The context: I have a complex simple Lie algebra $\mathfrak{g}$. The book I'm reading states
"Let $G$ be the connected simply-connected Lie group with Lie algebra $\mathfrak{g}$."
What is meant by these words? I have read about Lie's third theorem, but that holds for real Lie algebra. How does this work for complex Lie algebras? This is what I think might work:
- I have read somewhere that every (semi)simple Lie algebra is the complexification of a real one, let's denote it by $\mathfrak{g}_{\mathbb{R}}$
- by Lie's third theorem we have a connected simply-connected Lie group $G_{\mathbb{R}}$ whose Lie algebra is $\mathfrak{g}_{\mathbb{R}}$
- now what? I have read that the notion of complexification of a Lie group is not as "trivial" as the algebra case, so I wouldn't know how to continue.
Is this (at least the first two points) correct?
Related question: I had a course in which we defined $G = \operatorname{Int}(\mathfrak{g})$ as the group generated by $e^{\operatorname{ad}x}$ for all $\operatorname{ad}$-nilpotent $x\in \mathfrak{g}$. I suppose this is related to my question (the professor vaguely mentioned something about it being "the smallest Lie group which has $\mathfrak{g}$ as its Lie algebra" and also an algebraic group), but I can't connect the dots.
Thanks in advance for your answers, I'm sure they will be helpful.