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

Autor: Thorsten Holm, Peter Jørgensen
Rok vydání: 2016
Předmět:
Zdroj: Bulletin des Sciences Mathématiques. 140:112-131
ISSN: 0007-4497
Popis: 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 τ c → b → c be an Auslander–Reiten triangle. The map X has the salient property that X ( τ 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 ( τ c ) ρ ( c ) − ρ ( 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 .
Databáze: OpenAIRE
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