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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
DILOGARITHM IDENTITIES FOR CONFORMAL FIELD THEORIES AND CLUSTER ALGEBRAS: Simply Laced Case
, 2010
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Categorical tinkertoys for N = 2 gauge theories
"... In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abe ..."
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Cited by 7 (6 self)
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In view of classification of the quiver 4d N = 2 supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential (Q,W) associated to aN = 2 QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category rep(Q,W) of (finite–dimensional) representations of the Jacobian algebra CQ/(∂W) should enjoy what we call the Ringel property of type G; in particular, rep(Q,W) should contain a universal ‘generic ’ subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. More precisely, there is a family of ‘light ’ subcategories Lλ ⊂ rep(Q,W), indexed by points λ ∈ N, where N is a projective variety whose irreducible components are copies of P1 in one–to–one correspondence with the simple factors of G. If λ is the generic point of the i–th irreducible component, Lλ is the universal subcategory corresponding to the i–th simple factor of G. Matter, on the contrary, is encoded in the subcategories Lλa where {λa} is a finite set of closed points in N. In particular, for a Gaiotto theory there is one such family of subcategories, Lλ∈N, for each maximal degeneration of the corresponding surface Σ, and the index variety N may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed–point subcategories to ‘fixtures ’ (spheres with three punctures of various kinds) and higher–order generalizations. The rules for ‘gluing ’ categories are more general that the geometric gluing of surfaces, allowing for a few additional exceptional N = 2 theories which are not of the Gaiotto class. We include several examples and some amusing consequence, as the characterization in terms of quiver combinatorics of asymptotically free theories.
Periodicities Of T and YSystems, DILOGARITHM IDENTITIES, AND CLUSTER ALGEBRAS I: Type Br
, 2010
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Dilogarithm Identities for SineGordon and Reduced SineGordon YSystems
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... We study the family of Ysystems and Tsystems associated with the sineGordon models and the reduced sineGordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and t ..."
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Cited by 6 (4 self)
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We study the family of Ysystems and Tsystems associated with the sineGordon models and the reduced sineGordon models for the parameter of continued fractions with two terms. We formulate these systems by cluster algebras, which turn out to be of finite type, and prove their periodicities and the associated dilogarithm identities which have been conjectured earlier. In particular, this provides new examples of periodicities of seeds.
Intermediate cotstructures, twoterm silting objects, τ tilting modules, and torsion classes
"... Abstract. If (A,B) and (A′,B′) are cotstructures of a triangulated category, then (A′,B′) is called intermediate if A ⊆ A ′ ⊆ ΣA. Our main results show that intermediate cotstructures are in bijection with twoterm silting subcategories, and also with support τtilting subcategories under some ..."
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Cited by 3 (0 self)
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Abstract. If (A,B) and (A′,B′) are cotstructures of a triangulated category, then (A′,B′) is called intermediate if A ⊆ A ′ ⊆ ΣA. Our main results show that intermediate cotstructures are in bijection with twoterm silting subcategories, and also with support τtilting subcategories under some assumptions. We also show that support τtilting subcategories are in bijection with certain finitely generated torsion classes. These results generalise work by Adachi, Iyama, and Reiten. The aim of this paper is to discuss the relationship between the following objects. • Intermediate cotstructures. • Twoterm silting subcategories. • Support τtilting subcategories.
CLUSTER ALGEBRAS: AN INTRODUCTION
, 2013
"... To the memory of Andrei Zelevinsky Abstract. Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced ..."
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To the memory of Andrei Zelevinsky Abstract. Cluster algebras are commutative rings with a set of distinguished generators having a remarkable combinatorial structure. They were introduced