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.

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 ? (locally surjective means here an open map $L(G)\to G$)

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$7more comments