Proposition 1

Tag 0A6M, permanent. Cite it as [Bourbaki, Tag 0A6M].

Lie Groups and Lie Algebras, chapter II, § 7. Convergence of the Hausdorff series (real or complex case)

The formal power series $\tilde{H}$ is a convergent series (Differentiable and Analytic Manifolds, R, 3.1.1); its domain of absolute convergence (Differentiable and Analytic Manifolds, R, 3.1.4) contains the open set

\Omega = \left\{ (x, y) \in g \times g \mid \|x\| + \|y\| < \frac{1}{M} \log 2 \right\}.

Let $u, v$ be two real numbers $> 0$ such that $u + v < \frac{1}{M} \log 2$ ; then (Lemma 1)

\text{(12)} \quad M \sum_{r,s \geq 0} \|\tilde{H}_{r,s}\| u^r v^s \\ \leq \sum_{r,s \geq 0} \eta_{r,s} M^{r+s} u^r v^s = -\log(2 - \exp M(u + v)) < +\infty

by (9).

Let $h : \Omega \to g$ denote the analytic function (Differentiable and Analytic Manifolds, R, 3.2.9) defined by $\tilde{H}$ , that is by the formula

\text{(13)} \quad h(x, y) = \sum_{r,s \geq 0} \tilde{H}_{r,s}(x, y) = \sum_{r,s \geq 0} H_{r,s}(x, y) \quad \text{for } (x, y) \in \Omega.

This function is called the Hausdorff function of $g$ relative to $M$ (or simply the Hausdorff function of $g$ if no confusion can arise). Note that $H_{r,s}(U, -U) = 0$ if $r + s \geq 2$ and hence

\text{(14)} \quad h(x, -x) = 0 \quad \text{for } \|x\| < \frac{1}{2M} \log 2.

Similarly

\text{(15)} \quad h(0, x) = h(x, 0) = x \quad \text{for } \|x\| < \frac{1}{M} \log 2.

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

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Cites

Lemma 1