Israel Journal of Mathematics

, Volume 218, Issue 1, pp 445–487

Exotic cluster structures on SLn with Belavin–Drinfeld data of minimal size, II. Correspondence between cluster structures and Belavin–Drinfeld triples

Article

DOI: 10.1007/s11856-017-1470-6

Cite this article as:
Eisner, I. Isr. J. Math. (2017) 218: 445. doi:10.1007/s11856-017-1470-6
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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 SLn, the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data.

This paper proves the rest of the conjecture: the corresponding upper cluster algebra \(\overline {{A_\mathbb{C}}} \left( C \right)\) is naturally isomorphie to O (SLn), the torus determined by the BD triple generates the action of \({\left( {\mathbb{C}*} \right)^{2{k_T}}}\) on C (SLn), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael