Exotic cluster structures on SLn with Belavin–Drinfeld data of minimal size, I. The structure

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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, we give an algorithm for constructing an initial seed ∑ in O (SLn). 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) is naturally isomorphic to O (SLn), and the correspondence between Belavin-Drinfeld classes and cluster structures is one to one.

Original languageEnglish
Pages (from-to)391-443
Number of pages53
JournalIsrael Journal of Mathematics
Volume218
Issue number1
DOIs
StatePublished - 1 Mar 2017

Bibliographical note

Publisher Copyright:
© 2017, Hebrew University of Jerusalem.

ASJC Scopus subject areas

  • General Mathematics

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