Exotic cluster structures on SL_n with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples

2015/11/25 by Idan Eisner, Eisner, Idan
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA

paper · pdf · doi:10.48550/arxiv.1511.08234

arXiv admin note: text overlap with arXiv:1412.5352; text overlap with arXiv:1101.0015 by other authors

arxiv created 2015/11/25 · arxiv updated 2015/11/30

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, the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin--Drinfeld data. This paper proves the rest of the conjecture: the corresponding upper cluster algebra A_ℂ(C) is naturally isomorphic to O(SL_n), the torus determined by the BD triple generates theaction of (ℂ^*)^2k_T on ℂ(SL_n), and the correspondence between Belavin--Drinfeld classes and cluster structures is one to one.

Citations

• Exotic Cluster Structures on SL_n with Belavin-Drinfeld Data of Minimal Size, I. The Structure

Cited by

• Exotic Cluster Structures on SL_n with Belavin-Drinfeld Data of Minimal Size, I. The Structure

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