Abstract
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 \(\overline {{A_\mathbb{C}}} \left( C \right)\) is naturally isomorphic to O (SL n ), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.
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Eisner, I. Exotic cluster structures on SL n with Belavin–Drinfeld data of minimal size, I. The structure. Isr. J. Math. 218, 391–443 (2017). https://doi.org/10.1007/s11856-017-1469-z
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DOI: https://doi.org/10.1007/s11856-017-1469-z