# A Maurer-Cartan form can be used to define a local Lie group structure

I have heard that a Maurer-Cartan form can be used to define a local Lie group structure on a manifold. A such form is defined as follows:

Definition. Let $$N$$ be a $$n$$-dimensional smooth manifold and $$\mathfrak g$$ a Lie algebra. A smooth 1-form $$\omega\in\Omega^1(N,\mathfrak g)$$ is said to be a Maurer-Cartan form if $$d\omega+\frac{1}{2}[\omega,\omega]=0.$$

If one fix a point $$x\in N$$, one would expect to find a local neighbourhood aroung $$x$$ in $$N$$ that admits a Lie group structure.

Is there any classical reference where I can see the proof of this?

This is only true if $$\dim(N)=\dim(\mathfrak g)$$. The result is based on the characterization of solutions of the Maurer-Cartan equations as left logarithmic derivatives. For a Maurer-Cartan Form $$\omega\in\Omega^1(N,\mathfrak g)$$ and any point $$x\in N$$, there exists a unique smooth map from a neighborhood $$U$$ of $$x$$ in $$M$$ to a nighborhood of $$e$$ in any Lie group $$G$$ with Lie algebra $$\mathfrak g$$ such that $$\omega=f^*\omega_{MC}$$, where $$\omega_{MC}$$ is the left Maurer Cartan form on $$G$$. If $$N$$ and $$G$$ have the same dimension, it follows readily that $$f$$ has to be a local diffeomorphism, which leads to the local Lie group structure that you are looking for.