Proposition 18

Tag 0A9W, permanent. Cite it as [Bourbaki, Tag 0A9W].

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

Let $t \in \mathscr{T}^{(s)}(G)$ , $t' \in \mathscr{T}^{(s')}(G)$ and $f : X \to F$ a function of class $C^r$ , with $s + s' \leq r$ . Then

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

For all

x \in X

\begin{align*} \langle \varepsilon_x, (t * t') * f \rangle &= \langle ((t * t')^\vee * \varepsilon_x), f \rangle \quad \text{by (5)} \\ &= \langle {t'}^\vee * (t^\vee * \varepsilon_x), f \rangle \quad (\text{Propositions 2 and 7}) \\ &= \langle t^\vee * \varepsilon_x, t' * f \rangle \quad (\text{Proposition 17}) \\ &= \langle \varepsilon_x, t * (t' * f) \rangle \quad (\text{Proposition 17}). \end{align*}

If $r \geq \infty$ , we see that the set of functions of class $C^\infty$ on $X$ with values in $F$ is a left module over the algebra $\mathscr{T}^{(\infty)}(G)$ .

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

Cited by

§ 3. Passage from a Lie group to its Lie algebra

Cites

Proposition 17