TY - JOUR

T1 - Generalised friezes and a modified Caldero-Chapoton map depending on a rigid object, II

AU - Holm, Thorsten

AU - Jørgensen, Peter

N1 - Funding information: Part of this work was done while Peter Jørgensen was visiting the Leibniz Universität Hannover. He thanks Christine Bessenrodt, Thorsten Holm, and the Institut für Algebra, Zahlentheorie und Diskrete Mathematik for their hospitality. He gratefully acknowledges support from Thorsten Holm's grant HO 1880/5-1 , which falls under the research priority programme SPP 1388 Darstellungstheorie of the Deutsche Forschungsgemeinschaft (DFG).

PY - 2016/5/1

Y1 - 2016/5/1

N2 - 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.

AB - 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.

KW - Auslander-Reiten triangle

KW - Categorification

KW - Cluster algebra

KW - Cluster category

KW - Cluster tilting object

KW - Rigid object

UR - http://www.scopus.com/inward/record.url?scp=84929591094&partnerID=8YFLogxK

U2 - 10.1016/j.bulsci.2015.05.001

DO - 10.1016/j.bulsci.2015.05.001

M3 - Article

AN - SCOPUS:84929591094

VL - 140

SP - 112

EP - 131

JO - Bulletin des Sciences Mathematiques

JF - Bulletin des Sciences Mathematiques

SN - 0007-4497

IS - 4

ER -