Group invariance of integrable Pfaffian systems

TL;DR

This paper establishes a relationship between the invariance of integrable Pfaffian systems under Lie group actions and their variational cohomology, offering a new method to compute cohomology and detect symmetries.

Contribution

It demonstrates that the vertical variational cohomology of an invariant Pfaffian system equals the Lie algebra cohomology with horizontal cohomology coefficients, providing computational and obstruction detection tools.

Findings

Vertical variational cohomology equals Lie algebra cohomology for invariant systems.

Provides an effective algorithm for computing variational cohomology.

Identifies obstructions to symmetry actions on Pfaffian systems.

Abstract

Let $\mathcal{S}$ be an integrable Pfaffian system. If it is invariant under a transversally free infinitesimal action of a finite dimensional real Lie algebra $g$ and consequently invariant under the local action of a Lie group $G$, we show that the vertical variational cohomology of $\mathcal{S}$ is equal to the Lie algebra cohomology of $g$ with values in the space of the horizontal cohomology in maximum dimension. This result, besides giving an effective algorithm for the computation of the variational cohomology of an invariant Pfaffian system, provides a method for detecting obstructions to the existence of finite or infinitesimal actions leaving a given system invariant.