Maurer-Cartan equation in the DGLA of graded derivations

TL;DR

This paper constructs and classifies solutions to the Maurer-Cartan equation in the DGLA of graded derivations on a manifold, linking solutions to geometric structures like integrable distributions.

Contribution

It introduces a method to generate canonical solutions of the Maurer-Cartan equation using deformations of the exterior differential and classifies these solutions based on their type.

Findings

Canonical solutions are constructed via deformations depending on a 1-form ta.

Classification of solutions according to their finite type using the Fr46licher-Nijenhuis bracket.

Characterization of integrable distributions in terms of the finite type of associated solutions.

Abstract

Let M be a smooth manifold and $\Phi$ a differential 1-form on M with values in the tangent bundle TM. We construct canonical solutions $e_\Phi$ of Maurer-Cartan equation in the DGLA of graded derivations D*(M) of differential forms on M by means of deformations of d depending on $\Phi$. This yields to a classification of the canonical solutions of the Maurer-Cartan equation according to their type: $e_\Phi$ is of finite type r if there exists $r\in N$ such that $\Phi^r[\Phi,\Phi]_{FN} = 0$ and r is minimal with this property, where $[.,.]_{FN}$ is the Fr\"olicher-Nijenhuis bracket. A distribution $\xi\subset TM$ of codimension k > 1 is integrable if and only if the canonical solution $e_\Phi$ associated to the endomorphism $\Phi$ of TM which is trivial on $\xi$ and equal to the identity on a complement of $\xi$ in TM is of finite type $\leq 1$, respectively of finite type 0 if k = 1.