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The aim of this paper is to define and describe a family of manifolds and provide them with cluster structure starting from any simple Lie group. We give a description of them as the so called {\em cluster  ${\cal X}$\!-manifolds} and give a canonical maps of these manifolds to the group manifold compatible with the Poisson structure both on the group and on the manifolds. The manifolds are numerated by pairs of elements of a subsemigroup of the braid group. In particular one of these manifolds maps to the group birationally and thus provides it with canonical coordinates. 

We start with a small recollection of the definition and properties of cluster manifolds.

\section{Cluster manifolds.}

A {\em cluster seed} or just {\em seed} ${\mathbf J}$ is a quadruple $(J,J_0,\varepsilon, d)$, where $J$ is a finite set, $J_0 \subset J$ be its subset, $\varepsilon_{ij}, i,j \in J$ is a matrix, such that $\varepsilon_{ij} \in {\mathbb Z}$ unless $i,j \in J_0$ and $d_i, i \in J$ is a set of integers, such that the matrix $\tilde{\varepsilon}_{ij}=\varepsilon_{ij}d_j$ is skew-symmetric. The elements of the set $J$ are called {\em vertices}, the elements of $J_0$ are called {\em frozen vertices}\footnote{Fomin and Zelevinsky used the term {\em tropical} instead of {\em frozen}, which we have changed in order not to make confusion with tropical mathematics.}. The matrix $\varepsilon$ is sometimes called {\em cluster function} and the numbers $\{d_i\}$ are called {\em multipliers}. We omit $\{d_i\}$ if all of them are equal to one, and therefore the matrix $\varepsilon$ is skew-symmetric, and we omit the set $J_0$ if it is empty.

The seed ${\mathbf I}=(I,I_0,\eta,c)$ is called a {\em subseed} of the seed ${\mathbf J}=(J,J_0,\varepsilon, d)$ if $I\subset J$, $I_0\subset J_0$ and the functions $\eta$ and $c$ are the restrictions of $\varepsilon$ and $d$, respectively. In this case we denote ${\mathbf I}$ by ${\mathbf J}\vert_I$.

Fix a semifield ${\mathbb K}$. Then for such seed ${\mathbf J}$ we can associate a Poisson algebraic manifold ${\mathcal X}_{\mathbf J}({\mathbb K})=({\mathbb K}^\times)^J$ with the Poisson structure given by $\{x_i,x_j\}=\tilde{\varepsilon}_{ij}x_ix_j$, where $\{x_i| i\in J\}$ are the standard coordinates on the ${\mathbb K}^\times$ factors. We shall call this manifold the {\em seed ${\cal X}$\!-manifold} and omit the argument ${\mathbb K}$ if the semifield is clear from the context or inessential.

Let ${\mathbf J}=(J,J_0,\varepsilon, d)$ and ${\mathbf I}=(I,I_0,\eta, c)$ be two seeds, and $k\in J$. A {\em mutation at the vertex $k$} is an isomorphism $\mu_k:J\rightarrow I$ satisfying the following conditions:
\begin{trivlist}
\item[\ 1.] $\mu_k(J_0)=I_0,$ 
\item[\ 2.] $c_{\mu_k(i)}=d_i,$
\item[\ 3.] $\eta_{\mu_k(i)\mu_k(j)}=
\left\{ \begin{array}{lll} -\varepsilon_{ij} & \mbox{ if } & i=k \mbox{ or } j=k\\
\varepsilon_{ij} + \varepsilon_{ik}\max(0,\varepsilon_{kj}) &\mbox{ if }& \varepsilon_{ik}\geq 0\\
\varepsilon_{ij} + \varepsilon_{ik}\max(0,-\varepsilon_{kj}) &\mbox{ if }& \varepsilon_{ik}<0
\end{array}\right.$
\end{trivlist}

A {\em symmetry} of a seed ${\mathbf J}=(J,J_0,\varepsilon, d)$ is an automorphism $\sigma$ of the set $J$ preserving the subset $J_0$, the matrix $\varepsilon$ and the numbers $d_i$. In other words it satisfies the conditions:
\begin{trivlist}
\item[ \ 1. ] $ \sigma(J_0)=I_0,$
\item[ \ 2. ] $ c_{\sigma(i)}=d_i,$ 
\item[ \ 3. ] $ \eta_{\mu_k(i)\mu_k(j)}=\varepsilon_{ij}.$
\end{trivlist}

Symmetries and mutations induce maps between respective seed ${\cal X}$-manifolds denoted by the same symbols $\mu_k$ and $\sigma$ and given by the formulae

$$x_{\sigma(i)}=x_i$$
and 
$$
x_{\mu_k(i)} = \left\{\begin{array}{lll} x_k^{-1}& \mbox{ if } & i=k \\
    x_i(1+x_k)^{\varepsilon_{ik}} & \mbox{ if } & \varepsilon_{ik}\geq 0 \mbox{ and } i\neq k \\
 x_i(1+(x_k)^{-1})^{\varepsilon_{ik}} & \mbox{ if } & \varepsilon_{ik}\leq 0 \mbox{ and } i\neq k
\end{array} \right..
$$

Symmetries and mutations of cluster $\mathcal X$-manifolds are Poisson maps. Moreover, this property and the mutation rules for coordinates imply the mutation rules for the matrix $\varepsilon$.

A {\em cluster transformation} between two seeds (and between two seed
${\cal X}$-ma\-ni\-folds) is a composition of symmetries and mutations. If the source and the target of a cluster transformation coincide we call this map a {\em cluster automorphism}. Two seeds are called {\em equivalent} if they are related by a cluster transformation. The equivalence class of a seed ${\mathbf J}$ is denoted by $|{\mathbf J}|$

Remark, that a cluster transformation may be trivial on the level of seeds, and nontrivial on the level of seed ${\cal X}$-manifolds.

A {\em cluster ${\cal X}$-manifold} is a union of all seed ${\cal X}$-manifolds related to the given seed ${\mathbf J}$ by cluster transformation and is denoted by ${\cal X}^*_{|{\mathbf J}|}$ since cluster ${\cal X}$-manifolds corresponding to equivalent seeds are isomorphic. Every particular seed ${\cal X}$-manifold provide our cluster manifold with almost everywhere defined coordinate system, called {\em cluster coordinates}. 

The cluster transformations have the following basic properties:
\begin{trivlist}
\item[\ 1.] Every seed is related to other seeds by exactly $\sharp (J\backslash J_0)$ mutations.
\item[\ 2.] Cluster transformations form a groupoid. In particular the inverse of a mutation is a mutation. Cluster automorphisms form a group. The groups of cluster automorphisms of equivalent seeds are isomorphic.
\item[\ 3.] Cluster transformations preserve the Poisson structure. In  particular a cluster ${\cal X}$-manifold has a canonical Poisson structure and the automorphism group of this manifold acts on it by Poisson transformations.
\item[\ 4.] Cluster transformations are given by rational functions with positive real coefficients.
\end{trivlist}

\paragraph{Restricted product.} We shall need an operation on cluster seeds called {\em restricted product} given by the following rule. Let $\mathbf J = (J,J_0,\varepsilon,d)$ and $\mathbf I= (I,I_0,\zeta,c)$ be two seeds and let $L$ be a set embedded into both $I_0$ and $J_0$ in a such a way that for any $i,j \in L$ we have $\varepsilon_{ij}+\eta_{ij}\in \mathbb Z$ and $c(i)=d(i)$. Then the restricted product of  $\mathbf J$ and $\mathbf I$ is a seed $\mathbf K = (K,K_0,\zeta,b)$, such that $K=I \cup_L J$, $K_0 = (I_0\cup_L J_0) \backslash L$ and 
$$\zeta_{ij}=\left\{\begin{array}{lcl}
               0 &\mbox{if}& i\in I\backslash L \mbox{ and } j\in J\backslash L\\
               0 &\mbox{if}& i\in J\backslash L \mbox{ and } j\in I\backslash L\\
               \eta_{ij}&\mbox{if}&i\in I\backslash L \mbox{ or } j\in I\backslash L\\
               \varepsilon_{ij}&\mbox{if}&i\in J\backslash L \mbox{ or } j\in J\backslash L\\
               \eta_{ij}+\varepsilon_{ij}&\mbox{if}&i,j\in L
                    \end{array}
             \right.
$$
This operation induces the map between the corresponding seed ${\cal X}$-manifolds:${\cal X}_{\mathbf J} \times {\cal X}_{\mathbf I} \to {\cal X}_{\mathbf K}$ given by the rule:
\begin{equation}\label{restrx}
z_i = \left\{\begin{array}{lcl}x_i&\mbox{if}&i\in I\backslash L\\
                               y_i&\mbox{if}&i\in J\backslash L\\
                               x_iy_i&\mbox{if}&i\in L 
              \end{array}
       \right.
\end{equation}

It is easy to check, that the restricted product is an associative operation, it respects the Poisson structure and that it commutes with cluster transformations, thus is defined for the cluster ${\cal X}$-manifolds, and not only for the seeds.



\section{Construction of the manifolds.}
\paragraph{Example.}

Consider the map $ev_0({\mathbb C}^\times)^3\rightarrow PSL(2,{\mathbb C})$ given by
$$
ev_x:(x_0,x_1,x_2) \mapsto 
$$
$$ \mapsto
\begin{pmatrix}x_0^{1/2}&0\\0&x_0^{-1/2}\end{pmatrix}
\begin{pmatrix}1&0\\1&1\end{pmatrix}
\begin{pmatrix}x_1^{1/2}&0\\0&x_1^{-1/2}\end{pmatrix}
\begin{pmatrix}1&1\\0&1\end{pmatrix}
\begin{pmatrix}x_2^{1/2}&0\\0&x_2^{-1/2}\end{pmatrix}=
$$
$$
=\begin{pmatrix}x_0^{1/2}x_1^{1/2}x_2^{1/2}+x_0^{1/2}x_1^{-1/2}x_2^{1/2}&x_0^{1/2}x_1^{-1/2}x_2^{-1/2}\\x_0^{-1/2}x_1^{-1/2}x_2^{1/2}&x_0^{-1/2}x_1^{-1/2}x_2^{-1/2}\end{pmatrix}
$$
Consider also another map $ev_y({\mathbb C}^\times)^3\rightarrow PSL(2,{\mathbb C})$
$$
ev_y:(y_0,y_1,y_2) \mapsto
$$
$$ \mapsto 
\begin{pmatrix}y_0^{1/2}&0\\0&y_0^{-1/2}\end{pmatrix}
\begin{pmatrix}1&1\\0&1\end{pmatrix}
\begin{pmatrix}y_1^{1/2}&0\\0&y_1^{-1/2}\end{pmatrix}
\begin{pmatrix}1&0\\1&1\end{pmatrix}
\begin{pmatrix}y_2^{1/2}&0\\0&y_2^{-1/2}\end{pmatrix}=
$$
$$
=\begin{pmatrix}y_0^{1/2}y_1^{1/2}y_2^{1/2}&y_0^{1/2}y_1^{1/2}y_2^{-1/2}\\
                y_0^{-1/2}y_1^{1/2}y_2^{1/2}&y_0^{-1/2}y_1^{-1/2}y_2^{-1/2}+y_0^{-1/2}y_1^{1/2}y_2^{-1/2}
\end{pmatrix}
$$
One can see that these maps enjoy the following properties:
\begin{enumerate}
\item The maps $ev_0$ and $ev_1$ are single valued on $({\mathbb  C}^\times)^3$ since the matrix $\begin{pmatrix}x_1^{1/2}&0\\0&x_1^{-1/2}\end{pmatrix}$ is well defined as an element of the group $PSL(2,{\mathbb C})$
\item $ev_x(x_0,x_1,x_2)=ev_y(x_0(1+x_1^{-1})^{-1},x_1^{-1},x_2(1+x_1^{-1})^{-1})$
\item $ev_y(y_0,y_1,y_2)=ev_x(y_0(1+y_1),y_1^{-1},y_2(1+y_1))$
\item Both maps are birational isomorphisms.
\item The standard Drinfeld-Jimbo Poisson-Lie bracket on the group $PSL(2,{\mathbb C})$ reads as
$$\{x_0,x_1\}=x_0x_1;\qquad  \{x_1,x_2\}=x_1x_2;\ \quad\{x_0,x_2\}=0;$$
$$\{y_0,y_1\}=-y_0y_1; \quad\{y_1,y_2\}=-y_1y_2;\quad\{y_0,y_1\}=0;$$ 
\end{enumerate}

One thus can see, that $ev_x$ and $ev_y$ provide the group manifold $PSL(2,{\mathbb C})$ with two coordinate systems. The transition between this coordinates is given by a mutation and thus the union of the images of $ev_x$ and $ev_y$ is a cluster manifold corresponding to two equivalent seeds $(\{0,1,2\},\{0,2\},\varepsilon)$ and $(\{0,1,2\},\{0,2\},\eta)$, where $\varepsilon = -\eta =\begin{pmatrix}0&-1&0\\1&0&1\\0&-1&0\end{pmatrix}$ (columns correspond to the first index).

One can also consider the maps $ev_2, ev_3: ({\mathbb C}^\times)^2 \rightarrow PSL(2,{\mathbb C})$ and $ev_4:{\mathbb C}^\times \rightarrow PSL(2,{\mathbb C})$ given by:
$$
ev_z:(z_0,z_1) \mapsto
\begin{pmatrix}z_0^{1/2}&0\\0&z_0^{-1/2}\end{pmatrix}
\begin{pmatrix}1&1\\0&1\end{pmatrix}
\begin{pmatrix}z_1^{1/2}&0\\0&z_1^{-1/2}\end{pmatrix}=
$$
$$
=\begin{pmatrix}z_0^{1/2}z_1^{1/2}&z_0^{1/2}z_1^{-1/2}\\0&z_0^{-1/2}z_1^{-1/2}\end{pmatrix}
$$
$$
ev_w:(w_0,w_1) \mapsto 
\begin{pmatrix}w_0^{1/2}&0\\0&w_0^{-1/2}\end{pmatrix}
\begin{pmatrix}1&0\\1&1\end{pmatrix}
\begin{pmatrix}w_1^{1/2}&0\\0&w_1^{-1/2}\end{pmatrix}
$$
$$
=\begin{pmatrix}w_0^{1/2}w_1^{1/2}&w_0^{1/2}w_1^{-1/2}\\0&w_0^{-1/2}w_1^{-1/2}\end{pmatrix}
$$
$$
ev_t:(x_4) \mapsto 
\begin{pmatrix}t_0^{1/2}&0\\0&t_0^{-1/2}\end{pmatrix}
$$

and satisfying the properties:
\begin{enumerate}
\addtocounter{enumi}{5}
\item The images of $ev_z$,$ev_w$,$ev_t$ and the union of the images of $ev_x$ and $ev_y$ are pairwise disjoint. Their union covers the whole group but a codimension two.
\item The images of $ev_z$,$ev_w$,$ev_t$ are Poisson submanifolds. The Poisson bracket is given by:
$$\{z_0,z_1\}=z_0z_1;\qquad  \{w_0,w_1\}=w_0w_1;$$
\end{enumerate}


Our aim now is a generalization of this picture in two directions.  We consider groups of higher rank and simultaneously we construct manifolds of not necessarily injects into groups, but possibly maps onto them respecting the Poisson structure.

\paragraph{Fixing notations.}

Let $G$ be a semisimple adjoint Lie group of rank $r$ over a field ${\mathbb K}$, ${\mathfrak g}$ be the corresponding Lie algebra, $W$ be its Weyl group, $\Pi$ be the set of its positive simple roots, $C_{\alpha\beta}$, where $\alpha,\beta \in \Pi$ be its Cartan matrix and $d^\alpha, \alpha \in \Pi$ be the multipliers, i.e., the numbers satisfying the requirement that $\tilde{C}_\alpha\beta=C_{\alpha\beta}d^\alpha$ is symmetric. 

Let $G$ be equipped with the standard Drinfeld-Jimbo Poisson-Lie group structure.

Let $\Pi^-$ be the set of negative simple roots and let ${\mathfrak W}$ be a semigroup freely generated by $\Pi$ and $\Pi^-$. Any element $D$ of ${\mathfrak W}$ is thus a word $\mu_1\cdots \mu_{l(D)}$ of the letters from the alphabet $\Pi\cup\Pi^-$, where $l(D)$ is its length. For $\alpha \in \Pi$ we shall denote by $\bar{\alpha}$ the opposite element from $\Pi^-$.

Let now ${\mathfrak B}$ be the quotient of the semigroup ${\mathfrak W}$ by:
\begin{equation}\label{c1}
\alpha\bar{\beta}=\bar{\beta}\alpha
\end{equation}
and  the braid relations:
\begin{equation}\label{c2}
\begin{array}{c}\alpha(\beta\alpha)^{-\tilde{C}_{\alpha\beta}} = (\beta\alpha)^{-\tilde{C}_{\alpha\beta}}\beta;\quad \bar{\alpha}(\bar{\beta}\bar{\alpha})^{-\tilde{C}_{\alpha\beta}} = (\bar{\beta}\bar{\alpha})^{-\tilde{C}_{\alpha\beta}}\bar{\beta}; \mbox{ for } \tilde{C}_{\alpha\beta} \mbox{ odd}\\
(\beta\alpha)^{-\tilde{C}_{\alpha\beta}}=(\alpha\beta)^{-\tilde{C}_{\alpha\beta}};\qquad (\bar{\beta}\bar{\alpha})^{-\tilde{C}_{\alpha\beta}}=(\bar{\alpha}\bar{\beta})^{-\tilde{C}_{\alpha\beta}}  \mbox{ for } \tilde{C}_{\alpha\beta} \mbox{ even}\end{array}
\end{equation}
${\mathfrak B}$ is obviously a subsemigroup of the braid group. We shall denote by $p:{\mathfrak W}\rightarrow{\mathfrak B}$ the canonical projection.
  
Another semigroup appropriate in our context is the further quotient of ${\mathfrak B}$ by the relations:
\begin{equation}
\alpha^2 = \alpha; \quad \bar{\alpha}^2=\bar{\alpha}.
\end{equation}
denoted by ${\mathfrak H}$. This semigroup is isomorphic as a set to the square of the Weyl group of the group $G$. Analogously with the Weyl groups we shall call an element of ${\mathfrak W}$ or of ${\mathfrak B}$ {\em reduced} if its length is minimal among the elements having the same image in ${\mathfrak H}$.

\paragraph{Required properties.}
Now we are going define a cluster ${\cal X}$-manifold ${\cal X}^*(B)$ to any element $B\in {\mathfrak B}$. For this purpose we are going to define a seed $\mathbf J(D)=(J(D),J_0(D),\varepsilon(D),d(D))$ for any $D \in {\mathfrak W}$ in such a way that the seeds, corresponding to different elements of $p^{-1}(B)$ are related by cluster transformations. Moreover we are going to define the Poisson map $ev: {\cal X}^*(B) \rightarrow G$ and the maps $m:{\cal X}^*(B_1)\times{\cal X}^*(B_2)\rightarrow {\cal X}^*(B_1B_2)$. The correspondence $B \mapsto {\cal X}^*(B)$, and the maps $m$ and $ev$ are to satisfy the properties:
\begin{trivlist}
\item[\ 1.] $ev$ is a Poisson map. 
\item[\ 2.] $m$ is a restricted product of cluster manifolds (and thus is also a Poisson map).
\item[\ 3.] $ev(m({\cal X}^*(B_1),{\cal X}^*(B_2)))=ev({\cal X}^*(B_1))ev({\cal X}^*(B_2))$.\label{mevcomm}
\item[\ 4.] If $B$ is reduced, then the map $ev$ is an embedding.
\item[\ 5.] If $B$ projects onto the longest element of ${\mathfrak H}$ then the image of $ev$ is Zarisky dense in $G$.
\end{trivlist}


\paragraph{The set of vertices.} We need to define the set $J(D)$ and the subset $J_0(D)$ starting from a word $D \in {\mathfrak W}$. Fix a root $\alpha\in \Pi$ and denote by $n^\alpha(D)$ the number of occurrences of $\alpha$ and $\bar{\alpha}$ in the word $D$.  Denote by $J^\alpha(D)$ the set $\{(^\alpha_i)|\alpha \in \Pi, 0\leq i \leq n^\alpha\}$ and denote by $J^\alpha_0(D)$ the set $\{(^\alpha_0)\}\cup \{(^\alpha_{n_\alpha})\}$. Now denote by $J(D)$ (resp. $J_0(D)$) the disjoint union of $J^\alpha(D)$ (resp. $J^\alpha_0(D)$) for all $\alpha \in \Pi$.

Remark, that if a root $\alpha$ does not enter the word $D$ then $J^\alpha_0(D)=J^\alpha(D)=\{(^\alpha_0)\}$ is a one element set. 

One can imagine the elements of the set $J(D)$ as associated to the intervals between walls made by $\alpha$, $\bar{\alpha}$ or the ends of the word $D$ for some root $\alpha$. If at least one wall is just the end $D$, the corresponding element of $J(D)$ belongs to $J_0(D)$.

We shall denote these elements by braces connecting the walls with the name of the corresponding elements in the middle.

\noindent  {\em Example.} Let $r=\mathrm{rk}\ G=3$ and $\Pi=\{\alpha,\beta,\gamma\}$. Take $D=\alpha\bar{\beta}\bar{\alpha}\bar{\alpha}\beta$. Then 
$$n^\alpha(D)=3,\  n^\beta(D)=2,\ n^\gamma(D)=0$$  
$$J(D) = \{(^\alpha_0), (^\alpha_1),(^\alpha_2),(^\alpha_3),(^\beta_0),(^\beta_1),(^\beta_2),(^\gamma_0)\}, $$
$$J_0(D) = \{(^\alpha_0), (^\alpha_3),(^\beta_0),(^\beta_2),(^\gamma_0)\}, $$
In brace notations the set $J(D)$ can be shown as:
$$ \underbrace{
\lefteqn{\overbrace{\phantom{\hat{\beta}}\hspace{9ex}}^{(_0^\beta)}\overbrace{\phantom{\hat{\beta}}\hspace{16.7ex}}^{(_1^\beta)}\overbrace{\phantom{\hat{\beta}}}^{(_2^\beta)}}
\lefteqn{\underbrace{\phantom{y}}_{(_0^\alpha)}\underbrace{\phantom{y}\hspace{4.5ex}}_{(_1^\alpha)}\underbrace{\phantom{y}\hspace{10.5ex}}_{(_2^\alpha)}\underbrace{\phantom{y}\hspace{9.5ex}}_{(_3^\alpha)}}  
\;\;\quad\alpha\qquad\bar{\beta}\qquad\bar{\alpha}\qquad\bar{\alpha}\qquad\beta\quad\;\;\; }_{(_0^\gamma)}
$$

\paragraph{Map to the group.}
Denote by ${\cal X}_{\mathbf J(D)}$ the manifold $({\mathbb K}^\times)^{J(D)}$ and by $\{x^\alpha_i\}$ the natural coordinates on it. Now define the map $ev: {\cal X}_{\mathbf J(D)}\rightarrow G$.

In order to define it we are going to construct a sequence of group elements, each of which is either a constant or depends on just one coordinate of ${\cal X}_{\mathbf J(D)} $. The product of the elements of the sequence will give the desired map.

Let $f_\alpha,h_\alpha,e_\alpha$ be the Cartan generators of the Lie algebra of the group $G$. Let $h^\alpha$ be another basis of the Cartan subalgebra defined by the property:
$$
[h^\alpha,e_\beta]=\delta^\alpha_\beta e_j,\quad [h^\alpha,f_\beta]=-\delta^\alpha_\beta f_\beta.
$$
This basis is obviously related to the basis $h_\alpha$ by the action of the Cartan matrix $\sum_\beta C_{\alpha \beta} h^\beta=h_\alpha$. 

Now let us introduce the group elements ${\mathbf E}^\alpha=\exp e_\alpha$, ${\mathbf F}^\alpha=\exp f_\alpha $ and $H^\alpha(x) = \exp(\log(x)h^\alpha)$. It is obvious, that $H^\alpha(x)H^\alpha(y)=H^\alpha(xy)$.

Replace the letters in $D$ by the group elements using the rule $\alpha \rightarrow {\mathbf E}^\alpha, \bar{\alpha}\rightarrow {\mathbf F}^\alpha$. For $\alpha \in \Pi$ and for any $(^\alpha_i)\in J(D)$ insert $H^\alpha(x^\alpha_i)$ somewhere between the corresponding walls.  The choice in placing every $H$'s is unessential since it commutes with all $E$'s and $F$'s unless they are marked by the same root. 

In other words the sequence of group elements is defined by the following requirements:
\begin{trivlist}
\item[\ $\bullet$] The sequence of ${\mathbf E}$'s and ${\mathbf F}$'s reproduce the sequence of the letters in the word $D$.
\item[\ $\bullet$] Any $H$ depends on its own variable $x^\alpha_i$.
\item[\ $\bullet$] There is at least one ${\mathbf E}^\alpha$ or ${\mathbf F}^\alpha$ between any two $H^\alpha$'s.
\item[\ $\bullet$] The number of $H^\alpha$'s is equal to the total number of ${\mathbf E}^\alpha$'s and  ${\mathbf F}^\alpha$'s plus one.
\end{trivlist}

\noindent {\em Example:} Using once more the word $\alpha\bar{\beta}\bar{\alpha}\bar{\alpha}\beta$, it is mapped by $ev$ to 
$$
H^\alpha(x^\alpha_0)H^\beta(x^\beta_0){\mathbf E}^\alpha H^\alpha(x^\alpha_1) {\mathbf F}^\beta H^\beta(x^\beta_1) {\mathbf F}^\alpha  H^\alpha(x^\alpha_2){\mathbf F}^\alpha {\mathbf E}^\beta H^\alpha(x^\alpha_3)H^\beta(x^\beta_2)H^\gamma(x^\gamma_0)
$$



\paragraph{Construction of $\varepsilon$ and $d$.}.

Define the multipliers just by the rule $d_{(^\alpha_i)}(D)=d^\alpha$

In order to give an explicit formula for the matrix $\varepsilon_{(^\alpha_i)(^\beta_j)}$ introduce some more notations. Let $n^\alpha(k)$ be the number of letters $\alpha$ or $\bar{\alpha}$ among the first $k-1$ letters of the word $D$. Let $\mu_k$ be the $k$-th letter of $D$ and let $sgn(\mu)$ for $\mu \in \Pi \cup -\Pi$ be $+1$ if $\mu \in \Pi$ and $-1$ otherwise. Let finally $|\mu|= sgn(\mu)\mu\in \Pi$.
$$
P=\sum_{(^\alpha_i)(^\beta_j)}\tilde{\varepsilon}_{(^\alpha_i)(^\beta_j)}\frac{\partial}{\partial  \log x^\alpha_i}\wedge\frac{\partial}{\partial \log x^\beta_j} = 
$$
$$
=\frac{1}{2}\sum_{k=1}^{l(D)}\left(\frac{\partial}{\partial \log x^{|\mu_k|}_{n^{|\mu_k|}(k)}} -\frac{\partial}{\partial \log x^{|\mu_k|}_{n^{|\mu_k|}(k)+1}}\right)\wedge sgn(\mu_k)\sum_\alpha \tilde{C}_{|\mu(k)|\alpha}\frac{\partial}{\partial \log x^\alpha_{n^\alpha(k)}}.
$$

{\em Example.} For the word $D=\alpha\bar{\beta}\bar{\alpha}\bar{\alpha}\beta$ considered above one can easily compute that all nonvanishing elements of $\varepsilon$ are given by:
$$
\tilde{\varepsilon}_{(^\alpha_0)(^\alpha_1)}=
-\tilde{\varepsilon}_{(^\alpha_1)(^\alpha_2)}=
-\tilde{\varepsilon}_{(^\alpha_2)(^\alpha_3)}=\tilde{C}_{\alpha \alpha}/2
$$
$$
-\tilde{\varepsilon}_{(^\beta_0)(^\beta_1)}=
\tilde{\varepsilon}_{(^\beta_1)(^\beta_2)}=\tilde{C}_{\beta \beta}/2
$$
$$
\tilde{\varepsilon}_{(^\alpha_0)(^\beta_0)}=
\tilde{\varepsilon}_{(^\alpha_3)(^\beta_2)}=\tilde{C}_{\alpha \beta}/2
$$
$$
\tilde{\varepsilon}_{(^\alpha_0)(^\gamma_0)}=
-\tilde{\varepsilon}_{(^\alpha_1)(^\gamma_0)}/2=
\tilde{\varepsilon}_{(^\alpha_3)(^\gamma_0)}=\tilde{C}_{\alpha \gamma}/2
$$
$$
-\tilde{\varepsilon}_{(^\beta_0)(^\gamma_0)}=
\tilde{\varepsilon}_{(^\beta_1)(^\gamma_0)}/2=
-\tilde{\varepsilon}_{(^\beta_2)(^\gamma_0)}=\tilde{C}_{\beta \gamma}/2
$$
Now we are going to show, that the association of the seed ${\cal X}$-manifold to the words is compatible with natural operations on the words. 
\begin{proposition}
The map $ev$ is a Poisson map and there exist the canonical Poisson birational maps commuting with the map $ev$:
\begin{enumerate}
\item ${\cal X}_{\mathbf J(A\bar{\alpha}\beta B)} \rightarrow {\cal
  X}_{\mathbf J(A\beta\bar{\alpha} B)}$ if $\alpha \neq \beta$
\item ${\cal X}_{\mathbf J(A\alpha\beta B)} \rightarrow {\cal X}_{\mathbf J(A\beta\alpha B)}$ and ${\cal X}_{\mathbf J(A\bar{\alpha}\bar{\beta} B)} \rightarrow {\cal X}_{\mathbf J(A\bar{\beta}\bar{\alpha} B)}$\\ if $C_{\alpha\beta}=0$.
\item ${\cal X}_{\mathbf J(A\alpha\beta\alpha B)} \rightarrow {\cal X}_{\mathbf J(A\beta\alpha\beta B)}$ and ${\cal X}_{\mathbf J(A\bar{\alpha}\bar{\beta}\bar{\alpha} B)} \rightarrow {\cal X}_{\mathbf J(A\bar{\beta}\bar{\alpha}\bar{\beta} B)}$ \\ if $C_{\alpha\beta}=-1$.
\item ${\cal X}_{\mathbf J(A\alpha\beta\alpha\beta B)} \rightarrow {\cal
  X}_{\mathbf J(A\beta\alpha\beta\alpha B)}$ and ${\cal X}_{\mathbf J(A\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}  B)} \rightarrow {\cal X}_{\mathbf J(A\bar{\beta}\bar{\alpha}\bar{\beta}\bar{\alpha} B)}$ \\ if $C_{\alpha\beta}=-2$.
\item  ${\cal X}_{\mathbf J(A\alpha\beta\alpha\beta\alpha\beta B)} \rightarrow {\cal X}_{\mathbf J(A\beta\alpha\beta\alpha\beta\alpha B)}$ and ${\cal X}_{\mathbf J(A\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta} B)} \rightarrow {\cal X}_{\mathbf J(A\bar{\beta}\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}\bar{\alpha} B)}$\\ if $C_{\alpha\beta}=-3$.
\item ${\cal X}_{\mathbf J(A\alpha\alpha B)} \rightarrow {\cal X}_{\mathbf J(A\alpha B)}$
and ${\cal X}_{\mathbf J(A\bar{\alpha}\bar{\alpha}B)} \rightarrow {\cal X}_{\mathbf J(A\bar{\alpha}B)}$
\item ${\cal X}_{\mathbf J(A)}\times {\cal X}_{\mathbf J(B)}\to {\cal X}_{\mathbf J(AB)}$
\end{enumerate}
Here $A,B$ are arbitrary words and $\alpha,\beta\in \Pi$.

The maps 1,2 are isomorphisms. The maps 3,4,5 are cluster transformations. The maps $6$ are compositions of a cluster transformation and a projection. The map 7 is a restricted product. 
\end{proposition}
  
In order to prove the proposition let us first construct and prove the properties of the map 7. Once it is done it is sufficient to construct the canonical maps for the shortest word where this map is defined, and then extend it using the multiplicativity property 7 for the general case. 

Let $D=AB$. Define the map $:{\cal X}_{\mathbf J(A)}\times{\cal X}_{\mathbf J(B)}\rightarrow {\cal X}_{\mathbf J(AB)}$. Let $\{x_i^\alpha\}$, $\{y_i^\alpha\}$ and $\{z_i^\alpha\}$ be the coordinates on ${\mathcal X}_{\mathbf J(A)}$, ${\mathcal X}_{\mathbf J(B)}$ and ${\mathcal X}_{\mathbf J(AB)}$, respectively. Then the map is defined by the formula:
$$
z^\alpha_i = \left\{\begin{array}{lcl}
             x^\alpha_i &\mbox{if} & i<n^\alpha(A)\\
             x^\alpha_{n^\alpha(D_1)}y^\alpha_0 &\mbox{if} & i=n^\alpha(B)\\
             y^\alpha_{i+n^\alpha(D_1)} &\mbox{if} & i>n^\alpha(AB)
             \end{array}
             \right.
$$

Just by construction this map is compatible with the map to the group $ev$ and given by the same formulae as the ones for the restricted product (\ref{restrx}). 
 
The maps 1-6 follow from the identities between the generators $\mathbf E^\alpha,H^\alpha(x),\mathbf F^\alpha$
\begin{trivlist}
\item[\ $\underline{\alpha\alpha\to\alpha}:$] $ {\mathbf E}^\alpha H^\alpha(x){\mathbf E}^\alpha = H^\alpha(1+x) {\mathbf E}^\alpha H^\alpha(1+x^{-1})^{-1} $
\item[\ $\underline{\alpha\beta\to\beta\alpha, C_{\alpha\beta}=0}:$] $\mathbf E^\alpha \mathbf E^\beta = \mathbf E^\beta \mathbf E^\alpha$
\item[\ $\underline{\alpha\beta\alpha\to\beta\alpha\beta, C_{\alpha\beta}=-1}:$]$\mathbf E^\alpha H^\alpha(x) \mathbf E^\beta \mathbf E^\alpha = \\
= H^\alpha(1+x)H^\beta(1+x^{-1})^{-1} \mathbf E^\beta H^\alpha(x)^{-1}
\mathbf E^\alpha \mathbf E^\beta H^\alpha(1+x^{-1})^{-1} H^\beta(1+x)$ 
\item[\ $\underline{\alpha\beta\alpha\beta\to\beta\alpha\beta\alpha, C_{\alpha\beta}=-2}:$]
$\mathbf E^\alpha \mathbf E^\beta H^\alpha(x) H^\beta(y) \mathbf E^\alpha \mathbf E^\beta = \\ =
 H^\alpha\left(\frac{1+x+2xy+xy^2}{1+x+xy}\right)H^\beta\left(\frac{xy^2}{1+x+2xy+xy^2}\right)\mathbf E^\beta \mathbf E^\alpha H^\alpha\left(\frac{y}{1+x+2xy+xy^2}\right) \times \\ \times H^\beta\left(\frac{(1+x+xy)^2}{xy^2}\right)\mathbf E^\beta \mathbf E^\alpha 
H^\alpha\left(1+x+xy\right)H^\beta\left(\frac{x(1+x+2xy+xy^2)}{(1+x+xy)^2}\right)$,
\item[\ $\underline{\bar{\alpha}\alpha\to\alpha\bar{\alpha}}:$] $\mathbf F_\alpha H^\alpha(x) \mathbf E^\alpha =\\= \left(\prod\limits_{\beta \neq  \alpha} H^\beta(1+x)^{-C_{\alpha\beta}}\right) H^\alpha(1+x^{-1})^{-1}  \mathbf E^\alpha H^\alpha(x^{-1})\mathbf F^\alpha H^\alpha(1+x^{-1})^{-1}$ 
\item[\ $\underline{\bar{\alpha}\beta\to\beta\bar{\alpha}}:$] $\mathbf F^\alpha \mathbf E^\beta =  \mathbf E^\beta \mathbf F^\alpha$ for $\alpha \neq \beta$.
\item[\ $\underline{\bar{\alpha}\bar{\alpha}\to\bar{\alpha}}:$] $ {\mathbf F}^\alpha H^\alpha(x){\mathbf F}^\alpha = H^\alpha(1+x^{-1})^{-1} {\mathbf F}^\alpha H^\alpha(1+x)$
\item[\ $\underline{\bar{\alpha}\bar{\beta}\to\bar{\beta}\bar{\alpha}, C_{\alpha\beta=0}}:$] $\mathbf F^\alpha \mathbf F^\beta = \mathbf F^\beta \mathbf F^\alpha$,
\item[\ $\underline{\bar{\alpha}\bar{\beta}\bar{\alpha}\to\bar{\beta}\bar{\alpha}\bar{\beta}, C_{\alpha\beta}=-1}:$] $ \mathbf F^\alpha H^\alpha(x) \mathbf F^\beta \mathbf F_\alpha = \\
= H^\alpha(1+x^{-1})^{-1}H_\beta(1+x) \mathbf F^\beta H^\alpha(x)^{-1}
\mathbf F^\alpha \mathbf F^\beta H^\alpha(1+x) H^\beta(1+x^{-1})^{-1} 
$,
\item
\end{trivlist}

The map 1 is a corollary of the obvious property $\underline{\bar{\alpha}\beta\to\beta\bar{\alpha}}$. The map 2 follows from $\underline{\alpha\beta\to\beta\alpha}$ and $\underline{\bar{\alpha}\bar{\beta}\to\bar{\beta}\bar{\alpha}}$. The map $6$ follows from $\underline{\alpha\alpha\to\alpha}$ and $\underline{\bar{\alpha}\bar{\alpha}\to\bar{\alpha}}$. Each of these maps is obviously a composition of a mutation $\mu_{(^\alpha_{n^\alpha(A)+1})}$ and the projection along the corresponding coordinate. The maps $3$ follows from  $\underline{\alpha\beta\alpha\to\beta\alpha\beta}$ and $\underline{\bar{\alpha}\bar{\beta}\bar{\alpha}\to\bar{\beta}\bar{\alpha}\bar{\beta}}$ and are given by the mutations  $\mu_{(^\alpha_{n^\alpha(A)+1})}$. The maps $4$ follow form $\underline{\alpha\beta\alpha\beta\to\beta\alpha\beta\alpha}$ and $\underline{\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}\to\bar{\beta}\bar{\alpha}\bar{\beta}\bar{\alpha}}$ and can be easily shown to be given by composition of three mutations: $\mu_{(^\alpha_{n^\alpha(A)+1})}$, $\mu_{(^\beta_{n^\beta(A)+1})}$ and $\mu_{(^\alpha_{n^\alpha(A)+1})}$.

In order to prove the identities remark first that the transposition interchanges $\mathbf E^\alpha$ and $\mathbf F^\alpha$ and does not change $H^\alpha$, and thus the formulae  $\underline{\bar{\alpha}\bar{\alpha}\to\bar{\alpha}}$, $\underline{\bar{\alpha}\bar{\beta}\to\bar{\beta}\bar{\alpha}}$, $\underline{\bar{\alpha}\bar{\beta}\bar{\alpha}\to\bar{\beta}\bar{\alpha}\bar{\beta}}$ and $\underline{\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}\to\bar{\alpha}\bar{\beta}\bar{\alpha}\bar{\beta}}$ follow from the respective formulae for positive roots.  

The main tool to prove these identities is the Campbell-Hausdorff formula:
\begin{equation} \label{CH}
\begin{array}{c}
\exp(ae_\alpha)\exp(be_\beta)=\exp(ae_\alpha+be_\beta+\frac{1}{2}ab[e_\alpha,e_\beta]+\\
+\frac{1}{6}(a^2b[e_\alpha,[e_\alpha,e_\beta]]+ab^2[e_\beta,[e_\beta,e_\alpha]])-\\
-\frac{1}{24}a^2b^2([e_\alpha,[e_\beta,[e_\alpha,e_\beta]]]+[e_\beta,[e_\alpha,[e_\beta,e_\alpha]]])+\ldots).
\end{array}
\end{equation}

 Let us introduce a more traditional generator $H_\alpha(x)=\exp(\log(x)h_\alpha)$.

\noindent{$\underline{\alpha\alpha\to\alpha}:$}. It is easy to show using, for example, computations with $2\times2$ matrices, that
$$
\mathbf E^\alpha H_\alpha(x)\mathbf E^\alpha = H_\alpha(1+x^2)^{1/2} \mathbf E^\alpha H_\alpha(1+x^{-2})^{-1/2}
$$
Substituting the expression for $H_\alpha(x)=\prod_\beta H^\beta(c)^{C_{\alpha \beta}}$ and taking into account that $\mathbf E^\alpha$ commute with  $H^\beta$ when $\alpha\neq \beta$, that $C_{\alpha \alpha}=2$ and making a substitution of the independent variable $x^2 \rightarrow x$ one gets the identity {$\underline{\alpha\alpha\to\alpha}$}.

\noindent {$\underline{\bar{\alpha}\beta\to\beta\bar{\alpha}}$}. 
The proof is analogous to the previous one and is based on the easily verifiable identity for $2\times2$ matrices
$$
\mathbf F^\alpha H_\alpha(x) \mathbf E^\alpha = H_\alpha(1+x^{-2})^{-1/2}\mathbf E^\alpha H_\alpha(x)^{-1} \mathbf F^\alpha H_\alpha(1+x^{-2})^{-1/2}
$$

\noindent{$\underline{\alpha\beta\alpha\to\beta\alpha\beta}$.} This identity can be easily derived from the well known identity: 
$$e^{ae_\alpha}e^{be_\beta}e^{ce_\alpha}=e^{\frac{bc}{a+c}e_\beta}e^{(a+c)e_\alpha}e^{\frac{ab}{a+c}e_\beta}.$$ 
Taking into account that 
\begin{equation} \label{newold}
H^\alpha(a)\mathbf E^\alpha H^\alpha(a)^{-1}=e^{a e_\alpha}
\end{equation} 
for any $\alpha$ and making the substitution $a/c \rightarrow x$ one obtains the desired identity. Although one can meet this identity in many papers we would like to give another derivation here by the methods suitable for generalization for the cases when $C_{\alpha \beta} < -1$. Indeed, since the algebra generated by $e_\alpha$ and $e_\beta$ has dimension three, the identity 
\begin{equation}\label{a2}
e^{ze_\alpha}e^{ye_\beta}e^{xe_\alpha}=e^{ce_\beta}e^{be_\alpha}e^{ae_\beta}
\end{equation}
must hold for certain functions $a,b,c$ of $x,y,z$. Due to the Campbell-Hausdorff  formula both sides are actually series in the commutators: $1+c_1e_\alpha+c_2e_\beta + c_3[e_\alpha,e_\beta]+\ldots$\quad . Expressing $c_1\ldots c_3$ in terms of $a,b,c$ and of $x,y,z$, respectively, gives three equations: (Actually instead of using Campbell-Hausdorff  formula it is much easier just to take the series expansion for the exponents and finding the $c_i$'s as the coefficients at the terms $e_\alpha$, $e_\beta$ and $e_\alpha e_\beta$, respectively.)
\begin{eqnarray*}
y&=&a+c\\
x+z&=&b\\
yz&=&ab
\end{eqnarray*}
The equations are obviously homogeneous w.r.t. the rescaling $(a,b,c,x,y,z) \rightarrow (\lambda a, \mu b, \lambda c, \mu x, \lambda y, \mu z)$. Using this homogeneity one can make $y=z=1$ and obtain the equations:
\begin{eqnarray*}
a+c&=&1\\
b&=&x+1\\
ab&=&1,
\end{eqnarray*}
i.e., 
$$a=(1+x)^{-1}, b=(1+x), c=(1+x^{-1})^{-1}.$$
Substituting these expressions into (\ref{a2}) and using (\ref{newold}) one obtains the desired identity.

\noindent{$\underline{\alpha\beta\alpha\beta\to\beta\alpha\beta\alpha}$}. 
Let now $C_{\alpha \beta}=-2$ then we have the identity
\begin{equation}\label{b2}
e^{we_\alpha}e^{ze_\beta}e^{ye_\alpha}e^{xe_\beta}=e^{de_\beta}e^{ce_\alpha}e^{be_\beta}e^{ae_\alpha}
\end{equation}
holding for certain functions $a,b,c,d$ of $x,y,z,w$. Both sides are actually series in the commutators: $1+c_1e_\alpha+c_2e_\beta+c_3[e_\alpha,e_\beta]+c_4[e_\alpha,[e_\alpha,e_\beta]]+\ldots$\quad. Expressing analogously $c_1\ldots c_4$ in terms of $a,b,c,d$ and of $x,y,z,w$ and using the homogeneity to set $z=w=1$ one obtains four equations: 
\begin{eqnarray*}
a+c&=&y+1\\
b+d&=&x+1\\
ab+ad+cd&=&y\\
abc&=&y
\end{eqnarray*}
The solution for these equations is
\begin{eqnarray*}
a&=&\frac{y}{1+x+xy}\\
b&=&\frac{(1+x+xy)^2}{1+x+2xy+xy^2}\\
c&=&\frac{1+x+2xy+xy^2}{1+x+xy}\\
d&=&\frac{xy^2}{1+x+2xy+xy^2}
\end{eqnarray*}
Substituting these expressions into (\ref{b2}) and using (\ref{newold}) one obtains the desired identity.

\noindent{\bf EEEEEE=EEEEEE}. 
Let now $C_{\alpha\beta}=-3$ then we have the identity
$$
e^{te_\alpha}e^{re_\beta}e^{we_\alpha}e^{ze_\beta}e^{ywe_\alpha}e^{xze_\beta}=e^{ge_\beta}e^{fe_\alpha}e^{de_\beta}e^{ce_\alpha}e^{bde_\beta}e^{ace_\alpha}
$$
hold for certain functions $a,b,c,d,f,g$ of $x,y,z,w,r,t$. Both sides are actually series in the commutators:
$1+c_1e_\alpha+c_2e_\beta+c_3[e_\alpha,e_\beta]+c_4[e_\alpha,[e_\alpha,e_\beta]]+c_5[e_\alpha,[e_\alpha,[e_\alpha,e_\beta]]]+c_6[e_\beta,[e_\alpha,[e_\alpha,[e_\alpha,e_\beta]]]]+\ldots$\quad.
Expressing analogously $c_1\ldots c_6$ in terms of $a,b,c,d,f,g$ and of $x,y,z,w,r,t$ and using the homogeneity to set $r=t=1$ one obtains six equations:
\begin{eqnarray*}
ac+c+f&=&yw+w+1\qquad(e_\alpha)\\
bd+d+g&=&xz+z+1\qquad (e_\beta)\\
acbd+acd+acg+cd+cg+fg&=&ywz+yw+w\qquad (e_\beta e_\alpha)\\
abc^2d+abcdf+acdf+cdf&=&yzw^2+yzw+yw+w\qquad (e_\alpha e_\beta e_\alpha)\\
abc^3d+abcdf^2+2abc^2df+acdf^2+cdf^2&=&yzw^3+ywz+yw+w+2yzw^2\qquad
(e_\alpha e_\alpha e_\beta e_\alpha)\\
abc^2d^2f&=&yzw^2\qquad (e_\alpha e_\beta e_\alpha e_\beta e_\alpha)
\end{eqnarray*}







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