Proposition 8

Tag 0125, permanent. Cite it as [Bourbaki, Tag 0125].

Lie Groups and Lie Algebras, chapter VIII, § 5. Automorphisms of a semi-simple Lie algebra

Let V be the set of endomorphisms of the vector space $\mathfrak{g}$ . Then Aut( $\mathfrak{g}$ ) is closed in V for the Zariski topology (Chap. VII, App. I).

Let K be the Killing form of $\mathfrak{g}$ . If $s\in$ Aut( $\mathfrak{g}$ ),

[sx, sy] = [x, y] \tag{2}

K(sx, sy) = K(x, y) \tag{3}

for all $x, y\in \mathfrak{g}$ . Conversely, let $s$ be an element of V satisfying (2) and (3) for all $x, y\in \mathfrak{g}$ . Then Ker( $s$ ) $= 0$ , so $s$ is bijective and $s\in$ Aut( $\mathfrak{g}$ ). But, for all $x, y\in \mathfrak{g}$ , the maps $s \rightarrow [sx, sy]$ and $s \rightarrow K(sx, sy)$ from V to $\mathfrak{g}$ and $k$ are polynomial.

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

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Cites

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