Algebras-def}
----------------

In this section, we will review the definitions and some basic results on cluster algebras, or more precisely, on cluster algebras of geometric type over the field of complex numbers $\CC$. Denote by $\mathfrak{F}=\CC(x_1,\ldots,
x_n)$ the field of fractions in $n$ indeterminates. Recall that a $m
\times
m$-integer matrix $B'$ is called skew-symmetrizable if there exists a $m \times m$-diagonal matrix $D$ with positive integer entries such that $B' \cdot D$ is skew-symmetric. Now, let $B$ be a $m\times n$-integer matrix such that its principal $m\times m$-submatrix is skew-symmetrizable. We call the tuple $(x_1,\ldots,x_n, B)$ the [*initial seed*]{} of the cluster algebra and $ (x_1,\ldots x_m)$ a cluster, while ${\bf x}=(x_1,\ldots x_n)$ is called an extended cluster. The cluster variables $x_{m+1},\ldots,x_n$ are called [*coefficients*]{}. We will now construct more clusters, $(y_1,\ldots,
y_m)$ and extended clusters ${\bf
y}=(y_1,\ldots, y_n)$, which are transcendence bases of $\mathfrak{F}$, and the corresponding seeds $({\bf y}, \tilde B)$ in the following way.

Define for each real number $r$ the numbers $r^+={\rm max}(r,0)$ and $r^-={\rm
min}(r,0)$. Given a skew-symmetrizable integer $m \times n$-matrix $B$, we define for each $1\le i\le m$ the [*exchange polynomial*]{} $$P_i = \prod_{k=1}^n x_k^{b_{ik}^+}+ \prod_{k=1}^n  x_k^{-b_{ik}^-}\ .$$

We can now define 