The Role of the Jacobi Identity in Solving the Maurer-Cartan Structure Equation
arXiv:1506.00048 · doi:10.2140/pjm.2016.282.487
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.
16 pages. Minor Corrections and additions. Final version accepted for publication in Pacific Journal of Mathematics