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

About this Title

M. Gekhtman, Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556, M. Shapiro, Department of Mathematics, Michigan State University, East Lansing, Michigan 48823 and A. Vainshtein, Department of Mathematics & Department of Computer Science, University of Haifa, Haifa, Mount Carmel 31905, Israel

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
MSC: Primary 53D17, 13F60

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Table of Contents


  • 1. Introduction
  • 2. Cluster structures and Poisson–Lie groups
  • 3. Main result and the outline of the proof
  • 4. Initial cluster
  • 5. Initial quiver
  • 6. Regularity
  • 7. Quiver transformations
  • 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 Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on $\mathcal {G}$ corresponds to a cluster structure in $\mathcal {O}(\mathcal {G})$. We have shown before that this conjecture holds for any $\mathcal {G}$ in the case of the standard Poisson–Lie structure and for all Belavin-Drinfeld classes in $SL_n$, $n<5$. In this paper we establish it for the Cremmer–Gervais Poisson–Lie structure on $SL_n$, which is the least similar to the standard one.

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