## 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 A_{ℂ}¯ (C) is naturally isomorphic to O (SL_{n}), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.

Original language | English |
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Pages (from-to) | 391-443 |

Number of pages | 53 |

Journal | Israel Journal of Mathematics |

Volume | 218 |

Issue number | 1 |

DOIs | |

State | Published - 1 Mar 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Hebrew University of Jerusalem.

## ASJC Scopus subject areas

- General Mathematics