  Using the notion of compatibility between Poisson brackets and cluster
structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and
Vainshtein conjectured a correspondence between the two. Poisson Lie groups are
classified by the Belavin-Drinfeld classification of solutions to the classical
Yang Baxter equation. For any non trivial Belavin-Drinfeld data of minimal size
for $SL_{n}$, we give an algorithm for constructing an initial seed $\Sigma$ in
$\mathcal{O}(SL_{n})$. The cluster structure $\mathcal{C}=\mathcal{C}(\Sigma)$
is then proved to be compatible with the Poisson bracket associated with that
Belavin-Drinfeld data, and the seed $\Sigma$ is locally regular.
  This is the first of two papers, and the second one proves the rest of the
conjecture: the upper cluster algebra
$\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to
$\mathcal{O}(SL_{n})$, and the correspondence of Belavin-Drinfeld classes and
cluster structures is one to one.
