English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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 clustercategories to elements of cluster algebras.  Let \tau c --> b --> c be an Auslander-Reiten triangle. The map X has thesalient property that X(\tau c)X(c) - X(b) = 1. This is part of the definitionof a so-called frieze.  The construction of X depends on a cluster tilting object. In a previouspaper, we introduced a modified Caldero-Chapoton map \rho depending on a rigidobject; these are more general than cluster tilting objects. The map \rho sendsobjects of sufficiently nice triangulated categories to integers and has thekey property that \rho(\tau c)\rho(c) - \rho(b) is 0 or 1. This is part of thedefinition of what we call a generalised frieze.  Here we develop the theory further by constructing a modifiedCaldero-Chapoton map, still depending on a rigid object, which sends objects ofsufficiently nice triangulated categories to elements of a commutative ring A.We derive conditions under which the map is a generalised frieze, and show howthe conditions can be satisfied if A is a Laurent polynomial ring over theintegers.  The new map is a proper generalisation of the maps X and \rho.

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