Lemma 2

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

Lie Groups and Lie Algebras, chapter VIII, § 1. The Lie algebra sl(2, k) and its representations

In the enveloping algebra of $\mathfrak{s}\mathfrak{l}(2, k)$ ,

[H, X_+^n] = 2nX_+^n[H, X_-^n] =-2nX_-^n

for any integer $n\geq 0$ , and

[X_-, X_+^n] =nX_+^{n-1}(H+n-1) =n(H-n+ 1)X_+^{n-1}

[X_+, X_-^n] =nX_-^{n-1}(-H+n-1) =n(-H-n+ 1)X_-^{n-1}

if $n >0$ .

The first and third relations follow from Lemma 1. The others can be deduced from them by using the canonical involution of $\mathfrak{s}\mathfrak{l}(2, k)$ .

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

Lemma 4, § 1

Cites

Lemma 1