Quivers with potentials associated to triangulated surfaces, Part II: Arc representations

Abstract

This paper is a representation-theoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by Fomin-Shapiro-Thurston, the mutation theory of quivers with potentials initiated by Derksen-Weyman-Zelevinsky, and string modules associated to arcs on unpunctured surfaces by Assem-Brustle-Charbonneau-Plamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation.

51 pages; 37 figures. v2: 52 pages, 37 figures; a reference added; proof of Corollary 6.7 added; minor editorial changes

References in corpus (5)

• Gentle algebras arising from surface triangulations
• Positivity for cluster algebras from surfaces
• Quivers with potentials and their representations II: Applications to cluster algebras
• Mutations for quivers with potentials: Oberwolfach talk, April 2007
• Cluster expansion formulas and perfect matchings

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• Quivers with potentials associated to triangulated surfaces, part IV: Removing boundary assumptions
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• On Jacobian algebras from closed surfaces
• On Generalized Cluster Categories
• On the c-vectors and g-vectors of the Markov cluster algebra
• Categorified canonical bases and framed BPS states
• From groups to clusters
• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces
• Caldero-Chapoton algebras
• Quivers with potentials associated to triangulations of closed surfaces with at most two punctures
• On Jacobian algebras associated with the once-punctured torus