On a Non-Abelian Poincaré lemma

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 dω + ω 2 = 0 is gaugeequivalent to a constant, ω = 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 nonlinear 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. © 2011 American Mathematical Society.