Complexification of a Lie group

Summary

The complexification of a real Lie group $G$ is a complex Lie group $G_\C$ with a homomorphism $G \to G_\C$. Every continuous homomorphism of $G$ to a complex Lie group factors through it, in a unique holomorphic way. It is the group analogue of passing from a real Lie algebra $\mathfrak{g}$ to $\mathfrak{g} \otimes_\R \C$. For example $\mathrm{SL}_n(\C)$ is the complexification of both $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$. It links real groups, compact groups and complex groups, as in Weyl's unitary trick.