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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
Mutation of clustertilting objects and potentials
 Amer. Journal Math. (2008
"... Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that cl ..."
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Cited by 56 (10 self)
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Abstract. We prove that mutation of clustertilting objects in triangulated 2CalabiYau categories is closely connected with mutation of quivers with potentials. This gives a close connection between 2CYtilted algebras and Jacobian algebras associated with quivers with potentials. We show that clustertilted algebras are Jacobian and also that they are determined by their quivers. There are similar results when dealing with tilting modules over 3CY algebras. The nearly Morita equivalence for 2CYtilted algebras is shown to hold for the finite length modules over Jacobian algebras.
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 55 (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
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 39 (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.
Quantum cluster variables via Serre polynomials
, 2010
"... Abstract. For skewsymmetric acyclic quantum cluster algebras, we express the quantum Fpolynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the po ..."
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Cited by 35 (3 self)
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Abstract. For skewsymmetric acyclic quantum cluster algebras, we express the quantum Fpolynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.
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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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.
GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY
"... Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same i ..."
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Abstract. Inspired by recent work of Geiss–Leclerc–Schröer, we use Homfinite cluster categories to give a good candidate set for a basis of (upper) cluster algebras with coefficients arising from quivers. This set consists of generic values taken by the cluster character on objects having the same index. If the matrix associated to the quiver is of full rank, then we prove that the elements in this set are linearly independent. If the cluster algebra arises from the setting of Geiss–Leclerc–Schröer, then we obtain the basis found by these authors. We show how our point of view agrees with the spirit of conjectures of Fock–Goncharov concerning the parametrization of a basis of the upper cluster