Theodore Voronov

On a Non-Abelian Poincaré Lemma

It is well known that every Lie algebra valued 1-form satisfying the Maurer-Cartan equation has a logarithmic primitive (locally). I will show how this fact extends to a statement ("Non-Abelian Poincaré Lemma") concerning arbitrary odd forms, possibly inhomogeneous, taking values in a Lie superalgebra. Such a form can be interpreted as Quillen's superconnection. Time permitting, I will explain applications to Lie algebroids and their non-linear analogs.