Proposition 2

Tag 0A6V, permanent. Cite it as [Bourbaki, Tag 0A6V].

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

The formal power series $\tilde{H}$ is a convergent series (Differentiable and Analytic Manifolds, R, 4.1.1). If G is the ball $\{ x \in g \mid \| x \| < a^\theta \}$ , the domain of absolute convergence of $\tilde{H}$ (Differentiable and Analytic Manifolds, R, 4.1.3) contains $G \times G$ .

If $u$ and $v$ are two real numbers $> 0$ such that $u < a^\theta$ and $v < a^\theta$ , then (Lemma 3)

\| \tilde{H}_{r,s} \| u^r v^s \leq a^\theta (u a^{-\theta})^r (v a^{-\theta})^s

and

\| H_{r,s} \| u^r v^s

tends to 0 when

r + s

tends to infinity.

We denote by $h : G \times G \to g$ the analytic function (Differentiable and Analytic Manifolds, R, 4.2.4) defined by $\tilde{H}$ , that is by the formula

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 G \times G.

This function is called the Hausdorff function of

g

Let $(x, y) \in G \times G$ . Then

\| \tilde{H}_{r,s}(x, y) \| \leq \sup(\| x \|, \| y \|) \\ \| h(x, y) \| \leq \sup(\| x \|, \| y \|).

(17) follows immediately from (16) and (16) is trivial for $r = s = 0$ ; if $r \geq 1$ , then

\| \tilde{H}_{r,s}(x, y) \| \leq \| \tilde{H}_{r,s} \| \| x \|^{r} \| y \|^{s}

\leq \| x \| \left( \frac{\| x \|}{a^{\theta}} \right)^{r-1} \left( \frac{\| y \|}{a^{\theta}} \right)^{s}

\leq \| x \|;

we argue similarly if

s \geq 1

In particular, $\| h(x, y) \| < a^{\theta}$ for $(x, y) \in G \times G$ .

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

Cited by

Cites

Lemma 3