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

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

2009-09-23

arxiv.org/abs/0909.4100v2

Mathematics

Representation Theory

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

Scientific paper

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.

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