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
Canonical name	Complexification
Date of creation	2013-03-22 13:53:55
Last modified on	2013-03-22 13:53:55
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 22E15