PlanetMath: complexification

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complexification

(Definition)

Let be a real Lie group. Then the complexification $G_\mathbb{C}$ of is the unique complex Lie group equipped with a map $\varphi :G\to G_\mathbb{C}$ such that any map $G\to H$ where 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 is $\mathbb{C}^*$ .

The map $\varphi \colon G\to G_\mathbb{C}$ is not always injective. For example, if 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 $G\to \mathrm{SL}_{n} \mathbb{R}$ .

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Cross-references: covering, factors, fundamental group, universal cover, injective, subgroup, discrete, universal property, identity, image, inclusion, induced, obvious, groups, simply connected, Lie algebras, holomorphic, map, complex, Lie group, real
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This is version 4 of complexification, born on 2003-08-23, modified 2004-03-27.
Object id is 4646, canonical name is Complexification.
Accessed 4332 times total.

Classification:

AMS MSC:

22E15 (Topological groups, Lie groups :: Lie groups :: General properties and structure of real Lie groups)

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