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14
RIGID MODULES OVER PREPROJECTIVE ALGEBRAS II: THE Kacmoody Case
, 2007
"... Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials ..."
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Cited by 44 (8 self)
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Let Q be a finite quiver without oriented cycles, and let Λ be the associated preprojective algebra. We construct many Frobenius subcategories of mod(Λ), which yield categorifications of large classes of cluster algebras. This includes all acyclic cluster algebras. We show that all cluster monomials can be realized as elements of the dual of Lusztig’s semicanonical basis of a universal enveloping algebra U(n), where n is a maximal nilpotent subalgebra of the symmetric KacMoody Lie algebra g associated to the quiver Q.
Quiver varieties and cluster algebras
, 2009
"... Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a FominZelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric KacMoody Lie algebra, studied earlier by the author via perverse sheaves on ..."
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Cited by 15 (0 self)
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Motivated by a recent conjecture by Hernandez and Leclerc [31], we embed a FominZelevinsky cluster algebra [21] into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric KacMoody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties [49]. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The positivity and linearly independence (and probably many other) conjectures of cluster monomials [21] follow as consequences, when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 10 (8 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Clustertilting theory
 CONTEMPORARY MATHEMATICS
"... Cluster algebras were introduced by Fomin and Zelevinsky in order to understand the dual canonical basis of the quantized enveloping algebra of a quantum group and total positivity for algebraic groups. A cluster category is obtained by forming an appropriate quotient of the derived category of rep ..."
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Cited by 6 (1 self)
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Cluster algebras were introduced by Fomin and Zelevinsky in order to understand the dual canonical basis of the quantized enveloping algebra of a quantum group and total positivity for algebraic groups. A cluster category is obtained by forming an appropriate quotient of the derived category of representations of a quiver. In this survey article, we describe the connections between cluster categories and cluster algebras, and we survey the representationtheoretic applications of cluster categories, in particular how they provide an extended version of classical tilting theory. We also describe a number of interesting new developments linking cluster algebras, cluster categories, representation theory and the canonical basis.
Denominators of cluster variables
"... Abstract. Associated to any acyclic cluster algebra is a corresponding triangulated ..."
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Abstract. Associated to any acyclic cluster algebra is a corresponding triangulated
CLUSTER COMPLEXES VIA SEMIINVARIANTS
, 2008
"... We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semiinvariants on these spaces which we call virtual semiinvariants and prove that they satisfy the three basic ..."
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Cited by 3 (0 self)
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We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semiinvariants on these spaces which we call virtual semiinvariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semiinvariants and the
Partial tilting modules over mreplicated algebras
, 810
"... Abstract Let A be a hereditary algebra over an algebraically closed field k and A (m) be the mreplicated algebra of A. Given an A (m)module T, we denote by δ(T) the number of non isomorphic indecomposable summands of T. In this paper, we prove that a partial tilting A (m)module T is a tilting A ( ..."
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Abstract Let A be a hereditary algebra over an algebraically closed field k and A (m) be the mreplicated algebra of A. Given an A (m)module T, we denote by δ(T) the number of non isomorphic indecomposable summands of T. In this paper, we prove that a partial tilting A (m)module T is a tilting A (m)module if and only if δ(T) = δ(A (m)), and that every partial tilting A (m)module has complements. As an application, we deduce that the tilting quiver KA (m) of A (m) is connected. Moreover, we investigate the number of complements to almost tilting modules over duplicated algebras. Key words: contravariantly finite categories, partial tilting modules, mreplicated algebras, tilting quivers
The cluster complex of an hereditary Artin algebra
, 2008
"... We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the clustertilting objects form one equivalence class under mutation. ..."
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We show that the cluster complex of an arbitrary hereditary artin algebra has the structure of an abstract simplicial polytope. In particular, the clustertilting objects form one equivalence class under mutation.
UNIVERSAL DERIVED EQUIVALENCES OF POSETS OF CLUSTER TILTING OBJECTS
, 710
"... Abstract. We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flipflop. We deduce that the posets of cluster tilting objects of derived equivalent path algeb ..."
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Abstract. We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their cluster tilting objects are related by a simple combinatorial construction, which we call a flipflop. We deduce that the posets of cluster tilting objects of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. In particular, all Cambrian lattices corresponding to the various orientations of the same Dynkin diagram are universally derived equivalent. 1.