Proposition 4

Tag 0A6X, permanent. Cite it as [Bourbaki, Tag 0A6X].

Lie Groups and Lie Algebras, chapter II, § 8. Convergence of the Hausdorff series (ultrametric case)

Let $R$ be a real number such that $0 < R \leq a^\theta$ . The mapping $\exp_A$ defines an analytic isomorphism of $G_R$ onto $1 + G_R$ and the inverse isomorphism is the restriction of $\log_A$ to $1 + G_R$ .

$e(l(X)) = l(e(X)) = X$ . By (20), (21) and Differentiable and Analytic Manifolds, R, 4.1.5, we deduce that $e_A(l_A(x)) = l_A(e_A(x))$ for $x \in G_R$ . Then

\exp_A(\log_A x) = x \quad \text{for } x \in 1 + G_R \\ \log_A(\exp_A x) = x \quad \text{for } x \in G_R

which completes the proof.

If $A$ is given the bracket $[x, y] = xy - yx$ , $A$ becomes a complete normed Lie algebra, for $\|xy - yx\| \leq \sup(\|xy\|, \|yx\|) \leq \|x\| \cdot \|y\|$ . Proposition 2 of no. 3 implies that the domain of absolute convergence of $\tilde{H}$ contains $G \times G$ and $\tilde{H}$ therefore defines an analytic function $h : G \times G \to A$ ; then

h(x, y) = \sum_{r, s \geq 0} H_{r,s}(x, y).

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

Cited by

Nothing.

Cites

Proposition 2