Maurer-Cartan equation in the DGLA of graded derivations

Abstract

Abstract Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d * and i * components of D. Under the hypothesis that IdT ( M ) + Ψ is invertible we prove that Ψ = b ( Ψ ) = - 1 2 _ ( I d T M + Ψ ) - 1 ∘ [ Ψ , Ψ ] ℱ 𝒩 {\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}} , where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions e Ψ = ℒ Ψ +𝒥b ( Ψ ) of the Maurer-Cartan equation according to their type: e Ψ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒡𝒩 = 0 and r is minimal with this property, where [·, ·]𝒡𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution e Ψ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.