How does studying the complexified Lie algebra $\mathfrak{g}_\mathbb{C}$ help us understand the original Lie algebra $\mathfrak{g}$?

Question

I have recently finished studying a course on Lie algebras which included Cartan's classification. The main process we took when studying a particular semi-simple Lie algebra was to first complexify it, form a Cartan-Weyl basis, find the roots and then build up a representation of highest weight. I have a few questions regarding this

1. How does studying the complexified Lie algebra $\mathfrak{g}_\mathbb{C}$ help us understand the original Lie algebra $\mathfrak{g}$? The method of highest weight only works for the complexified Lie algebra, so what could roots and weights tell me about the original Lie algebra?

2. Do $ \mathfrak{g} $ and $\mathfrak{g}_\mathbb{C} $ exponentiate to the same Lie group?

Answer 1

This is too broad to really answer it, but I can try to give some indications: The Lie algebras $\mathfrak g$ and $\mathfrak g_{\mathbb C}$ do not exponentiate to the same group. Keep in mind standard examples like $\mathfrak{sl}(n,\mathbb R)$ whose complexification ist $\mathfrak{sl}(n,\mathbb C)$ and the natural groups are $SL(n,\mathbb R)$ and $SL(n,\mathbb C)$ here. So in this case you get a subgroup, but in general the situation on the group level is more complicated than the one on the Lie algebra level.

A simple indication why $\mathfrak g_{\mathbb C}$ tells you a lot about $\mathfrak g$ is that the two Lie algebras have the same complex representations. Also, knowing the list of complex semisimple Lie algebra, the fact that $\mathfrak g_{\mathbb C}$ is in that list contains substantial information, for example that the dimension of $\mathfrak g$ cannot be arbitrary. The actual way to understand real Lie algebras is to study the possible real forms of a complex simple Lie algebra, and there is a substantial amount of theory available, which leads to a full classification of real simple Lie algebras.But this usally does not fit into a first course.