complexification

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

For simply connected groups, the construction is obvious: we simply take thesimply connected complex group with Lie algebra ${𝔤}_{ℂ}$, and $\phi$ to be themap induced by the inclusion $𝔤\to {𝔤}_{ℂ}$.

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 $\phi$, and thus must be theidentity by uniqueness half of the universal property. Thus, if$\mathrm{\Gamma }\subset G$ is a discrete central subgroup, then we get a map$G/\mathrm{\Gamma }\to G_{ℂ}/\phi (\mathrm{\Gamma })$, which gives a complexification for$G/\mathrm{\Gamma }$. Since every Lie group is of this form, this shows existence.

Some easy examples: the complexification both of ${\mathrm{SL}}_nℝ$ and $\mathrm{SU}(n)$ is${\mathrm{SL}}_nℂ$. The complexification of $ℝ$ is $ℂ$ and of $S^1$ is ${ℂ}^{*}$.

The map $\phi :G\to G_{ℂ}$ is not always injective. For example, if $G$ isthe universal cover of ${\mathrm{SL}}_nℝ$ (which has fundamental group $\mathbb{Z}$), then$G_{ℂ}\cong {\mathrm{SL}}_nℂ$, and $\phi$ factors through the covering $G\to {\mathrm{SL}}_nℝ$.