Proposition 41

Tag 0AB4, permanent. Cite it as [Bourbaki, Tag 0AB4].

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

Let $G$ be a Lie group and $\pi_1, \pi_2, \ldots, \pi_n, \pi$ analytic linear representation of $G$ on complete normable spaces $E_1, E_2, \ldots, E_n, E$ . Let

(x_1, x_2, \ldots, x_n) \mapsto x_1 x_2 \ldots x_n

be a continuous multilinear mapping of

E_1 \times E_2 \times \cdots \times E_n

into

E

. Suppose that

\pi(g)(x_1 x_2 \ldots x_n) = (\pi_1(g)x_1)(\pi_2(g)x_2) \ldots (\pi_n(g)x_n)

for all

g \in G, x_1 \in E_1, \ldots, x_n \in E_n

. Then

(L(\pi)a)(x_1 x_2 \ldots x_n) = \sum_{i=1}^n x_1 x_2 \ldots x_{i-1} ((L(\pi_i)a)x_i) x_{i+1} \ldots x_n

for all

a \in L(G), x_1 \in E_1, \ldots, x_n \in E_n

As an example we perform the calculation for $n = 2$ .

\begin{align*} (L(\pi)a)(x_1 x_2) &= \langle a, g \mapsto \pi(g)(x_1 x_2) \rangle & \text{(Proposition 38)} \\ &= \langle a, (g \mapsto \pi_1(g)x_1)(g \mapsto \pi_2(g)x_2) \rangle \\ &= \langle a, g \mapsto \pi_1(g)x_1 \rangle \cdot x_2 + x_1 \cdot \langle a, g \mapsto \pi_2(g)x_2 \rangle & \text{(*Diff.* \& *Anal.* *Man.*, R, 5.5.6)} \\ &= ((L(\pi_1)a)x_1) \cdot x_2 + x_1 \cdot ((L(\pi_2)a)x_2) & \text{(Proposition 38)}. \end{align*}

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

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Cites

Proposition 38