Proposition 33

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

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

The mapping $t \mapsto \langle t, f \rangle$ is a morphism of the algebra $\mathscr{T}^{(\infty)}(G)$ into the algebra $A$ .

It suffices to verify that, if $t$ and $t'$ are point distributions on $G$ , then $\langle t * t', f \rangle = \langle t, f \rangle \langle t', f \rangle$ . But

\begin{align*} \langle t * t', f \rangle &= \langle t \otimes t', (g, g') \mapsto f(gg') \rangle \\ &= \langle t \otimes t', (g, g') \mapsto f(g)f(g') \rangle \\ &= \langle t, f \rangle \langle t', f \rangle \end{align*}

(Diff. & Anal. Man., R, 13.4.3).

The morphism of Proposition 33 is said to be associated with $f$ .

Take $G$ to be the group $A^*$ itself and $f$ to be the identity mapping $i$ of $A^*$ . We obtain a morphism, called canonical, of the algebra $\mathscr{T}^{(\infty)}(A^*)$ into the algebra $A$ . The tangent space $T_e(A^*)$ is canonically identified with $A$ ; and if $t \in T_e(A^*)$ , the definition of this identification is such that $\langle t, i \rangle = t$ . Then Proposition 33 implies the following corollary:

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

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