---
tag: DRYYK2WX0M
kind: concept
name: Complexification of a Lie group
msc: ["22E15", "22E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Complexification_(Lie_group)
same_as:
  wikidata: Q17005573
  wikipedia: Complexification (Lie group)
---

## Summary

The complexification of a real [[Lie group|7JBRGCVJRB]] $G$ is a complex Lie group $G_\C$ with a homomorphism $G \to G_\C$.
Every continuous homomorphism of $G$ to a complex Lie group factors through it, in a unique holomorphic way.
It is the group analogue of passing from a real Lie algebra $\mathfrak{g}$ to $\mathfrak{g} \otimes_\R \C$.
For example $\mathrm{SL}_n(\C)$ is the complexification of both $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$.
It links real groups, compact groups and complex groups, as in Weyl's unitary trick.

## Explanations

### L1 human

A real Lie group is described by real parameters, such as angles or real matrix entries.
Its complexification lets these parameters become complex numbers.
For the circle of angles $e^{i\theta}$, letting $\theta$ be complex gives all nonzero complex numbers $\C^\times$.
So $\C^\times$ is the complexification of the circle group.
Any continuous homomorphism from the circle to a complex Lie group extends in exactly one way to a holomorphic homomorphism on $\C^\times$.