Let $G$ be a real Lie group. Then the complexification $G_\C$ of $G$ is the unique complex Lie group equipped with a map $\vp:G\to G_\C$ such that any map $G\to H$ where $H$ is a complex Lie group, extends to a holomorphic map $G_\C\to H$ . If $\fr g$ and $\fr g_\C$ are the respective Lie algebras, $\fr g_\C\cong\fr g\otimes_\R\C$ .

For simply connected groups, the construction is obvious: we simply take the simply connected complex group with Lie algebra $\fr g_\C$ , and $\vp$ to be the map induced by the inclusion $\fr g\to\fr g_\C$ .

If $\gamma\in G$ is central, then its image is in central in $G_\C$ since $g\mapsto \gamma g\gamma^{-1}$ is a map extending $\vp$ , and thus must be the identity by uniqueness half of the universal property. Thus, if $\Gamma\subset G$ is a discrete central subgroup, then we get a map $G/\Gamma\to G_\C/\vp(\Gamma)$ , which gives a complexification for $G/\Gamma$ . Since every Lie group is of this form, this shows existence.

Some easy examples: the complexification both of $\SL n\R$ and $\SU n$ is $\SL n\C$ . The complexification of $\R$ is $\C$ and of $S^1$ is $\C^*$ .

The map $\vp\colon G\to G_\C$ is not always injective. For example, if $G$ is the universal cover of $\SL n\R$ (which has fundamental group $\Z$ ), then $G_\C\cong\SL n\C$ , and $\vp$ factors through the covering $G\to \SL n\R$ .