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Exotic cluster structures on $SL_n$: the Cremmer–Gervais case

About this Title

M. Gekhtman, M. Shapiro and A. Vainshtein

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1165
ISBNs: 978-1-4704-2258-5 (print); 978-1-4704-3639-1 (online)
Published electronically: December 1, 2016
Keywords:Poisson–Lie group, cluster algebra, Belavin–Drinfeld triple

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Cluster structures and Poisson–Lie groups
  • Chapter 3. Main result and the outline of the proof
  • Chapter 4. Initial cluster
  • Chapter 5. Initial quiver
  • Chapter 6. Regularity
  • Chapter 7. Quiver transformations
  • Chapter 8. Technical results on cluster algebras


This is the second paper in the series of papers dedicated to the study of natural cluster structures in the rings of regular functions on simple complex Lie groups and PoissonâĂŞLie structures compatible with these cluster structures. According to our main conjecture, each class in the BelavinâĂŞDrinfeld classification of PoissonâĂŞLie structures on corresponds to a cluster structure in . We have shown before that this conjecture holds for any in the case of the standard PoissonâĂŞLie structure and for all Belavin-Drinfeld classes in , . In this paper we establish it for the CremmerâĂŞGervais PoissonâĂŞLie structure on , which is the least similar to the standard one.

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