TITLE: Relationship between real Lie groups/algebras, complex Lie groups/algebras and complexification
QUESTION [5 upvotes]: Having read posts such as this and the links within, as well as the Wikipedia articles on complex Lie groups, complex Lie algebras and complexification, I am still rather confused about what they are and the relationship between them.
The context of this question is in trying to understand the representation theory of the Lorentz group. My current understanding is as follows:
Every real Lie group has an associated real Lie algebra, and every real Lie algebra $\mathfrak{g}$ has a unique associated simply connected real Lie group $G$. The basis of $\mathfrak{g}$ can be chosen to be some set of $d$ "generators" $\{t_a\}$, where $d$ is the dimensionality of the algebra, and the elements of $G$ are obtained from the exponentiation of real linear combinations of $\{t_a\}$.
In this language, is the complexification of the algebra $\mathfrak{g}$ a complex Lie algebra $\mathfrak{g}_\mathbb{C}$, and is the corresponding Lie group, where we instead exponentiate complex linear combinations of the generators, the complexification $G_\mathbb{C}$ of the Lie group $G$?

REPLY [5 votes]: Expanding out my comment into a full answer. There are two ideas of complexification of Lie groups that are used. Because of the ambiguity and technical awkwardness we tend to sweep it under the rug and talk about complexified Lie algebras and their Lie groups rather than complexified Lie groups.
The first idea is perhaps the obvious one. A complexification of a real Lie group $G$ is a complex Lie group $G_\mathbb{C}$ which contains $G$ as a real Lie subgroup and whose Lie algebra $\mathfrak{g}_\mathbb{C}$ is a complexification of the Lie algebra of $G$. As an example of this $SL_n(\mathbb{C})$ is the complexification $SL_n(\mathbb{R})$.
See here for more detail:  https://encyclopediaofmath.org/wiki/Complexification_of_a_Lie_group
This works quite well in many cases and nicely interfaces with the Lie group/algebra correspondence. However it has a major flaw. Not every real Lie group has a complexification even among the semisimple groups. As I understand having a complexification in this sense is equivalent to being linear (so if you're into linear algebraic groups you can stop here). In particular the universal cover of $SL_n(\mathbb{R})$ has no complexification in this sense
To fill in the gaps we define the universal complexification.  This is the definition that Wikipedia uses https://en.wikipedia.org/wiki/Complexification_(Lie_group). This one is defined by a universal property. The complexification of $G$ is a complex Lie group $G_\mathbb{C}$ together with a continuous homomorphism $\varphi:G \to G_\mathbb{C}$ such that for any homomorphism to a complex Lie group $f:G\to H$ there is a unique complex analytic homomorphism $F: G_\mathbb{C}\to H$ such that $ f= F \circ \varphi$.
Constructively, we can build this as follows. Take the simply connected group $\tilde{G}$ covering $G$ and the natural homomorphism $\Phi:\tilde{G}\to \tilde{G}_\mathbb{C}$ from this into the simply connected complex group (which we find by complexifying the Lie algebra). The natural projection $\pi:\tilde{G}\to G$ has kernel equal to the fundamental group of $G$. Now we define the universal complexification as  $\tilde{G}_\mathbb{C}/N$ where $N$ is the smallest normal subgroup which contains $\Phi(\ker \pi)$.
As you can see the second is much more complicated and can even produce a group whose Lie algebra isn't even the complexification of the original group's but a quotient of it. One neat thing is the universal complexification of a simply connected group is exactly the corresponding simply connected complex group.