# complexification

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

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

If $\gamma\in G$ is central, then its image is in central in $G_{\mathbb{C}}$ since $g\mapsto\gamma g\gamma^{-1}$ is a map extending $\varphi$, 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 $G/\Gamma\to G_{\mathbb{C}}/\varphi(\Gamma)$, 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 $\mathrm{SL}_{n}\mathbb{R}$ and $\mathrm{SU}(n)$ is $\mathrm{SL}_{n}\mathbb{C}$. The complexification of $\mathbb{R}$ is $\mathbb{C}$ and of $S^{1}$ is $\mathbb{C}^{*}$.

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

Title complexification Complexification 2013-03-22 13:53:55 2013-03-22 13:53:55 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 22E15