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CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. 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). I ..."
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Abstract. 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. Contents
A geometric description of mcluster categories
"... Abstract. We show that the mcluster category of type An−1 is equivalent to ..."
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Abstract. We show that the mcluster category of type An−1 is equivalent to
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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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.
A geometric model for cluster categories of type Dn
, 2006
"... We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices. ..."
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We give a geometric realization of cluster categories of type Dn using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
Cluster expansion formulas and perfect matchings
 J. Algebraic Combin
"... Abstract. We study cluster algebras with principal 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 perfect matchings of a certain graph GT,γ that is constructed f ..."
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Abstract. We study cluster algebras with principal 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 perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula for these Laurent polynomial expansions in terms of subgraphs of the graph GT,γ. 1.
Quivers with potentials associated to triangulated surfaces
 Proc. London Math. Soc
"... To the memory of José Guadalupe RamírezRocha. Abstract. This paper is a representationtheoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by FominShapiroThurston, the mutation theory of quivers with potentials initiated by DerksenWe ..."
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To the memory of José Guadalupe RamírezRocha. Abstract. This paper is a representationtheoretic extension of Part I. It has been inspired by three recent developments: surface cluster algebras studied by FominShapiroThurston, the mutation theory of quivers with potentials initiated by DerksenWeymanZelevinsky, and string modules associated to arcs on unpunctured surfaces by AssemBrüstleCharbonneauPlamondon. Modifying the latter construction, to each arc and each ideal triangulation of a bordered marked surface we associate in an explicit way a representation of the quiver with potential constructed in Part I, so that whenever two ideal triangulations are related by a flip, the associated representations are related by the corresponding mutation. Contents
New graphs of finite mutation type
"... To a directed graph without loops or 2cycles, we can associate a skewsymmetric matrix with integer entries. Mutations of such skewsymmetric matrices, and more generally skewsymmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of ..."
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To a directed graph without loops or 2cycles, we can associate a skewsymmetric matrix with integer entries. Mutations of such skewsymmetric matrices, and more generally skewsymmetrizable matrices, have been defined in the context of cluster algebras by Fomin and Zelevinsky. The mutation class of a graph Γ is the set of all isomorphism classes of graphs that can be obtained from Γ by a sequence of mutations. A graph is called mutationfinite if its mutation class is finite. Fomin, Shapiro and Thurston constructed mutationfinite graphs from triangulations of oriented bordered surfaces with marked points. We will call such graphs “of geometric type”. Besides graphs with 2 vertices, and graphs of geometric type, there are only 9 other “exceptional ” mutation classes that are known to be finite. In this paper we introduce 2 new exceptional finite mutation classes. Cluster algebras were introduced by Fomin and Zelevinsky in [5, 6] to create an algebraic framework for total positivity and canonical bases in semisimple algebraic groups. An n × n matrix B = (bi,j) is called skew symmetrizable if there exists nonzero d1, d2,..., dn such that dibi,j = −djbj,i for all i, j. An exchange matrix is a skewsymmetrizable matrix with integer entries. A seed is a pair (x, B) where B is an exchange matrix and x = {x1, x2,..., xn} is a set of n algebraically independent elements. For any k with 1 ≤ k ≤ n we define another
Categorification of acyclic cluster algebras: an introduction
 IN THE PROCEEDINGS OF THE CONFERENCE ‘HIGHER STRUCTURES IN GEOMETRY AND PHYSICS 2007’, BIRKHÄUSER
"... This is a concise introduction to FominZelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster v ..."
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This is a concise introduction to FominZelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
Higher dimensional cluster combinatorics and representation theory
"... Abstract. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study i ..."
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Abstract. Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of evendimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type A which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any drepresentation finite algebra we introduce a certain ddimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type A we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occuring in the mutation of cluster tilting objects. 1.
Teichmüller theory of bordered surfaces
, 2006
"... We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of ..."
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We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliationshear coordinates), mappingclass group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braidgroup relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss briefly the relation to the Thurston theory. 1