptic involutive structures on compact Lie groups can be computed algebraically using Lie algebras, simplifying the analytical complexity.

Contribution

It extends classical results by linking cohomology of elliptic structures on Lie groups to Lie algebra cohomology via spectral sequences, under new algebraic and topological conditions.

Findings

Cohomology can be computed algebraically for certain structures.

Leray spectral sequence connects classical and modern cohomology results.

Reduction of analytical problems to algebraic computations.

Abstract

Inspired by the work of Chevalley and Eilenberg on the de Rham cohomology on compact Lie groups, we prove that, under certain algebraic and topological conditions, the cohomology associated to left-invariant elliptic, and even hypocomplex, involutive structures on compact Lie groups can be computed by using only Lie algebras, thus reducing the analytical problem to a purely algebraic one. The main tool is the Leray spectral sequence that connects the result obtained Chevalley and Eilenberg to a result by Bott on the Dolbeault cohomology of a homogeneous manifold.