Proposition 3

Tag 012S, permanent. Cite it as [Bourbaki, Tag 012S].

Lie Groups and Lie Algebras, chapter VIII, § 6. Modules over a split semi-simple Lie algebra

Let V be a simple $\mathfrak{g}$ -module with a highest weight $\omega$ .

(i) The primitive elements of V are the non-zero elements of $V^{\omega}$ .

(ii) V is semi-simple as an $\mathfrak{h}$ -module.

(iii) We have $V =\bigoplus_{\lambda\in\mathfrak{h}^*}V^{\lambda}$ . For all $\lambda \in \mathfrak{h}^*,V^{\lambda}$ is finite dimensional. We

have dim $V^{\omega}= 1$ .

(iv) The $\mathfrak{g}$ -module V is absolutely simple.

Assertions (i), (ii) and (iii) follow from Prop. 1 and Lemma 2. Assertion (iv) follows from Prop. 1 (iii) and Algebra, Chap. VIII, §7, no. 3.

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

Remark 1, § 7
Theorem 1, § 7
Corollary 2, § 7

Cites

Algebra, Chap. VIII, §7, no. 3
Lemma 2
Prop. 1