On a non-Abelian Poincaré lemma

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 $dø+{ø}^2=0$ is gauge-equivalent to a constant, $$ø=gC{g}^{-1}-dg{g}^{-1}.$$ 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.

LaTeX, 21 pages. Apart from minor editing, the new version contains more about the "abstract" setup (such as the statement about homotopy of odd elements in a dLie algebra)

References in corpus (2)

• Deformation theory and rational homotopy type
• Feynman diagrams and minimal models for operadic algebras

Cited by in corpus (8)

• Gauge PDE and AKSZ-type Sigma Models
• Graded Geometry, $Q$-Manifolds, and Microformal Geometry
• Microformal geometry and homotopy algebras
• Higher Poincare Lemma and Integrability
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• The Poincare lemma, antiexact forms, and fermionic quantum harmonic oscillator
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Inverting covariant exterior derivative