Proposition 6

Tag 01L4, permanent. Cite it as [Bourbaki, Tag 01L4].

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

Let E be a finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$ -module, and $t\in k^*$ . Let $E_p$ be the set of elements of E of weight $p$ .

(i) $\theta (t)_E|E_p$ is a bijection from $E_p$ to $E_{-p}$ .

(ii) $h(t)_E|E_p$ is the homothety with ratio $t^p$ on $E_p$ .

If $E = V(n)$ , the proposition follows from formulas (4) and (6). The general case follows from Th. 1.

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 1, § 5

Cites

Th. 1