Proposition 16

Tag 0A9T, permanent. Cite it as [Bourbaki, Tag 0A9T].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let G, G' be Lie groups, X a manifold of class C^r and suppose that G (resp. G') operates on X on the left (resp. right), with $(gx)g' = g(xg')$ for all $x \in X, g \in G, g' \in G'$ . Let $t \in \mathscr{T}^{(s)}(G)$ , $t' \in \mathscr{T}^{(s')}(G')$ , $t'' \in T^{(s'')}(X)$ , with $s + s' + s'' \leq r$ . Then

(t * t'') * t' = t * (t'' * t').

(t * t'') * t'

(resp.

t * (t'' * t')

) is the image of

t \otimes t'' \otimes t'

under the mapping

(g, x, g') \mapsto (gx)g'

(resp.

g(xg')

) of

G \times X \times G'

into X.

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

Cited by

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Cites

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