The role of the Jacobi identity in solving the Maurer-Cartan structure equation (Q265543)

The realization problem for Lie algebras concerns an \(n\)-dimensional Lie algebra \(\mathfrak{g}\) and searches for a \(\mathfrak{g}\)-valued \(1\)-form \(\phi \in \Omega ^1(U, \mathfrak{g})\) defined on some open neighborhood \(U\subset \mathfrak{g}\) of the origin such that \(\phi \) is pointwise an isomorphism and satisfies the Maurer-Cartan equation: \(d\phi +\frac{1}{2}[\phi , \phi ]=0\). A solution to this problem induces a local Lie group structure on some open subset of \(U\) and then, we can think of this realization problem as the problem of locally integrating Lie algebras.NEWLINENEWLINEThe present paper gives a two-step method for solving the realization problem. The first step is performed in Theorem 1.2 which provides a weaker version of the realization problem, which admits a unique solution given any pre-Lie algebra. The second step is described in Theorem 1.4 which proves that the solution of the weak realization problem is a solution of the complete realization problem if and only if the Jacobi identity is satisfied.