nLab
compact Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Definition

A semisimple Lie algebra 𝔤\mathfrak{g} is compact if its Killing form ,:𝔤𝔤\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)

Revised on September 14, 2011 18:35:53 by Urs Schreiber (82.113.121.85)