Proposition 3

Tag 00Y1, permanent. Cite it as [Bourbaki, Tag 00Y1].

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

For any integer $m\geq 0$ , $V(m)$ is an absolutely simple $\mathfrak{s}\mathfrak{l}(2, k)$ -module. In this module, $e^{(m)}_0=u^m$ is primitive of weight $m$ .

We have $X_+u^m= 0$ and $Hu^m=mu^m$ , so $u^m$ is primitive of weight $m$ . The submodule of $V(m)$ generated by $u^m$ is of dimension $m+ 1$ (Prop. 2 (i)) and so is equal to $V(m)$ . By Prop. 2 (v), $V(m)$ is absolutely simple.

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

Nothing.

Cites

Prop. 2