Quivers with potentials associated to triangulated surfaces

2008/11/14 by Daniel Labardini-Fragoso · 8 citations
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology

paper · pdf · doi:10.1112/plms/pdn051

Abstract

We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin–Shapiro–Thurston, and quivers with potentials (QPs) and their mutations introduced by Derksen–Weyman–Zelevinsky. To each ideal triangulation of a bordered surface with marked points, we associate a QP, in such a way that whenever two ideal triangulations are related by a flip of an arc, the respective QPs are related by a mutation with respect to the flipped arc. We prove that if the surface has non-empty boundary, then the QPs associated to its triangulations are rigid and hence non-degenerate.

Cited by

• Skew-symmetrizable cluster algebras from surfaces and symmetric quivers
• The locally complexified-gentle algebras
• Stable Brauer-Thrall II' conjecture for finite-dimensional Jacobian algebras
• The derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure
• Cluster theory of topological Fukaya categories. Part II: Higher Teichmüller theory
• Induction in perverse schobers and cluster tilting theory
• Derksen-Weyman-Zelevinsky mutations of infinite-dimensional modules I: Foundations
• Mutations of quivers with 2-cycles

Related

• Quivers with potentials associated to triangulated surfaces, Part II: Arc representations
• On triangulations, quivers with potentials and mutations
• Quivers with potentials associated to triangulations of closed surfaces with at most two punctures
• Quivers from non-orientable surfaces
• On cluster algebras from once punctured closed surfaces