Coloured quivers for rigid objects and partial triangulations: The unpunctured case

TL;DR

This paper introduces coloured quivers associated with rigid objects and partial triangulations in unpunctured Riemann surfaces, establishing their equivalence in certain categories and exploring mutation and reduction operations.

Contribution

It defines coloured quivers for rigid objects and partial triangulations, demonstrating their equivalence in surface-related categories and interpreting Iyama-Yoshino reduction geometrically.

Findings

Coloured quivers coincide for rigid objects and partial triangulations in surface categories.

Mutation operations are explicitly described for disks.

Iyama-Yoshino reduction corresponds to cutting along an arc.

Abstract

We associate a coloured quiver to a rigid object in a Hom-finite 2-Calabi--Yau triangulated category and to a partial triangulation on a marked (unpunctured) Riemann surface. We show that, in the case where the category is the generalised cluster category associated to a surface, the coloured quivers coincide. We also show that compatible notions of mutation can be defined and give an explicit description in the case of a disk. A partial description is given in the general 2-Calabi-Yau case. We show further that Iyama-Yoshino reduction can be interpreted as cutting along an arc in the surface.