On a non-Abelian Poincar\'e lemma

TL;DR

This paper extends the non-Abelian Poincaré lemma to arbitrary odd forms in Lie superalgebras, showing they are gauge-equivalent to constants using a chain homotopy formula with multiplicative integrals, with applications to Lie algebroids.

Contribution

It generalizes the non-Abelian Poincaré lemma to inhomogeneous odd forms and introduces a chain homotopy formula using multiplicative integrals for Lie superalgebras.

Findings

Any odd form satisfying the Maurer--Cartan equation is gauge-equivalent to a constant.

The chain homotopy formula with multiplicative integrals facilitates this generalization.

Applications to Lie algebroids and differential Lie superalgebras are provided.

Abstract

We show that a well-known result on solutions of the Maurer--Cartan equation extends to arbitrary (inhomogeneous) odd forms: any such form with values in a Lie superalgebra satisfying $d\o+\o^2=0$ is gauge-equivalent to a constant, $$\o=gCg^{-1}-dg\,g^{-1}\,.$$ This follows from a non-Abelian version of a chain homotopy formula making use of multiplicative integrals. An application to Lie algebroids and their non-linear analogs is given. Constructions presented here generalize to an abstract setting of differential Lie superalgebras where we arrive at the statement that odd elements (not necessarily satisfying the Maurer--Cartan equation) are homotopic\,---\,in a certain particular sense\,---\,if and only if they are gauge-equivalent.