A Caldero-Chapoton map for infinite clusters

TL;DR

This paper extends the Caldero-Chapoton map to infinite cluster categories, establishing its properties and applications to cluster algebras, especially in the context of Dynkin type A infinity, revealing new structural insights.

Contribution

It constructs a Caldero-Chapoton map for infinite cluster categories, demonstrating its weak cluster map properties and linking it to cluster variables and tilings.

Findings

The map is a weak cluster map in infinite settings.

It induces a surjection onto cluster variables.

An example shows limitations of extending the map to all objects.

Abstract

We construct a Caldero-Chapoton map on a triangulated category with a cluster tilting subcategory which may have infinitely many indecomposable objects. 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. Along the way, we study the interaction between Calabi-Yau reduction, cluster structures, and the Caldero-Chapoton map. 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. The case of D also permits us to illuminate results by Assem-Reutenauer-Smith on SL_2-tilings of the plane.