§ 7. Convergence of the Hausdorff series (real or complex case)

In this paragraph we assume that $K$ is one of the fields $\mathbf{R}$ or $\mathbf{C}$ with its usual absolute value. Recall that a normable algebra over $K$ is a (not necessarily associative) algebra over $K$ with a topology $\mathscr{T}$ with the following properties: (1) $\mathscr{T}$ can be defined by a norm: (2) the mapping $(x, y) \mapsto xy$ of $A \times A$ into $A$ is continuous. A normed algebra over $K$ is an algebra $A$ over $K$ with a norm such that $\|xy\| \leq \|x\|\ \|y\|$ for all $x, y$ in $A$ . We denote by $g$ a complete normable Lie algebra over $K$ . We choose a norm on $g$ and a number $M > 0$ such that

\|[x, y]\| \leq M \|x\|\ \|y\| \quad \text{for } x, y \text{ in } g.

1. Continuous-polynomials with values in $g$

Let $I$ be a finite set and let $P(g^I; g)$ (resp. $\hat{P}(g^I; g)$ ) be the vector space of continuous-polynomials (resp. formal power series with continuous components) on $g^I$ with values in $g$ . Recall (Differentiable and Analytic Manifolds, R, Appendix) that $P(g^I; g)$ has a graduation of type $\mathbf{N}^I$ and that $\hat{P}(g^I; g)$ is identified with the completion of the vector space $P(g^I; g)$ with the topology defined by the filtration associated with the graduation of $P(g^I; g)$ . Moreover, $P(g^I; g)$ is a graded Lie algebra with the bracket defined by $[f, g](x) = [f(x), g(x)]$ for f, g in P(g^I; g), x \in g^I; this Lie algebra structure can be extended by continuity to $\hat{P}(g^I; g)$ and makes it into a complete Hausdorff filtered Lie algebra.

By proposition 2 of § 6, no. 3, there exists one and only one continuous Lie algebra homomorphism $\phi_I : u \mapsto \tilde{u}$ of $\hat{L}(I)$ into $\hat{P}(g^I; g)$ mapping the indeterminate of index i to $\mathrm{pr}_i$ for all $i \in I$ , since $\mathrm{pr}_i \in P(g^I; g)$ . It follows that $\tilde{u} \in P(g^I; g)$ for $u \in L(I)$ ; more precisely, when $u \in L(I)$ , $\tilde{u}$ is just the polynomial mapping $(t_i) \mapsto u((t_i))$ of § 2, no. 4. On the other hand, clearly $\phi_I$ is compatible with the multigraduations of $L(I)$ and $P(g^I; g)$ . If $u = \sum_{v \in \mathbf{N}^I} u_v$ , where $u_v \in L^v(I)$ for $v \in \mathbf{N}^I$ , then

\tilde{u} = \sum_{v \in \mathbf{N}^I} \tilde{u}_v, \quad \text{where } \tilde{u}_v \in P_v(g^I; g).

Let $u = (u_j)_{j \in J}$ be a finite family of elements of $\hat{L}(I)$ , let $v \in \hat{L}(J)$ and let $w = v \circ u$ (§ 6, no. 3). We write $\tilde{u} = (\tilde{u}_j)_{j \in J} \in \mathfrak{g}$ . Then

(2)

\tilde{v} \circ \tilde{u} = (v \circ u)^{\sim}.

This follows by extending by continuity formula (7) of § 6, no. 3 and from Differentiable and Analytic Manifolds, R, Appendix, no. 6.

2. Group germ defined by a complete normed Lie algebra

Let $H = \sum_{r,s \geq 0} H_{r,s} \in \hat{L}(U, V)$ be the Hausdorff series (§ 6, no. 4, Definition 1). We shall show that the corresponding formal power series

(3)

\tilde{H} = \sum_{r,s \geq 0} \tilde{H}_{r,s} \in \hat{P}(g \times g, g)

is convergent (Differentiable and Analytic Manifolds, R, 3.1.1).

We introduce the following formal power series $\eta \in \mathbf{Q}[[U, V]]$

(4)

\eta(U, V) = -\log(2 - \exp(U + V))

(5)

= \sum_{m \geq 1} \frac{1}{m} (\exp(U + V) - 1)^m

(6)

= \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1, \ldots, r_m \\ s_1, \ldots, s_m}} \frac{U^{r_1} V^{s_1}}{r_1!} \frac{U^{r_2} V^{s_2}}{r_2!} \cdots \frac{U^{r_m} V^{s_m}}{s_m!}.

Hence

(7)

\eta(U, V) = \sum_{r,s \geq 0} \eta_{r,s} U^r V^s,

where

\eta_{r,s} = \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_m = r \\ s_1 + \cdots + s_m = s \\ r_i + s_i \geq 1}} \frac{1}{r_1! \ldots r_m! s_1! \ldots s_m!}.

Now let

u

and

v

be two positive real numbers such that

u + v < \log 2

; then

0 \leq \exp(u + v) - 1 < 1

; the series derived from (5) and (6) by substituting

u

for

U

and

v

for

V

are convergent and the above calculations imply that

\sum_{r, s \geq 0} \eta_{r,s} u^r v^s = -\log(2 - \exp(u + v)) < +\infty.

Let

r, s \geq 0

and let

\| \tilde{H}_{r,s} \|

be the norm of the continuous-polynomial

\tilde{H}_{r,s}

(Differentiable and Analytic Manifolds, R, Appendix, no. 2).

Lemma 1

\| \tilde{H}_{r,s} \| \leq M^{r+s-1} \eta_{r,s}.

Let

r_i, s_i

be in

\mathbf{N}

for

1 \leq i \leq m

, with

s_m = 1

; we write

r = \sum_i r_i, s = \sum_i s_i

and consider the following element of

L(\{U, V\})

Z = \left( \left( \sum_{i=1}^{m-1} (\mathrm{ad}\ U)^{r_i} (\mathrm{ad}\ V)^{s_i} \right) (\mathrm{ad}\ U)^{r_m} \right)(V).

Then

\tilde{Z} = f \circ p

, where

f

is the following

(r+s)

-linear mapping of

g^{r+s}

into

g

(x_1, \ldots, x_r, y_1, \ldots, y_s) \mapsto (\mathrm{ad}(x_1) \circ \cdots \circ \mathrm{ad}(x_r) \circ \mathrm{ad}(y_1) \circ \cdots \circ \mathrm{ad}(y_s))(y_s)

and where

p

is the following mapping of

g^2

into

g^{r+s}

(x, y) \mapsto (\underbrace{x, \ldots, x}_{r}, \underbrace{y, \ldots, y}_{s});

hence

\| \tilde{Z} \| \leq \| f \| \leq M^{r+s-1}

(Differentiable and Analytic Manifolds, R, Appendix). Applying these inequalities to the various terms on the right hand side of formula (9) of § 6, no. 4, we obtain:

\|(H'_{r,s})\sim\| \leq \frac{M^{r+s-1}}{r+s} \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_m = r \\ s_1 + \cdots + s_m = s \\ r_1 + s_1 \geq 1, \ldots, r_{m-1} + s_{m-1} \geq 1}} \frac{1}{r_1! \ldots r_m! s_1! \ldots s_m!}.

A similar argument gives

\|(H''_{r,s})\sim\| \leq \frac{M^{r+s-1}}{r+s} \sum_{m \geq 1} \frac{1}{m} \sum_{\substack{r_1 + \cdots + r_{m-1} = r-1 \\ s_1 + \cdots + s_{m-1} = s \\ r_1 + s_1 \geq 1, \ldots, r_{m-1} + s_{m-1} \geq 1}} \frac{1}{r_1! \ldots r_{m-1}! s_1! \ldots s_{m-1}!}

whence, by (8)

\|\tilde{H}_{r,s}\| < \eta_{r,s} \frac{M^{r+s-1}}{r+s} \leq \eta_{r,s} M^{r+s-1},

which proves the lemma.

Proposition 1

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.

Proposition 2

Let

\Omega' = \left\{ (x, y, z) \in g \times g \times g \mid \|x\| + \|y\| + \|z\| < \frac{1}{M} \log \frac{3}{2} \right\}.

If $(x, y, z) \in \Omega'$ , then

\text{(16)} \quad (x, y) \in \Omega, \quad (h(x, y), z) \in \Omega, \quad (y, z) \in \Omega, \quad (x, h(y, z)) \in \Omega

and

\text{(17)} \quad h(h(x, y), z) = h(x, h(y, z)).

Let $(x, y, z) \in \Omega'$ ; clearly $(x, y) \in \Omega$ and $(y, z) \in \Omega$ . Moreover:

\|h(x, y)\| \leq \sum_{r, s} \|H_{r,s}\| \|x|r\|y|s,

and hence by (13)

\|h(x, y)\| \leq -\frac{1}{M} \log(2 - \exp M(\|x\| + \|y\|)).

Now $M(\|x\| + \|y\|) < \log \frac{3}{2} - M\|z\|$ ; we write $u = \exp(M\|z\|)$ ; then $1 \leq u \leq \frac{3}{2}$ and

\begin{align*} M(\|h(x, y)\| + \|z\|) &< -\log(2 - \exp(\log \frac{3}{2} - M\|z\|)) + M\|z\| \\ &= -\log\left(2 - \frac{3}{2u}\right) + \log u = \log \frac{2u^2}{4u - 3} \\ &= \log\left(2 + \frac{2(u - 1)(u - 3)}{4u - 3}\right) \leq \log 2. \end{align*}

We see similarly that $(x, h(y, z)) \in \Omega$ .

We now prove (17). In the Lie algebra $\hat{L}(\{U, V, W\})$ ,

H(H(U, V), W) = H(U, H(V, W))

by Proposition 4 of § 6, no. 5. By no. 1, formula (2), we therefore have in $\hat{P}(g \times g \times g, g)$ the relation

\tilde{H} \circ (\tilde{H} \times \mathrm{Id}_g) = \tilde{H} \circ (\mathrm{Id}_g \times \tilde{H}).

By Differentiable and Analytic Manifolds, R, 3.1.9, there exists a number $\varepsilon > 0$ such that formula (17) is true when $\|x\|, \|y\|$ and $\|z\|$ are $\leq \varepsilon$ . But the functions $(x, y, z) \mapsto h(h(x, y), z)$ and $(x, y, z) \mapsto h(x, h(y, z))$ are analytic functions on $\Omega'$ with values in $g$ (Differentiable and Analytic Manifolds, R, 3.2.7). As $\Omega'$ is connected and they coincide in a neighbourhood of 0, they are equal (Differentiable and Analytic Manifolds, R, 3.2.5).

The above results imply:

Let $\alpha$ be a real number such that $0 < \alpha \leq \frac{1}{3M} \log \frac{3}{2}$ . Let

G = \{x \in g \mid \|x\| < \alpha\},

\Theta = \{(x, y) \in G \times G \mid h(x, y) \in G\}

and

m : \Theta \to G

be the restriction of

h

\Theta

. Then: (1)

\Theta

is open in

G \times G

and

m

is analytic. (2)

x \in G

implies

(0, x) \in \Theta

(x, 0) \in \Theta

and

m(0, x) = m(x, 0) = x

. (3)

x \in G

implies

-x \in G

(x, -x) \in \Theta

(-x, x) \in \Theta

and

m(x, -x) = m(-x, x) = 0.

(4) Let x, y, z be elements of G such that (x, y) ∈ Θ, (m(x, y), z) ∈ Θ, (y, z) ∈ Θ and (x, m(y, z)) ∈ Θ. Then m(m(x, y), z) = m(x, m(y, z)).

In other words (Chapter III, § 1), if we write −x = σ(x), the quadruple (G, 0, σ, m) is a Lie group germ over K.

3. Exponential in Complete Normed Associative Algebras

In this no. we denote by A a complete normed unital associative algebra (General Topology, Chapter IX, § 3, no. 7). Then \|x.y\| ≤ \|x\|. \|y\| for x, y in A.

Let I be a finite set and let $\hat{P}(A^I; A)$ be the vector space of formal power series with continuous components on $A^I$ with values in A (Differentiable and Analytic Manifolds, R, Appendix, no. 5) with the algebra structure obtained by writing

f.g = m \circ (f, g) \quad \text{for } f, g \text{ in } \hat{P}(A^I; A),

where $m : A \times A \to A$ denotes multiplication on A. Arguing as in no. 1 and using Proposition 1 of § 5, no. 1, we define a continuous homomorphism of unital algebras $u \mapsto \tilde{u}$ of $\hat{A}(I)$ into $\hat{P}(A^I; A)$ mapping the indeterminate of index i to $pr_i$ ; this homomorphism extends the Lie algebra homomorphism of $\hat{L}(I)$ into $\hat{P}(A^I; A)$ defined in no. 1. If $u = \sum_v u_v$ with $u_v \in A^v(I)$ for $v \in \mathbf{N}^I$ , then $\tilde{u} = \sum_v \tilde{u}_v$ , where $\tilde{u}_v$ is the polynomial mapping $(t_i)_{i \in I} \mapsto u_v((t_i))$ .

Let $u = (u_j)_{j \in I}$ be a finite family of elements of $\hat{A}(I)$ , let $v \in \hat{A}(J)$ and write $w = v \circ u$ (§ 5, no. 1). Then

(v \circ u)^{\sim} = \tilde{v} \circ \tilde{u}.

This follows by extending by continuity formula (2) of § 5, no. 1 and from Differentiable and Analytic Manifolds, R, Appendix, no. 6.

In particular we take I = {U}, identify A and $A^{(U)}$ and consider the images $\tilde{e}$ and $\tilde{l}$ of the series $e(U) = \sum_{n \geq 1} U^n/n!$ and $l(U) = \sum_{n \geq 1} (-1)^{n-1} U^n/n$ in $\hat{P}(A; A)$ . Then $\|\tilde{U}^n\| \leq 1$ for $\|x_1 \ldots x_n\| \leq \|x_1\| \ldots \|x_n\|$ for $x_1, \ldots, x_n$ in A. Therefore the radius of absolute convergence of $\tilde{e}$ (resp. $\tilde{l}$ ) is infinite (resp. $\geq 1$ ).

We shall denote by $e_A$ (resp. $l_A$ ) the analytic mapping of A into A (resp. of B into A, where B is the open unit ball of A) defined by the convergent series $\tilde{e}$ (resp. $\tilde{l}$ ) and we shall write $\exp_A(x) = 1 + e_A(x)$ (for $x \in A$ ) and

\log_A(x) = l_A(x - 1)

(for $x \in A, \|x - 1\| < 1$ ). Then

\exp_A x = \sum_{n \geq 0} \frac{x^n}{n!} \quad (x \in A)

(20) $\log_A x = \sum_{n \geq 1} (-1)^{n-1} \frac{(x-1)^n}{n} \quad (x \in A, \|x-1\| < 1).$

As $(e \circ l)(U) = (l \circ e)(U) = U$ (cf. § 6, no. 1), by (18) $\tilde{e} \circ \tilde{l} = \tilde{l} \circ \tilde{e} = \mathrm{Id}_A$ . Therefore (Differentiable and Analytic Manifolds, R, 3.1.9) (21) $\exp_A(\log_A(x)) = x \quad (x \in A, \|x-1\| \leq 1)$ (22) $\log_A(\exp_A(x)) = x \quad (x \in A, \|x\| < \log 2)$ for $\|x\| < \log 2$ implies $\|\exp_A(x) - 1\| \leq \exp \|x\| - 1 < 1$ .

Finally we consider A as a complete normed Lie algebra. Then

\|[x, y]\| = \|xy - yx\| \leq 2\|x\|\cdot\|y\|.

Proposition 1 of no. 2 implies that the domain of absolute convergence of the formal power series

\tilde{H}

contains the set

\Omega = \{x, y) \in A \times A \mid \|x\| + \|y\| < \frac{1}{2} \log 2\}.

Hence

\tilde{H}

defines an analytic function

h : \Omega \to A

. Then

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

(cf. § 3, no. 1, Remark 4).

Proposition 3

For $\|x\| + \|y\| < \frac{1}{2} \log 2$ ,* (23) $\exp_A x \cdot \exp_A y = \exp_A h(x, y).$ It follows from (18) and the relation $e^u e^v = e^{u+v}$ that

m \circ (1 + \tilde{e}, 1 + \tilde{e}) = (1 + \tilde{e}) \circ \tilde{H}

\hat{P}(A \times A; A)

. We therefore deduce from Differentiable and Analytic Manifolds, R, 3.1.9 that (23) is true for

(x, y)

sufficiently close to

(0, 0)

, whence the proposition follows by analytic continuation (Differentiable and Analytic Manifolds, R, 3.2.5).

Exercises

See the exercises for § 7.