Let G be a real Lie group. Then the complexification G of G is the unique complex Lie group equipped with a map φ:GG such that any map GH where H is a complex Lie group, extends to a holomorphic map GH. If 𝔤 and 𝔤 are the respective Lie algebras, 𝔤𝔤.

For simply connected groups, the construction is obvious: we simply take the simply connected complex group with Lie algebra 𝔤, and φ to be the map induced by the inclusion 𝔤𝔤.

If γG is central, then its image is in central in G since gγgγ-1 is a map extending φ, and thus must be the identityPlanetmathPlanetmathPlanetmathPlanetmath by uniqueness half of the universal propertyMathworldPlanetmath. Thus, if ΓG is a discrete central subgroup, then we get a map G/ΓG/φ(Γ), which gives a complexification for G/Γ. Since every Lie group is of this form, this shows existence.

Some easy examples: the complexification both of SLn and SU(n) is SLn. The complexification of is and of S1 is *.

The map φ:GG is not always injectivePlanetmathPlanetmath. For example, if G is the universal cover of SLn (which has fundamental groupMathworldPlanetmathPlanetmath ), then GSLn, and φ factors through the covering GSLn.

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