1,720,966 research outputs found
Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions
No full textWe prove that the quivers with potentials associated with triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly five punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related to a flip, their associated QPs are related to the corresponding QP-mutation. As a by-product, for (arbitrarily punctured) surfaces with non-empty boundary, we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called Popping Theorem, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence
Quivers with potentials associated to triangulated surfaces
No full textWe 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. © 2008 London Mathematical Society
On a family of Caldero–Chapoton algebras that have the Laurent phenomenon
No full textWe realize a family of generalized cluster algebras as Caldero–Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero–Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc j on the polygon with orbifold point a representation M(j) of the referred quiver with relations, and by proving that for every triangulation τ and every arc j∈τ the product of the Caldero–Chapoton functions of M(j) and M(j′), where j′ is the arc that replaces j when we flip j in τ equals the corresponding exchange polynomial of Chekhov–Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of E-rigid indecomposable decorated representations of Λ
On triangulations, quivers with potentials and mutations
No full textIn this survey article we give a brief account of constructions and results concerning the quivers with potentials associated to triangulations of surfaces with marked points. Besides the fact that the mutations of these quivers with potentials are compatible with the flips of triangulations, we mention some recent results on the representation type of Jacobian algebras and the uniqueness of non-degenerate potentials. We also mention how the quivers with potentials associated to triangulations give rise to CY2 and CY3 triangulated categories that have turned out to be useful in the subject of stability conditions and in theoretical physics
Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights
Full text linkLet Σ =(Σ, M,O) be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair (τ, ω) consisting of a triangulation τ of Σ and a function ω: O1,4, we define a chain complex C• (τ, ω) with coefficients in F_2=Z/2Z. Given Σ and ω , we define a colored triangulation of Σ _ω =(Σ, M,O,ω) to be a pair (τ, \xi) consisting of a triangulation of Σ and a 1-cocycle in the cochain complex that is dual to C• (τ, ω) ; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of Σ _ω as the graph whose vertices are the pairs (τ, x) consisting of a triangulation τ and a cohomology class x\in H1C(τ, ω)) , with an edge connecting two such pairs, (τ, x) and (Σ, z), if and only if there exist 1-cocycles xi in x and ζ in z such that (τ, xi) and (Σ, ζ) are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface Σ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair (τ, ω) over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the 2^|O| skew-symmetrizable matrices that Felikson-Shapiro-Tumarkin associated in [17] to any given triangulation of Σ . In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures
Species with potential arising from surfaces with orbifold points of order 2, part I: one choice of weights
No full textWe present a definition of mutations of species with potential that can be applied to the species realizations of any skew-symmetrizable matrix B over cyclic Galois extensions E / F whose base field F has a primitive [E : F]th root of unity. After providing an example of a globally unfoldable skew-symmetrizable matrix whose species realizations do not admit non-degenerate potentials, we present a construction that associates a species with potential to each tagged triangulation of a surface with marked points and orbifold points of order 2. Then we prove that for any two tagged triangulations related by a flip, the associated species with potential are related by the corresponding mutation (up to a possible change of sign at a cycle), thus showing that these species with potential are non-degenerate. In the absence of orbifold points, the constructions and results specialize to previous work by Labardini-Fragoso (Proc Lond Math Soc 98(3):797–839, 2009. arXiv:0803.1328; Sel Math New Ser 22(1):145–189, 2016. doi:10.1007/s00029-015-0188-8. arXiv:1206.1798). The species constructed here for each triangulation τ is a species realization of one of the several matrices that Felikson–Shapiro–Tumarkin have associated to τ in (Adv Math 231(5):2953–3002, 2012. arXiv:1111.3449), namely, the one that in their setting arises from choosing the number 12 for every orbifold point
Strongly primitive species with potentials I: Mutations
No full textMotivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations
The representation type of Jacobian algebras
No full textWe show that the representation type of the Jacobian algebra P(Q, S) associated to a 2-acyclic quiver Q with non degenerate potential S is invariant under QP-mutations. We prove that, apart from very few exceptions, P(Q, S) is of tame representation type if and only if Q is of finite mutation type. We also show that most quivers Q of finite mutation type admit only one non-degenerate potential up to weak right equivalence. In this case, the isomorphism class of P(Q, S) depends only on Q and not on S. (C) 2015 Published by Elsevier Inc
Derived categories of skew-gentle algebras and orbifolds
Full text linkSkew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of their q-Cartan matrices
Schemes of modules over gentle algebras and laminations of surfaces
Full text linkWe study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically τ-reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker–Schiffler–Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions (defined by Geiß–Leclerc–Schröer)
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