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Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

Part of: Algebraic combinatorics Abelian categories Representation theory of rings and algebras Arithmetic rings and other special rings

Published online by Cambridge University Press:  11 January 2016

Thorsten Holm  and
Peter Jørgensen
Thorsten Holm
Affiliation:
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathematik und Physik, Leibniz Universität Hannover, 30167 Hannover, Germany, [email protected]://www.iazd.uni-hannover.de/~tholm
Peter Jørgensen
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom, [email protected]://www.staff.ncl.ac.uk/peter.jorgensen
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Abstract

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The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

MSC classification

Secondary: 05E10: Combinatorial aspects of representation theory 13F60: Cluster algebras 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 218 , June 2015 , pp. 101 - 124
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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