Quivers with potentials associated to triangulated surfaces

2008/03/31 by Daniel Labardini-Fragoso · 116 citations
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Cluster (spacecraft) #Geometric and Algebraic Topology #Ideal (ethics) #Point set triangulation #Surface (topology) #Triangulation #math.RT #msc:16G99 #msc:16S99 #msc:57M50 #msc:57N05

paper · pdf · doi:10.1112/plms/pdn051

published in Proceedings of the London Mathematical Society 98(3), 797-839 (Wiley) · v3: 44 pages, 57 figures. Prop 29 of v2 generalized to Thm 36, some changes to References. In response to referee's comments: some examples added, more cases verified in proof of Thm 30 (formerly Thm 23). Submitted to Proc. London Math. Soc

arxiv created 2008/08/11 · openalex publication_date 2008/11/14 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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
• Finite-dimensional Jacobian algebras: Finiteness and tameness
• Arc representations
• Mutation classes ofÃn-quivers and derived equivalence classification of cluster tilted algebras of typeÃn
• Schemes of modules over gentle algebras and laminations of surfaces
• Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D
• On the cluster category of a marked surface without punctures
• Almost Gentle Algebras and their Trivial Extensions
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

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