Bourbaki

Definition 3

Tag 0AV2, permanent. Cite it as [Bourbaki, Tag 0AV2].

Lie Groups and Lie Algebras, chapter VI, § 1. Root systems

Let B be a basis of R. The Cartan matrix of R (relative to B) is the matrix (n(α,β))α,β∈B(n(\alpha, \beta))_{\alpha, \beta \in B}.

For all α∈B\alpha \in B, n(α,α)=2n(\alpha, \alpha) = 2. For α,β∈B\alpha, \beta \in B,

n(α,β)=2(α∣β)(β∣β)=−2∥α∥∥β∥cos⁡πm(α,β), n(\alpha, \beta) = 2 \frac{(\alpha|\beta)}{(\beta|\beta)} = -2 \frac{\|\alpha\|}{\|\beta\|} \cos \frac{\pi}{m(\alpha, \beta)},

where m(α,β)m(\alpha, \beta) denotes as above the order of sαsβs_\alpha s_\beta. If α≠β\alpha \neq \beta, n(α,β)=0,−1,−2n(\alpha, \beta) = 0, -1, -2 or −3-3 (cf. no. 3).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Transcribed from the printed volume. Not translated.