Lie algebras MOC

Killing form

The Killing form is an invariant symmetric bilinear form form on a finite-dimensional Lie algebra defined as the trace of the composition of two linear endomorphisms¹ lie

Proof of symmetric bilinearity

is linear in the second argument, since

and symmetric since

by Properties.

Properties

1. Let be an ideal. Then the restriction of the Killing form of to is the Killing form of .

Proof

proof

Relation to Lie groups

If is the Lie algebra over of a Lie group with Adjoint representation then

1. for all and

Proof of 1

Since is a Lie algebra automorphism, for all and ,

hence

as required.

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §5.1, p. 21 ↩