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Exotic Cluster Structures on $SL_n$: The Cremmer-Gervais Case

M. Gekhtman University of Notre Dame, IN
M. Shapiro Michigan State University, East Lansing, MI
A. Vainshtein University of Haifa, Israel
Available Formats:
Electronic ISBN: 978-1-4704-3639-1
Product Code: MEMO/246/1165.E
List Price: $75.00 MAA Member Price:$67.50
AMS Member Price: $45.00 Click above image for expanded view Exotic Cluster Structures on$SL_n$: The Cremmer-Gervais Case M. Gekhtman University of Notre Dame, IN M. Shapiro Michigan State University, East Lansing, MI A. Vainshtein University of Haifa, Israel Available Formats:  Electronic ISBN: 978-1-4704-3639-1 Product Code: MEMO/246/1165.E  List Price:$75.00 MAA Member Price: $67.50 AMS Member Price:$45.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 2462016; 94 pp
MSC: Primary 53; 13;

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 $\mathcal{O}(\mathcal{G})$. The authors 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 the authors establish it for the Cremmer–Gervais Poisson–Lie structure on $SL_n$, which is the least similar to the standard one.

• Chapters
• 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

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Volume: 2462016; 94 pp
MSC: Primary 53; 13;

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 $\mathcal{O}(\mathcal{G})$. The authors 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 the authors establish it for the Cremmer–Gervais Poisson–Lie structure on $SL_n$, which is the least similar to the standard one.

• Chapters
• 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
Review Copy – for reviewers who would like to review an AMS book
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