Example 2

Tag 09W1, permanent. Cite it as [Bourbaki, Tag 09W1].

Lie Groups and Lie Algebras, chapter I, § 3. Representations

We preserve the notation of Example 1.

The $g$ -module structure on $M$ defines on the $K$ -module $P = \mathscr{L}_K(M, M)$ of endomorphisms of $M$ a $g$ -module structure. By (6), for all $x \in g$ and $u \in P$ :

(11)

x_P \cdot u = [x_M, u] = (\mathrm{ad}\, x_M) \cdot u

where

\mathrm{ad}\, x_M

denotes the image of

x_M

under the adjoint representation of

\mathrm{gl}(M)

. In other words:

(12)

x_P = \mathrm{ad}\, x_M

\mathscr{L}(\mathscr{L}(M, M)) = \mathscr{L}(\mathrm{gl}(M))

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Example 1