satisfying for all $a,b,c\in A$:

1.  skew-symmetry: $\{a,b\}=-\{b,a\}$

2.  Jacobi identity: $\{a,\{b,c\}\}+\{c,\{a,b\}\}+\{b,\{c,a\}\}=0$,

3.  Leibniz rule: $a\{b,c\}=\{a,b\}c+b\{a,c\}$.

If there is no room for confusion we will refer to a Poisson algebra $(A,\{\cdot,\cdot\})$ simply as $A$. A [*Poisson Ideal*]{} $\II$ in a Poisson algebra $A$ is an ideal such that $\{\II,A\}\subset \II$, and if $k$ is of characteristic zero, then a Poisson prime ideal is a prime ideal which is also Poisson.

Gekhtman, Shapiro and Vainshtein showed in [@GSV] that one can associate Poisson structures to cluster algebras in the following way. Let $\mathfrak{A} \subset \CC[x_1^{\pm 1}, \ldots,
x_n^{\pm 1}] \subset \mathfrak{F}$ be a cluster algebra. A Poisson structure $\{\cdot, \cdot\}$ on $\CC [x_1, \ldots, x_n]$ is called log-canonical if $\{
x_i,x_j\}=\lambda_{ij} x_ix_j$ with $\lambda_{ij}\in \CC$ for all $1\le i,j\le n$.

The Poisson structure can be naturally extended to $\mathfrak{F}$ by using the identity $0=\{f\cdot f^{-1},g\}$ for all $f,g\in\CC [x_1, \ldots, x_n]$. We thus obtain that $\{f^{-1},g\}=-f^{-2}\{f,g\}$ for all $f,g\in \mathfrak{F}$. We call $\Lambda=\left( \lambda_{ij}\right)_{i,j=1}^n$ the [*coefficient matrix*]{} of the Poisson structure. We say that a Poisson structure on $\mathfrak{F}$ is compatible with $\mathfrak{A}$ if it is log-canonical with respect to each cluster $(y_1,\ldots,
y_n)$; i.e., it is log canonical on $\CC[y_1, 