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58
Quivers with potentials and their representations I: Mutations
, 2007
"... We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representationtheoretic interpretation of quiver mutations at arbitrary vertices. This gives a farreaching generalization of BernsteinGelfandPono ..."
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Cited by 178 (3 self)
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We study quivers with relations given by noncommutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representationtheoretic interpretation of quiver mutations at arbitrary vertices. This gives a farreaching generalization of BernsteinGelfandPonomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, CalabiYau algebras, cluster algebras.
Cluster algebras, quiver representations and triangulated categories
, 2009
"... This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In additi ..."
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Cited by 106 (6 self)
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This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.
Cluster algebras and quantum affine algebras
, 2009
"... Let C be the category of finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C ..."
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Cited by 71 (10 self)
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Let C be the category of finitedimensional representations of a quantum affine algebra Uq(̂g) of simplylaced type. We introduce certain monoidal subcategories Cℓ (ℓ ∈ N) of C
CLUSTER ALGEBRAS VIA CLUSTER CATEGORIES WITH INFINITEDIMENSIONAL MORPHISM SPACES
"... Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an e ..."
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Cited by 49 (3 self)
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Abstract. We apply our previous work on cluster characters for Hominfinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky’s Cluster algebras IV for skewsymmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the Einvariant and show that an arbitrary decorated representation with vanishing Einvariant is characterized by its gvector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid
Positivity for cluster algebras from surfaces
, 2009
"... We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our for ..."
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Cited by 43 (11 self)
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We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
The periodicity conjecture for pairs of Dynkin diagrams
, 2010
"... We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories. ..."
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Cited by 36 (0 self)
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We prove the periodicity conjecture for pairs of Dynkin diagrams using FominZelevinsky’s cluster algebras and their (additive) categorification via triangulated categories.
On cluster algebras arising from unpunctured surfaces II
, 2008
"... We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an im ..."
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Cited by 30 (11 self)
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We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras. Furthermore, we obtain direct formulas for Fpolynomials and gvectors and show that Fpolynomials have constant term equal to 1. As an application, we compute the EulerPoincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type Ã.
On the combinatorics of rigid objects in 2CalabiYau categories
 INT. MATH. RES. NOT. IMRN, (11):ART. ID RNN029
, 2008
"... Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposab ..."
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Cited by 26 (2 self)
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Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a clustertilting subcategory T ′ form a basis of the Grothendieck group of T and that, if T and T ′ are related by a mutation, then the indices with respect to T and T ′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given clustertilting subcategory T. Conjecturally, these indices coincide with FominZelevinsky’s gvectors.
Cluster characters for cluster categories with infinitedimensional morphism spaces
"... We prove the existence of cluster characters for Hominfinite cluster categories. For this purpose, we introduce a suitable mutationinvariant subcategory of the cluster category. We sketch how to apply our results in order to categorify any skewsymmetric cluster algebra. More applications and a c ..."
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Cited by 25 (1 self)
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We prove the existence of cluster characters for Hominfinite cluster categories. For this purpose, we introduce a suitable mutationinvariant subcategory of the cluster category. We sketch how to apply our results in order to categorify any skewsymmetric cluster algebra. More applications and a comparison to DerksenWeymanZelevinsky’s results will be given in a future paper.
Cluster mutationperiodic quivers and associated laurent sequences
 Journal of Algebraic Combinatorics
"... We consider quivers/skewsymmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequenc ..."
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Cited by 23 (5 self)
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We consider quivers/skewsymmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which have higher periodicity. The periodicity means that sequences given by recurrence relations arise in a natural way from the associated cluster algebras. We present a number of interesting, new nonlinear recurrences, necessarily with the Laurent property, of both the real line and the plane. In particular, we show that some of these recurrences can be linearised and, with certain initial conditions, give integer sequences which contain all solutions of some particular Pell equations.