Theodore Voronov

Non-Abelian Poincaré lemma and Lie algebroids

I shall discuss a generalization of the well-known statement that a Lie algebra valued 1-form satisfying the Maurer-Cartan equation is a "pure gauge". If one considers arbitrary odd (pseudo)differential forms with values in a Lie superalgebra, there is a non-Abelian version of the homotopy identity. In particular, an odd form satisfying the Maurer-Cartan equation on a contractible domain is gauge-equivalent to a constant. This will be applied to Lie algebroids and their non-linear analogs.