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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Caldero-Chapoton map for infinite clusters


Authors: Peter Jørgensen and Yann Palu
Journal: Trans. Amer. Math. Soc. 365 (2013), 1125-1147
MSC (2010): Primary 13F60, 16G10, 16G20, 16G70, 18E30
Published electronically: November 6, 2012
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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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Additional Information

Peter Jørgensen
Affiliation: School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom
Email: peter.jorgensen@ncl.ac.uk

Yann Palu
Affiliation: Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
Address at time of publication: LAMFA, Université de Picardie Jules Verne, 33, Rue Saint-Leu, 80039 Amiens, France
Email: ypalu@maths.leeds.ac.uk, yann.palu@u-picardie.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05464-X
PII: S 0002-9947(2012)05464-X
Keywords: Calabi-Yau reduction, cluster category, cluster map, cluster structure, cluster tilting subcategory, coindex, Dynkin type $A_{\infty}$, exchange pair, exchange triangle, Fomin-Zelevinsky mutation, Grassmannian, index, $K$-theory, $\operatorname{SL}_{2}$-tiling
Received by editor(s): May 4, 2010
Received by editor(s) in revised form: September 8, 2010
Published electronically: November 6, 2012
Article copyright: © Copyright 2012 American Mathematical Society