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

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.

29 pages, 17 figures. Discussion in Section 6 clarified and expanded. Some minor corrections, clarification of notation

References in corpus (5)

• Deformed Calabi-Yau Completions
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• Gentle algebras arising from surface triangulations
• From triangulated categories to module categories via localisation
• Coloured quiver mutation for higher cluster categories

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• Cotorsion pairs and t-structures in a $2-$Calabi-Yau triangulated category
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• Extensions in Jacobian algebras via punctured skein relations