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Keywords
(4)
Cluster Algebra
Cluster Category
Triangulated Category
calabi yau
Related Publications
(18)
Cluster categories and duplicated algebras
On the structure of CalabiYau categories with a cluster tilting subcategory
Cluster algebras as Hall algebras of quiver representations
Generalized associahedra via quiver representations
Clustertilted algebras
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From triangulated categories to cluster algebras II
From triangulated categories to cluster algebras II,10.1016/j.ansens.2006.09.003,Annales Scientifiques De L Ecole Normale Superieure,Bernhard Keller
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From triangulated categories to cluster algebras II
(
Citations: 128
)
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Bernhard Keller
In the acyclic case, we establish a onetoone correspondence between the tilting objects of the
cluster category
and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi–Yau property of the cluster category.
Journal:
Annales Scientifiques De L Ecole Normale Superieure  ANN SCI ECOLE NORM SUPER
, vol. 39, no. 6, pp. 9831009, 2006
DOI:
10.1016/j.ansens.2006.09.003
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Citation Context
(46)
...Together with cluster categories they provide an algebraic understanding (see [
11
,
14
,
15
]) of combinatorics of cluster algebras defined and studied by Fomin and Zelevinsky in [
17
]...
Bin Zhu
.
ClusterTilted Algebras and Their Intermediate Coverings
...Remark 2.3 (i) In recent years, many examples of categorifications of cl uster algebras have been constructed (see e.g. [MRZ, BMRRT, CC, GLS2, BIRS,
CK
, GLS4])...
...Taking into ac count the additivity properties of the Euler characteristics and the results of [
CK
], this gives for χq(L(m))6 2 a formula similar to (49), in which the indecomposable representation M[τ−(β)] is replaced by a generic representation of C (or equivalently a representation without selfextension )...
David Hernandez
,
et al.
Cluster algebras and quantum affine algebras
...The connection between cluster algebras and cluster categories was deepened by various authors and expanded over the original limit of finite type to hereditary finitedimensional algebras (over an algebraically closed field) in general, see for example [
16
], [14]...
Michael Barot
,
et al.
From iterated tilted algebras to clustertilted algebras
... category is the category of finite dimensional representations of an acyclic quiver Q. (When Q is a quiver with underlying graph An, the cluster category was independently defined in [11].) It has then been shown that the indecomposable rigid objects are in bijection with the cluster variables in the cluster algebra AQ associated with the same quiver, and under this bijection the clusters correspond to the maximal rigid objects ([
12
] based ...
Aslak Bakke Buan
,
et al.
Cluster structures from 2Calabi–Yau categories with loops
...(3) If � = CQ for some acyclic quiver Q, let XM be the image of the CalderoChapton map ([1]) for any �module M. Then Theorem 5.7 is equivalent to the fact XMXN = XM ⊕N for any �modules M and N ([
2
])...
Ming Ding
,
et al.
Realizing enveloping algebras via varieties of modules
References
(38)
Positivity and canonical bases in rank 2 cluster algebras of finite and affine types
(
Citations: 46
)
Paul Sherman
,
Andrei Zelevinsky
Published in 2003.
Triangulated categories in the representation theory of finitedimensional algebras
(
Citations: 289
)
D. Happel
Published in 1988.
Toric degenerations of Schubert varieties
(
Citations: 30
)
Philippe Caldero
Published in 2000.
AuslanderReiten sequences and representationfinite algebras
(
Citations: 109
)
Peter Gabriel
Cluster algebras and Poisson geometry
(
Citations: 53
)
Michael Gekhtman
,
Michael Shapiro
,
Alek Vainshtein
Published in 2002.
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Citations
(128)
Generic variables in acyclic cluster algebras
(
Citations: 15
)
Gregoire Dupont
Journal:
Journal of Pure and Applied Algebra  J PURE APPL ALG
, vol. 215, no. 4, pp. 628641, 2011
Quiver Grassmannians associated with string modules
(
Citations: 6
)
Giovanni Cerulli Irelli
Journal:
Journal of Algebraic Combinatorics  J ALGEBR COMB
, vol. 33, no. 2, pp. 259276, 2011
Contemporary Features, Risk Factors, and Prognosis of the PostPericardiotomy Syndrome
Massimo Imazio
,
Antonio Brucato
,
Maria Elena Rovere
,
Anna Gandino
,
Roberto Cemin
,
Stefania Ferrua
,
Silvia Maestroni
,
Alberto Barosi
,
Caterina Simon
,
Paolo Ferrazzi
,
Riccardo Belli
,
Rita Trinchero
http://academic.research.microsoft.com/io.ashx?type=5&id=49004977&selfId1=0&selfId2=0&maxNumber=12&query=
Journal:
American Journal of Cardiology  AMER J CARDIOL
, vol. 108, no. 8, pp. 11831187, 2011
Computational model of an inclined aerial ropeway and numerical method for analyzing nonlinear cablecar interaction
Danuta Bryja
,
Marta Knawa
Journal:
Computers & Structures  COMPUT STRUCT
, vol. 89, no. 21, pp. 18951905, 2011
Anabolic Status and Functional Impairment in Men With Mild Chronic Heart Failure
Francisco J. PastorPérez
,
Sergio ManzanoFernández
,
Iris P. Garrido Bravo
,
Francisco Nicolás
,
Pedro L. Tornel
,
Antonio Lax
,
Gonzalo de la Morena
,
Mariano Valdés
,
Domingo A. PascualFigal
Journal:
American Journal of Cardiology  AMER J CARDIOL
, vol. 108, no. 6, pp. 862866, 2011