Let be a real Lie group. Then the complexification of is the unique complex Lie group equipped with a map such that any map where is a complex Lie group, extends to a holomorphic map . If and are the respective Lie algebras, .
If is central, then its image is in central in since is a map extending , and thus must be the identity by uniqueness half of the universal property. Thus, if is a discrete central subgroup, then we get a map , which gives a complexification for . Since every Lie group is of this form, this shows existence.
Some easy examples: the complexification both of and is . The complexification of is and of is .
|Date of creation||2013-03-22 13:53:55|
|Last modified on||2013-03-22 13:53:55|
|Last modified by||mathcam (2727)|