complexification
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, .
For simply connected groups, the construction is obvious: we simply take the simply connected complex group with Lie algebra , and to be the map induced by the inclusion .
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 .
The map is not always injective. For example, if is
the universal cover of (which has fundamental group
), then
, and factors through the covering .
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 |