Proposition 19

Tag 0A9X, permanent. Cite it as [Bourbaki, Tag 0A9X].

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

Let $t \in \mathscr{T}^{(s)}(G)$ , with $s \leq r$ . Let $f$ (resp. $f'$ ) be a function of class $C^r$ on $X$ with values in a Hausdorff polynormed space $F$ (resp. $F'$ ). Let $(u, u') \mapsto uu'$ be a continuous bilinear mapping of $F \times F'$ into a Hausdorff polynormed space $F''$ , so that $ff'$ is a function of class $C^r$ on $X$ with values in $F''$ . Let

\sum_{i=1}^n t_i \otimes t'_i \text{ be the image of } t \text{ in } \mathscr{T}^{(s)}(G) \otimes \mathscr{T}^{(s)}(G) \text{ under the coproduct. Then}

t * (ff') = \sum_{i=1}^n (t_i * f)(t'_i * f').

Let $x \in X$ and let $\rho(x)$ denote the orbital mapping of $x$ . Then

\langle \varepsilon_x, t * (ff') \rangle = \langle t^\vee, (ff') \circ \rho(x) \rangle \quad \text{by (4)} \\ = \langle t^\vee, (f \circ \rho(x))(f' \circ \rho(x)) \rangle \\ = \sum_{i=1}^n \langle t_i^\vee, f \circ \rho(x) \rangle \langle t_{i'}^\vee, f' \circ \rho(x) \rangle \tag{Diff. \& Man. Anal., R, 13.5.2} \\ = \sum_{i=1}^n \langle \varepsilon_x, t_i * f \rangle \langle \varepsilon_x, t_i' * f' \rangle \quad \text{by (4).}

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

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