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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

From triangulated categories to cluster algebras II   (Citations: 128)
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In the acyclic case, we establish a one-to-one 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.
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    • ...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. Cluster-Tilted 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 self-extension )...

    David Hernandezet 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 finite-dimensional algebras (over an algebraically closed field) in general, see for example [16], [14]...

    Michael Barotet al. From iterated tilted algebras to cluster-tilted 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 Buanet al. Cluster structures from 2-Calabi–Yau categories with loops

    • ...(3) If � = CQ for some acyclic quiver Q, let XM be the image of the Caldero-Chapton 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 Dinget al. Realizing enveloping algebras via varieties of modules

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