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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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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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
  • MR Author ID: 857813
  • Email: ypalu@maths.leeds.ac.uk, yann.palu@u-picardie.fr
  • Received by editor(s): May 4, 2010
  • Received by editor(s) in revised form: September 8, 2010
  • Published electronically: November 6, 2012
  • © Copyright 2012 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 1125-1147
  • MSC (2010): Primary 13F60, 16G10, 16G20, 16G70, 18E30
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05464-X
  • MathSciNet review: 3003260