Notation : Let $X$ be a manifold, $\tilde X$ its universal cover, $G$ a Lie group and $\mathfrak g$ its Lie algebra. Finally, let $\alpha \in \Omega^1(X, \mathfrak g)$ a flat connexion, i.e $ d \alpha + [\alpha, \alpha]/2 = 0$.
How to compute the associated holonomy map $h : \tilde X \to G ?$
I would be very happy with a simple example (if possible with a non-trivial Lie algebra), where I can see all the computations, as I am not so familiar with differential geometry.