Corollary 2

Tag 0AJW, permanent. Cite it as [Bourbaki, Tag 0AJW].

Lie Groups and Lie Algebras, chapter III, § 9. Commutators, centralizers and normalizers in a Lie group

Let $\rho$ be a finite-dimensional semi-simple analytic linear representation of $G$ on a vector space $V$ , $S$ the symmetric algebra of $V$ and $S^G$ the subalgebra of $S$ consisting of the elements invariant under $(S\rho)(G)$ . Then $S^G$ is a finitely generated algebra.

This follows from Theorem 1, Chapter I, § 6, Theorem 6 (a) and Commutative Algebra, Chapter V, § 1, Theorem 2.

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

Cited by

Exercise 35, § 9

Cites

*Commutative Algebra*, Chapter V, § 1, Theorem 2 not yet on this site: AC V
Chapter I, § 6, Theorem 6
Theorem 1