Real involutive systems on compact Lie groups

TL;DR

This paper investigates the global solvability and cohomology of differential complexes associated with involutive structures on compact Lie groups, revealing conditions under which solvability in one degree implies solvability in all degrees.

Contribution

It establishes the equivalence of solvability across degrees under certain conditions and explores the impact of subgroup structures and real tube structures on solvability.

Findings

Solvability in the first degree implies solvability in all degrees.

Converse holds under a commutativity hypothesis, always true for tori.

Solvability is guaranteed when the structure derives from a subgroup's Lie algebra.

Abstract

On a compact connected Lie group $G$, we study the global solvability and the cohomology spaces of the differential complex associated with an essentially real involutive structure that is invariant under left translations. We prove that solvability in the first degree of the complex implies solvability in all other degrees, and furnish a converse for this fact under a certain commutativity hypothesis (that always holds when $G$ is a torus). Additionally, it is proved that the solvability holds when the structure comes from the Lie algebra of a closed subgroup of $G$. We also investigate real tube structures when $G$ is the base manifold.