Exotic cluster structures on SL_n with Belavin–Drinfeld data of minimal size, I. The structure

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman, Shapiro and Vainshtein conjectured the existence of a cluster structure for each Belavin-Drinfeld solution of the classical Yang-Baxter equation compatible with the corresponding Poisson-Lie bracket on the simple Lie group. 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 ∑ in O (SL_n). The cluster structure C = C (∑) is then proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data, and the seed ∑ is locally regular. This is the first of two papers. The second one proves the rest of the conjecture: the upper cluster algebra A_ℂ¯ (C) is naturally isomorphic to O (SL_n), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.