# Top-degree solvability for hypocomplex structures and the cohomology of left-invariant involutive structures on compact Lie groups

## TL;DR

This paper establishes conditions for top-degree solvability in hypocomplex structures and shows that cohomology of certain involutive structures on compact Lie groups can be computed algebraically using left-invariant forms.

## Contribution

It introduces a sufficient condition for top-degree solvability and demonstrates algebraic computation of cohomology for hypocomplex and elliptic involutive structures on compact Lie groups.

## Key findings

- Top-degree cohomology can be computed using only left-invariant forms.
- A sufficient condition for top-degree solvability is established.
- Cohomology of elliptic involutive structures can be algebraically determined.

## Abstract

We use the theory of dual of Fr\'echet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact Lie group can be computed only by using left-invariant forms, thus reducing the computation to a purely algebraic one. In the case of left-invariant elliptic involutive structures on compact Lie groups, under certain reasonable conditions, we prove that the cohomology associated to the involutive structure can be computed only by using left-invariant forms.