Exotic Cluster Structures on $SL_n$ with Belavin-Drinfeld Data of Minimal Size, I. The Structure
| Autor: | Eisner, Idan |
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| Rok vydání: | 2014 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Final version, to appear in Israel Journal of Mathematics. arXiv admin note: text overlap with arXiv:1511.08234; text overlap with arXiv:1101.0015 by other authors |
| Databáze: | arXiv |
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