Quivers with potentials associated to triangulated surfaces

By Daniel Labardini-Fragoso. 2008

250 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d3b2bcf6c2bce662a0d3a6e1051fdee8e1d8c484

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions (2012, methodology): “for example [1], [2], [8], [23]), with the compatibility between ideal flips and QP-mutations proved in [15] appearing as well.”
Dimer models and cluster categories of Grassmannians (2013, methodology): “and all terms in the potential containing them, we obtain the quiver associated to T in [10] with potential as in [21].”
A geometric model for the derived category of gentle algebras (2018, unknown): “erived algebras. 4 SEBASTIAN OPPER, PIERRE-GUY PLAMONDON, AND SIBYLLE SCHROLL Jacobian algebras of (ideal) triangulations of marked surfaces with all marked points in the boundary are gentle algebras [51, 5]. We note that the ribbon graph of such a gentle algebra corresponds exactly to the triangula”
Cluster Algebras and Triangulated Surfaces Part II: Lambda Lengths (2012, background): “. Another family of bases, closely related to dual canonical bases for quantum groups, was investigated in [50]. A construction of quivers with potentials associated to triangulations was proposed in [41], and further studied in [34]. A geometric construction that can be regarded as the orbifold ver”
A Geometric Model for the Module Category of a Gentle Algebra (2018, background): “[22]), and as Jacobian algebras associated to triangulations of unpunctured surfaces with marked points on the boundary [2, 31].”
Quadratic differentials as stability conditions (2013, background): “The case of flips is dealt with by the following result of Labardini-Fragoso [24, Theorem 30].”
Gentle algebras arising from surface triangulations (2009, background): “An even more general case is considered by Labardini in [25], where such an algebra A(Γ) is defined for a general marked surface (allowing punctures).”
Donaldson-Thomas theory and cluster algebras (2010, methodology): “ theory of the motivic Hall algebra in the formal setting, we can apply all the arguments in this paper. (2) A typical example of a ﬁnite potential is a potential associated to a triangulated surface [LF09]. We will apply the results in this paper for a triangulated surface in [Naga]. 7 (3) It is ex”
On the cluster category of a marked surface without punctures (2010, methodology): “Therefore, for each curve γ in (S,M) we have Ext(1)C0′ (T0′, μ̄i (γ ))= μi (Ext 1 C0 (T0, γ )) ∼= μi (M(0, γ ))∼= M( fτi (0), γ )= M(0 , γ ), where the last isomorphism is given by the main result in [Labardini-Fragoso 2009b].”
On cluster algebras arising from unpunctured surfaces II (2008, background): “Independently, these relations have also been defined in [29] using quivers with potentials introduced in [16].”
Trivial extensions of gentle algebras and Brauer graph algebras (2014, background): “In [1] and in [16], a finite dimensional algebra, the so-called Jacobian algebra, is associated to the triple (S,M, T ).”
A survey on maximal green sequences (2019, background): “I am grateful to Daniel Labardini–Fragoso for helpful discussions and for reference [84].”