The Role of the Jacobi Identity in Solving the Maurer-Cartan Structure Equation

TL;DR

This paper presents a method that highlights the Jacobi identity's role as an obstruction in solving the Maurer-Cartan equation, extending to Poisson structures and Lie algebroids with explicit solutions.

Contribution

It introduces a unified approach to solving the Maurer-Cartan equation, emphasizing the Jacobi identity's obstruction role and generalizing solutions to Lie algebroids.

Findings

Identifies the Jacobi identity as an obstruction in the Maurer-Cartan equation.

Provides an explicit formula for solutions in the Lie algebroid case.

Extends the method to Poisson structures and Lie algebroids.

Abstract

We describe a method for solving the Maurer-Cartan structure equation associated with a Lie algebra that isolates the role of the Jacobi identity as an obstruction to integration. We show that the method naturally adapts to two other interesting situations: local symplectic realizations of Poisson structures, in which case our method sheds light on the role of the Poisson condition as an obstruction to realization; and the Maurer-Cartan structure equation associated with a Lie algebroid, in which case we obtain an explicit formula for a solution to the equation which generalizes the well known formula in the case of Lie algebras.