Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

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• 作者：Thorsten Holm^a ; ^{holm@math.uni-hannover.de" class="auth_mail" title="E-mail the corresponding author} ; Peter Jø ; rgensen^b ; ^{peter.jorgensen@ncl.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05E10 ; 13F60 ; 16G70 ; 18E30
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：140
• 期：4
• 页码：112-131
• 全文大小：542 K

文摘

It is an important aspect of cluster theory that cluster categories are “categorifications” of cluster algebras. This is expressed formally by the (original) Caldero–Chapoton map X which sends certain objects of cluster categories to elements of cluster algebras.

Let τc→b→c$\tau c\to b\to c$ be an Auslander–Reiten triangle. The map X   has the salient property that X(τc)X(c)−X(b)=1$X(\tau c)X(c)-X(b)=1$. This is part of the definition of a so-called frieze, see [1].

The construction of X depends on a cluster tilting object. In a previous paper [14], we introduced a modified Caldero–Chapoton map ρ depending on a rigid object; these are more general than cluster tilting objects. The map ρ   sends objects of sufficiently nice triangulated categories to integers and has the key property that ρ(τc)ρ(c)−ρ(b)$\rho (\tau c)\rho (c)-\rho (b)$ is 0 or 1. This is part of the definition of what we call a generalised frieze.

Here we develop the theory further by constructing a modified Caldero–Chapoton map, still depending on a rigid object, which sends objects of sufficiently nice triangulated categories to elements of a commutative ring A. We derive conditions under which the map is a generalised frieze, and show how the conditions can be satisfied if A is a Laurent polynomial ring over the integers.

The new map is a proper generalisation of the maps X and ρ.