Killing form

The Killing form $𝜅 : 𝔤 \times 𝔤 \to 𝕂$ is an invariant symmetric bilinear form form on a finite-dimensional Lie algebra $𝔤$ defined as the trace of the composition of two linear endomorphisms¹ #m/def/lie

𝜅 (𝑋, 𝑌) = t r (a d 𝑋 \circ a d 𝑌)

Proof of symmetric bilinearity

$𝐾$ is linear in the second argument, since

𝜅 (𝑋, 𝛼 𝑌 + 𝛽 𝑍) = t r (a d 𝑋 \circ a d 𝛼 𝑌 + 𝛽 𝑍) = t r (a d 𝑋 \circ (𝛼 a d 𝑌 + 𝛽 a d 𝑍)) = t r (𝛼 a d 𝑋 \circ a d 𝑌 + 𝛽 a d 𝑋 \circ a d 𝑍) = 𝛼 t r (a d 𝑋 \circ a d 𝑌) + 𝛽 t r (a d 𝑋 \circ a d 𝑍) = 𝛼 𝜅 (𝑋, 𝑌) + 𝛽 𝜅 (𝑋, 𝑍)

and symmetric since

𝜅 (𝑋, 𝑌) - 𝜅 (𝑌, 𝑋) = t r (a d 𝑋 a d 𝑌) - t r (a d 𝑌 a d 𝑋) = t r (a d 𝑋 a d 𝑌 - a d 𝑌 a d 𝑋) = t r ([a d 𝑋, a d 𝑌]) = t r (a d [𝑋, 𝑌]) = 0

Properties

Let $𝔞 ⊴ 𝔤$ be an ideal. Then the restriction of the Killing form of $𝔤$ to $𝔞$ is the Killing form of $𝔞$ .

Proof

#missing/proof

If $𝔤$ is the Lie algebra over $ℝ$ of a Lie group $𝐺$ with Adjoint representation $A d$ then

Proof of 1

Since $A d 𝑔$ is a Lie algebra automorphism, for all $𝑋, 𝑍 \in 𝔤$ and $𝑔 \in 𝐺$ ,

a d A d 𝑔 (𝑋) (𝑍) = [A d 𝑔 (𝑋), 𝑍] = A d 𝑔 A d 𝑔 - 1 [A d 𝑔 (𝑋), 𝑍] = A d 𝑔 [𝑋, A d 𝑔 - 1 (𝑍)] = A d 𝑔 a d 𝑋 A d 𝑔 - 1 (𝑍)

hence

𝐾 (A d 𝑔 (𝑋), A d 𝑔 (𝑌)) = t r (a d A d 𝑔 (𝑋) a d A d 𝑔 (𝑌)) = t r (A d 𝑔 a d 𝑋 A d 𝑔 - 1 A d 𝑔 a d 𝑋 A d 𝑔 - 1) = t r (A d 𝑔 a d 𝑋 a d 𝑌 A d 𝑔 - 1) = t r (a d 𝑋 a d 𝑌)

as required.

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