Bourbaki

Lemma 3

Tag 00Z7, permanent. Cite it as [Bourbaki, Tag 00Z7].

Lie Groups and Lie Algebras, chapter VIII, § 2. Root system of a split semi-simple Lie algebra

For all α∈R\alpha \in R,

⟨Xα,X−α⟩=−12⟨Hα,Hα⟩ \langle X_{\alpha}, X_{-\alpha}\rangle =-\frac{1}{2}\langle H_{\alpha}, H_{\alpha}\rangle

Indeed,

2⟨Xα,X−α⟩=⟨α(Hα)Xα,X−α⟩=⟨[Hα,Xα],X−α⟩ 2\langle X_{\alpha}, X_{-\alpha}\rangle =\langle \alpha (H_{\alpha})X_{\alpha}, X_{-\alpha}\rangle =\langle [H_{\alpha}, X_{\alpha}], X_{-\alpha}\rangle
=⟨Hα,[Xα,X−α]⟩=−⟨Hα,Hα⟩ =\langle H_{\alpha},[X_{\alpha}, X_{-\alpha}]\rangle =-\langle H_{\alpha}, H_{\alpha}\rangle

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Cites

Nothing.

Transcribed from the printed volume. Not translated.