﻿ Killing-Form

# Killing-Form

The Killing form plays an important role in differential geometry and in the classification of semisimple Lie algebras. It is named after Wilhelm Killing.

## Definition

Let be a Lie algebra over the field and its adjoint representation.

Killing the shape is represented by

For defined symmetric bilinear form

## Properties

• Is a symmetric bilinear form.
• For all is skew-symmetric with respect to, that is, applies to all
• The Killing form is non- degenerate if and only if the Lie algebra is semi- simple.
• If the Lie algebra of a Lie group, then is invariant, ie applies to all
• If the Lie algebra of a semisimple Lie group, then the Killing form is negative definite if and only if is compact. In particular, a bi -invariant Riemannian metric defined on a compact semisimple Lie group. More generally, on the Lie algebra of a compact (not necessarily semisimple ) Lie group, the Killing form is always negative semidefinite.

## Examples

The Killing form nilpotent Lie algebras is identically zero.

For many classical Lie algebras can be the Killing form explicitly specify:

## Riemannian metric on symmetric spaces of type nichtkompaktem

A symmetric space of nichtkompaktem type is a manifold of the form

With a semi- simple Lie group and a maximum compact subgroup.

For a symmetric space has a Cartan decomposition

And one can identify the tangent space at the neutral element.

The Killing form is negative definite and positive definite on to. In particular, it defines an invariant scalar product on and thus a left -invariant Riemannian metric on. Up to multiplication with scalar, this is the only invariant metric.

The differential geometry of symmetric spaces is concerned with the properties of Riemannian manifolds.

## Classification of semisimple Lie algebras

The Killing form plays a key role in the classification of semi- simple Lie algebras over fields of characteristic.

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