A CALDERO-CHAPOTON MAP FOR INFINITE CLUSTERS

作者：Jorgensen Peter^*; Palu Yann

来源：Transactions of the American Mathematical Society, 2013, 365(3): 1125-1147.

• Calabi-Yau reduction
• cluster category
• cluster map
• cluster structure
• cluster tilting subcategory
• coindex
• Dynkin type A(infinity)
• exchange pair
• exchange triangle
• Fomin-Zelevinsky mutation
• Grassmannian
• index
• K-theory
• SL2-tiling

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. %26lt;br%26gt;The map is not necessarily defined on all objects of the triangulated category, but we show that it is a (weak) cluster map in the sense of Buan Iyama Reiten-Scott. As a corollary, it induces a surjection from the set of exceptional objects which can be reached from the cluster tilting subcategory to the set of cluster variables of an associated cluster algebra. %26lt;br%26gt;Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. %26lt;br%26gt;We apply our results to the cluster category D of Dynkin type A(infinity) which has a rich supply of cluster tilting subcategories with infinitely many indecomposable objects. We show an example of a cluster map which cannot be extended to all of D. %26lt;br%26gt;The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL2-tilings of the plane.

• 出版日期2013-3