ON A NON-ABELIAN POINCARE 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 d omega + omega(2) = 0 is gauge-equivalent to a constant, omega = 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.