Abstract
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.
| Originalsprog | Engelsk |
|---|---|
| Tidsskrift | Bulletin des Sciences Mathematiques |
| Vol/bind | 140 |
| Nummer | 4 |
| Sider (fra-til) | 112-131 |
| Antal sider | 20 |
| ISSN | 0007-4497 |
| DOI | |
| Status | Udgivet - 1 maj 2016 |
| Udgivet eksternt | Ja |