2014/12/17 by Idan Eisner, Eisner, Idan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.1412.5352
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
openalex publication_date 2014/12/17 · arxiv created 2016/10/05 · arxiv updated 2016/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 SLn, we give an algorithm for constructing an initial seed Σ in O(SLn). The cluster structure C=C(Σ) is then proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data, and the seed Σ is locally regular. This is the first of two papers, and the second one proves the rest of the conjecture: the upper cluster algebra Aℂ(C) is naturally isomorphic to O(SLn), and the correspondence of Belavin-Drinfeld classes and cluster structures is one to one.