Quivers with relations arising from clusters $(A_n$ case)

By Philippe Caldero, Frédéric Chapoton, Ralf Schiffler. 2004

496 citations recorded by the source corpus.

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6973957a32da3ff2ba3647f5ead33e04ece773c

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Citations in the corpus

Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, background): “As a categorification of cluster algebras, Buan, Marsh, Reineke, Reiten and Todorov [23] and Caldero, Chapoton and Schiffler (forAn case) [ 25 ] introduced certain triangulated categories called cluster categories, which were defined as orbit categories [47] of the derived categories of quiver repre”
Cluster algebras via cluster categories with infinite-dimensional morphism spaces (2010, background): “Schiffler in [7] in the An case) and generalized to any quiver with potential by C.”
Generic Variables in Acyclic Cluster Algebras (2010, methodology): “ derived category of H-mod with shift functor [1] and Auslander-Reiten translation τ. The cluster category CQ of Q was introduced in [BMR+06] in order to categorify the cluster algebra A(Q) (see also [CCS06] for Dynkin type A). It is deﬁned as the orbit category of the functor τ−1[1] in Db(H-mod). I”
Non-kissing and non-crossing complexes for locally gentle algebras (2018, methodology): “This was first done for triangulations of polygons in [CCS06], and then for any orientable surface with boundary in [ABCJP10, LF09].”
Wall and chamber structure for finite-dimensional algebras (2018, background): “ed by Fomin and Zelevinski in [18], prompting a lot of subsequent work on the subject. In particular, there are several representation theoretic categoriﬁcations of cluster algebras, see for instance [13, 12, 20]. More recently, Adachi, Iyama and Reiten introduced in [1] τ-tilting theory, an extensi”
A geometric model for cluster categories of type Dn (2006, methodology): “On the other hand, in the approach of [11], which is only valid in type An, the cluster category is realized by an ad-hoc method as a category of diagonals of a regular polygon with n+ 3 vertices.”
Acyclic Calabi–Yau categories (2006, methodology): “These were introduced in [BMRRT06] and, for Dynkin quivers of type An, in [CCS06].”
Cluster structures from 2-Calabi–Yau categories with loops (2008, background): “These categories have been widely studied for the case when the initial hereditary 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 th”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, unknown): “nderstanding of the ‘decorated quiver representations’ introduced by Reineke-Marsh-Zelevinsky [108]. For quivers of type A, an equivalent category was deﬁned independently by Caldero-Chapoton-Schiﬄer [29] using an entirely diﬀerent description. Clearly the category CQ is k-linear. It is not hard to ”
Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras (2006, methodology): “A similar type of program was carried out for acyclic cluster algebras, leading to the notions of cluster categories and cluster tilted algebras [BMRRT][BMR1] (see [ CCS ] for the An case), along with related work in [GLS].”