Bourbaki

Proposition 7

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

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

Let (Xα)α∈R(X_{\alpha})_{\alpha\in R} be a Chevalley system for (g,h)(\mathfrak{g},\mathfrak{h}). We retain the notation of Lemma 4. Then, N−α,−β=Nα,βN_{-\alpha ,-\beta}= N_{\alpha ,\beta} and Nα,β=±(q+1)N_{\alpha ,\beta}=\pm (q+ 1) for α,β,α+β∈R\alpha , \beta , \alpha +\beta \in R.

Let φ\varphi be the automorphism of g\mathfrak{g} considered in Def. 3 (iii). Then

N−α,−βX−α−β=[X−α,X−β]=[φ(Xα),φ(Xβ)]=φ([Xα,Xβ]) N_{-\alpha ,-\beta}X_{-\alpha-\beta}= [X_{-\alpha}, X_{-\beta}] = [\varphi (X_{\alpha}), \varphi (X_{\beta})] =\varphi ([X_{\alpha}, X_{\beta}])
=φ(Nα,βXα+β)=Nα,βX−α−β =\varphi (N_{\alpha ,\beta}X_{\alpha+\beta}) = N_{\alpha ,\beta}X_{-\alpha-\beta}

so N−α,−β=Nα,βN_{-\alpha ,-\beta}= N_{\alpha ,\beta}. Now Nα,β=±(q+1)N_{\alpha ,\beta}=\pm (q+ 1) by (5).

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.

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Transcribed from the printed volume. Not translated.