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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
TUBULAR CLUSTER ALGEBRAS II: EXPONENTIAL GROWTH
"... Abstract. Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding ..."
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Abstract. Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding
τ2stable tilting complexes over weighted projective lines. arXiv:1402.6036
, 2014
"... Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite ..."
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Abstract. Let X be a weighted projective line and cohX the associated categoy of coherent sheaves. We classify the tilting complexes T in Db(cohX) such that τ2T ∼ = T, where τ is the AuslanderReiten translation in Db(cohX). As an application of this result, we classify the 2representationfinite algebras which are derivedequivalent to a canonical algebra. This complements IyamaOppermann’s classification of the iterated tilted 2representationfinite algebras. By passing to 3preprojective algebras, we obtain a classification of the selfinjective clustertilted algebras of canonicaltype. This complements Ringel’s classification of the selfinjective clustertilted algebras. 1.
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"... Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields o ..."
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Abstract. Over any field of positive characteristic we construct 2CYtilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2CYtilted even if they do not arise from potentials. In another direction, we compute the fractionally CalabiYau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2. Introduction A 2CYtilted algebra is an endomorphism algebra of a clustertilting object in a 2CalabiYau triangulated category. There are close connections between 2CYtilted algebras and Jacobian algebras of quivers with potentials as introduced by Derksen, Weyman and Zelevinsky On the other hand, by the work of Amiot [2], any finitedimensional Jacobian algebra is 2CYtilted. It is therefore natural to ask whether any 2CYtilted algebra is a Jacobian algebra of a quiver with potential [3, Question 2.20]. The purpose of this note is twofold. First, we provide a negative answer to this question over any field of positive characteristic. Our examples are given by certain selfinjective Nakayama algebras which are also known as truncated cycle algebras. Second, we show that it is actually possible to slightly extend the notion of a potential in order to exclude this kind of examples. Let us explain the motivation behind such extension. Since 2CYtilted algebras have some remarkable homological and structural properties Consider for example the algebra Λ K = K[x]/(x n−1 ) over a field K for some n > 2, which could be described as a quiver with one vertex, one loop x and a relation x n−1 . Date: March 26, 2014.
ON FINITE DIMENSIONAL JACOBIAN ALGEBRAS
"... Abstract. We show that Jacobian algebras arising from a sphere with npunctures, with n ≥ 5, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional. 1. ..."
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Abstract. We show that Jacobian algebras arising from a sphere with npunctures, with n ≥ 5, are finite dimensional algebras. We consider also a family of cyclically oriented quivers and we prove that, for any primitive potential, the associated Jacobian algebra is finite dimensional. 1.