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On the combinatorics of rigid objects in 2CalabiYau categories
 INT. MATH. RES. NOT. IMRN, (11):ART. ID RNN029
, 2008
"... Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposab ..."
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Given a triangulated 2CalabiYau category C and a clustertilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a clustertilting subcategory T ′ form a basis of the Grothendieck group of T and that, if T and T ′ are related by a mutation, then the indices with respect to T and T ′ are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given clustertilting subcategory T. Conjecturally, these indices coincide with FominZelevinsky’s gvectors.
On a cluster category of infinite Dynkin type, and the relation to triangulations of the infinitygon
 Math. Z
"... This paper investigates a certain 2CalabiYau triangulated category D whose AuslanderReiten quiver is ZA∞. We show that the cluster tilting subcategories of D form a socalled cluster structure, and we classify these subcategories in terms of ..."
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This paper investigates a certain 2CalabiYau triangulated category D whose AuslanderReiten quiver is ZA∞. We show that the cluster tilting subcategories of D form a socalled cluster structure, and we classify these subcategories in terms of
THE CLUSTER CATEGORY OF A CANONICAL ALGEBRA
, 2010
"... Abstract. We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the clustertilting objects form a c ..."
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Abstract. We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the clustertilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is nonnegative, or equivalently, if A is of tame (domestic or tubular) representation type. 1.
QUANTUM UNIPOTENT SUBGROUP AND DUAL CANONICAL BASIS
, 2010
"... ... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture w ..."
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... cluster algebra structure on the coordinate ring C[N(w)] of the unipotent subgroup, associated with a Weyl group element w. And they proved cluster monomials are contained in Lusztig’s dual semicanonical basis S ∗. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras in [4] to the dual canonical basis B up. In particular, we prove that the quantum analogue Oq[N(w)] of C[N(w)] has the induced basis from B up, which contains quantum flag minors and satisfies a factorization property with respect to the ‘qcenter’ of Oq[N(w)]. This generalizes Caldero’s results [7, 8, 9] from ADE cases to an arbitary symmetrizable KacMoody Lie algebra.
CLUSTER CHARACTERS II: A MULTIPLICATION FORMULA
"... cluster tilting object. Under some constructibility assumptions on C which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, we prove a multiplication formula for the cluster character ..."
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cluster tilting object. Under some constructibility assumptions on C which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, we prove a multiplication formula for the cluster character associated with any cluster tilting object. This formula generalizes those obtained by Caldero–Keller for representation finite path algebras and by Xiao–Xu for finitedimensional path algebras. It is analogous to a formula obtained by Geiss–Leclerc–Schröer in the context of preprojective algebras.
Cluster structures from 2CalabiYau categories with loops
, 2008
"... We generalise the notion of cluster structures from the work of BuanIyamaReitenScott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Homfinite 2CalabiYau category, the set of maximal rigid objects satisfies these axioms whenever there are ..."
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We generalise the notion of cluster structures from the work of BuanIyamaReitenScott to include situations where the endomorphism rings of the clusters may have loops. We show that in a Homfinite 2CalabiYau category, the set of maximal rigid objects satisfies these axioms whenever there are no 2cycles in the quivers of their endomorphism rings.