# nLab compact Lie algebra

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Definition

A semisimple Lie algebra $\mathfrak{g}$ is compact if its Killing form $\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is a negative definite bilinear form.

## Properties

###### Proposition

The Lie algebra of a compact semisimple Lie group is compact.

A proof is spelled out for instance in (Woit, theorem 1).

## References

For instance

• Peter Woit, Topics in Representation Theory: The Killing Form, Reflections and Classification of Root Systems (pdf)

Last revised on September 14, 2011 at 18:35:53. See the history of this page for a list of all contributions to it.