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- W619796 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 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." @default.
- W619796 created "2016-06-24" @default.
- W619796 creator A5066021669 @default.
- W619796 date "2014-12-17" @default.
- W619796 modified "2023-09-27" @default.
- W619796 title "Exotic Cluster Structures on $SL_n$ with Belavin-Drinfeld Data of Minimal Size, I. The Structure" @default.
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