A Caldero–Chapoton map for infinite clusters

Jørgensen, Peter; Palu, Yann

doi:10.1090/S0002-9947-2012-05464-X

A Caldero–Chapoton map for infinite clusters
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by Peter Jørgensen and Yann Palu PDF

Trans. Amer. Math. Soc. 365 (2013), 1125-1147 Request permission

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 $\mathscr {D}$ of Dynkin type $A_{\infty }$ 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 $\mathscr {D}$.

The case of $\mathscr {D}$ also permits us to illuminate results by Assem–Reutenauer–Smith on $\operatorname {SL}_2$-tilings of the plane.

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