What does the classification of Complex Semi-simple Lie algebras buy us in terms of classifying Lie groups? Certainly it classifies complex semi-simple lie groups but can we get any better? I know we can take compact real forms of the semi-simple algebras and there are several theorems about topological similarities for Lie groups with the same Lie algebra. How far can we take this? What is the biggest class of Lie groups we can rope in by this method?
[Math] Lie algebras to classify Lie groups
lie-algebraslie-groupsrt.representation-theory
Best Answer
The point is, if $G$ is a real semisimple Lie group, then its Lie algebra $\mathfrak{g}$ is also semisimple and so is the complexification $\mathfrak{g} \otimes \mathbb{C}$. Or, given a complex $\mathfrak{g}$, any real form $\mathfrak{g}_\mathbb{R}$ (meaning, a pre-complexification) integrates to a real semisimple Lie group. The real forms have been classified and they are described by Satake diagrams, which are Dynkin diagrams with an extra decoration to describe the real form. So together with finding the real forms, the complex classification "buys" you everything about semi-simple real Lie groups. The great Élie Cartan not only reorganized the complex classification, he also did the real classification.
See also the Wikipedia page for real forms. As it points out, there are always two special real forms, the compact form and the completely split form, and often also others.