Abstract
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 -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 , it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.
Citation
Thorsten Holm. Peter Jørgensen. "Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object." Nagoya Math. J. 218 101 - 124, June 2015. https://doi.org/10.1215/00277630-2891495
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