---
tag: SADNTQAY3G
kind: concept
name: Real form of a Lie group
msc: ["22E15", "22E10"]
specializes: [NM8XCPS1Q3]
sources:
  - url: https://en.wikipedia.org/wiki/Real_form_(Lie_theory)
---

## Summary

A real form of a complex Lie group $G_\C$ is a closed real Lie subgroup $G$.
Its Lie algebra must be a real form of the Lie algebra of $G_\C$.
So $G$ has half the real dimension of $G_\C$, and complexifying its Lie algebra gives back that of $G_\C$.
One complex group has several real forms, for example $\mathrm{SL}_n(\R)$ and $\mathrm{SU}(n)$ in $\mathrm{SL}_n(\C)$.
The classification of real semisimple groups is largely the classification of real forms of complex semisimple groups.

## Explanations

### L1 human

A complex Lie group can be cut down to a real group in several ways.
Inside the complex $n \times n$ matrices of determinant $1$, one can keep the matrices with real entries.
One can also keep the unitary ones, which preserve lengths in $\C^n$.
Both choices have half the dimension, and both "complexify" back to the whole group.
Such real subgroups are the real forms.