Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being given) that there exists a neighbourhood $W$ of $1_G$ (in $G$) and $n$ *local coordinate* analytic functions
$$
W\rightarrow k,\ (t_i)_{1\leq i\leq n}
$$
such that, for all $g\in W$
$$
(*)\quad g=\prod_{1\leq i\leq n}^{\rightarrow} e^{t_i(g)b_i}=e^{t_1(g)b_1}e^{t_2(g)b_2}\dots e^{t_n(g)b_n}
$$
to see this, just remark that
$$
(t_1,t_2,\cdots t_n)\rightarrow exp(t_1b_1)exp(t_2b_2)\cdots exp(t_nb_n)
$$
is a *local* diffeomorphism from $k^n$ to $G$ in a neighbourhood of $0$ and take the inverse.

This is the local **Wei-Norman's theorem**.

My questions are the following

Let us loosely call

infinite dimensionala Lie group whose Lie algebra is not finite dimensional (this includes the example below and infinite dimensional Banach-Lie groups for instance).Q1) Can you provide examples of infinite dimensional Lie groups where the exponential map $L(G)\rightarrow G$ is locally surjective ?

Q2) If, in these examples, $L(G)$ admits a

topological basisin some sense, does it provide a formula like (*) ?

**The case of Schützenberger's factorization.-**

Below an example of an infinite dimensional Lie group and such a system of coordinates.

All definitions of : algebra (resp. large algebra) of a monoid, Lie algebra, enveloping algebra, used here are standard and can be taken e.g. from [1] and [2] (I can detail interactively on request).

Let $X$ be a set (of variables, or indeterminates, or an alphabet), $k$ a $\mathbb{Q}$-algebra and let $$ k\langle X\rangle, k\langle \langle X\rangle\rangle, \mathcal{L}_k\langle X\rangle, \mathcal{L}_k\langle\langle X\rangle\rangle $$ be respectively the free algebra (i.e. the algebra of noncommutative polynomials or the algebra of the free monoid $X^*$), the algebra of noncommutative formal power series (i.e. the large algebra of the free monoid $X^*$) [1], the free Lie algebra and the Lie algebra of Lie series [2]. We will use the natural pairing between $k\langle \langle X\rangle\rangle=k^{X^*}$ and $k\langle X\rangle=k^{(X^*)}$ given by the following sum on the words $$ \langle S|P\rangle=\sum_{w} coeff(S,w)coeff(P,w) $$ It is well known that $$ k\langle X\rangle=\mathcal{U}(\mathcal{L}_k\langle X\rangle)\ . $$ As such, it admits a structure of Hopf algebra $$ (k\langle X\rangle, conc, 1_{X^*}, \Delta_{shuffle},\epsilon,S) $$ $conc$ being the concatenation, $\Delta_{shuffle}$ being the dual law of the shuffle product, $\epsilon(P)=\langle P|1_{X^*}\rangle$ (constant term) and $S(a)=-a$ for all $a\in X$;

Every basis ($B=(b_i)_{i\in I};\ I$ totally ordered) of $\mathcal{L}_k\langle X\rangle$ (which is free, for all rings $k$) can be extended to a Poincaré-Birkhoff-Witt basis of $k\langle X\rangle$, parametrized by the multiindices of $\mathbb{N}^{(I)}$. The multi-index product is defined as follows. For every $\alpha\in \mathbb{N}^{(I)}$, we set

$$
B^\alpha=b_{i_1}^{\alpha_1}b_{i_2}^{\alpha_2}\cdots b_{i_m}^{\alpha_m}
$$
with $supp(\alpha)=\{i_1<i_2<\cdots i_m\}$. Now, if $B$ is multi-homogeneous (w.r.t. the $\mathbb{N}^{(X)}$-grading), so is $(B^\alpha)_{\alpha\in \mathbb{N}^{(I)}}$ and there is a unique family of polynomials $B_\alpha$ such that
$$
\langle B_\alpha|B^\beta \rangle=\delta_{\alpha,\beta}\qquad\qquad \mathrm{(Dual-Basis)}
$$

Now within the algebra of double series (whose support is $k^{X^*\otimes X^*}$ endowed with the law $shuffle\hat{\otimes} conc$, **Schützenberger** (see [3,4]) gave the beautiful formula
$$
(**)\ \sum_{w\in X^*}w\hat{\otimes} w=\prod_{i\in I}^{\rightarrow} e^{B_{e_i}\hat{\otimes} b_i}
$$
where $e_i$ are the irreducibles of the monoid $\mathbb{N}^{(I)}$ defined by $e_i(j)=\delta_{i,j}$ (in particular $B^{e_i}=b_i$).

This can be used to provide a system of local coordinates on the *Hausdorff group* [2]
$$
Haus_k(X)=\{e^L\}_{L\in \mathcal{L}_k\langle \langle X\rangle\rangle}=\{S\in k\langle \langle X\rangle\rangle|\epsilon(S)=1,\, \Delta_{shuffle}(S)=S\hat{\otimes}S\}
$$
because, in this case $S\otimes Id$ is compatible with the law of the double algebra and then, applying this operator to $(**)$, we get
$$
S=(S\hat{\otimes} Id)(\sum_{w\in X^*}w\hat{\otimes} w)=\prod_{i\in I}^{\rightarrow} e^{\langle S|B_{e_i} \rangle\,b_i}
$$
which is a *system of local coordinates* of the type $(*)$ for the group $Haus_k(X)$.

**Application to Riemann zeta functions** When one multiplies several zeta values
$$
\zeta(s)=\sum_{n\geq 1} \frac{1}{n^s}
$$
*multi-zeta values* do appear, they are defined by
$$
(***)\ \zeta(s_1,s_2,\cdots s_k)=\sum_{n_1>n_2>\cdots n_k\geq 1} \frac{1}{n_1^{s_1}n_2^{s_2}\cdots n_k^{s_k}}\ .
$$
When $s_1,s_2,\cdots s_k$ are integers, the link with the shuffle product is that the quantity $(***)$ converges when $s_1>1$ and, coding $(s_1,s_2,\cdots s_k)$ by the word (here $X=\{x_0,x_1\}$) $w=(x_0^{s_1-1}x_1x_0^{s_1-1}x_1\cdots x_0^{s_k-1}x_1)$ and recoding $(***)$ by $\tilde{\zeta}(w)=\zeta(s_1,s_2,\cdots s_k)$ one can prove that $\tilde{\zeta}$ can be extended uniquely as a shuffle character of $\mathbb{Q}\langle X\rangle$ satisfying
$\tilde{\zeta}(x_0)=\tilde{\zeta}(x_1)=0$ so that, applying (**) we get
$$
\tilde{\zeta}=(\tilde{\zeta}\hat{\otimes} Id)(\sum_{w\in X^*}w\hat{\otimes} w)=\prod_{i\in I}^{\rightarrow} e^{\tilde{\zeta}(B_{e_i})\,b_i}
$$
for every multihomogeneous basis $B$ of the free Lie algebra $\mathcal{L}_{\mathbb{Q}}\langle X\rangle$.

**Coda:** I guess (but have not set it technically yet) that, given a Lie $\mathfrak{g}$ algebra (finite or infinite dimensional), which is free as a $k$-module ($k$ is, as above, a $\mathbb{Q}$-algebra), given any ordered basis $B=(b_i)_{i\in I}$ of $\mathfrak{g}$, you can consider the convolution ($\ast$) algebra
$$
\mathcal{A}=span_k\{(B_\alpha)|\alpha\in \mathbb{N}^{(I)}\}\subset \mathcal{U}^*(\mathfrak{g})
$$
which is closed by $\ast$ and every character $\chi$ of $\mathcal{A}$ could be factorized in the same way
$$
\chi=\prod_{i\in I}^{\rightarrow} e^{\chi(B_{e_i})\,b_i}
$$
for the topology of pointwise convergence on $\mathcal{A}$ ($k$ being discrete and the notation of $B_{\alpha}$ being those of $\mathrm{(Dual-Basis)}$).

**References**

[1] Bourbaki, *Algebra*, Ch III par. 2, Springer

[2] Bourbaki, *Lie groups and Lie algebras*, Ch II, Springer

[3] Schützenberger, LITP Report

[4] Christophe Reutenauer, Free Lie Algebras, Oxford University Press.

[5] Georges Racinet, Séries génératrices non-commutatives de polyzetas et associateurs de Drinfeld, Thèse (2 Nov 2006)

someneigh. of 0 is a neigh. of 1? Does it mean something not only at 0? $\endgroup$ – YCor May 4 '15 at 18:086more comments