Proposition 8

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

Lie Groups and Lie Algebras, chapter VII, § 1. Primary decomposition of linear representations

Let $\mathfrak{h}$ be a Lie algebra, and $L,M,N\mathfrak{h}$ -modules. Denote by P the set of maps from $\mathfrak{h}$ to $k$ .

(i) The sum $\sum_{\lambda\in P}L^{\lambda}(P)$ is direct.

(ii) If $f: L\rightarrow M$ is a homomorphism of $\mathfrak{h}$ -modules, $f(L^{\lambda}(\mathfrak{h}))\subset M^{\lambda}(\mathfrak{h})$ for all $\lambda \in P$ .

(iii) If $f: L\times M\rightarrow N$ is a bilinear $\mathfrak{h}$ -invariant map,

f(L^{\lambda}(\mathfrak{h})\times M^\mu(\mathfrak{h}))\subset N^{\lambda+\mu}(\mathfrak{h})

for all $\lambda , \mu\in P$ .

This follows from Props. 2 and 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

§ 2. Root system of a split semi-simple Lie algebra

Cites

Nothing.