Quivers with Relations and Cluster Tilted Algebras

By Philippe Caldero, Frédéric Chapoton, Ralf Schiffler. 2004

135 citations recorded by the source corpus.

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6dc2196c1f80ab84a669b083a3fd05abf15f0dc2

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Citations in the corpus

Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, background): “The following result appeared in [10] and was generalized in [9] and [6].”
Derived Categories (2016, background): “• Representation theory of algebras and cluster categories (see [36], [33], [82], [84], [92] and Example 18.”
Clusters and seeds in acyclic cluster algebras (2005, methodology): “The multiplication rule is independently interpreted in the case of nite type in [ CCS2 ].”
Cluster‐tilted algebras as trivial extensions (2006, background): “Since then, these algebras have been the subject of many investigations, see, for instance, [7, 8, 9, 10, 11, 12, 13 , 21].”
Cluster expansion formulas and perfect matchings (2008, background): “In [10], the authors obtained a formula for the denominators of the cluster expansion in types A, D, and E, see also [4].”
On Cluster Algebras Arising from Unpunctured Surfaces (2007, background): “In [ CCS2 ], the authors obtained a formula for the denominators of the cluster expansion in types A, D and E, see also [BMR].”
Quivers with potentials and their representations I: Mutations (2007, methodology): “The elementary approach of [28] has not been further pursued until now, giving way to a more sophisticated approach via cluster categories and cluster-tilted algebras developed in [7, 8, 9, 10, 12, 13 , 14, 15] and many other publications.”
Cluster categories (2010, background): “This type of connection between denominators and indecomposable rigid objects was first shown in [48] for Dynkin diagrams with alternating orientation, then in [32] and [101] for the general Dynkin case, with another approach in [35].”
Representations of Algebras and Related Topics (2011, methodology): “a consequence of Theorem 2.9. Geometric description of the acyclic cluster category. Another description of the cluster category has been given independently by Caldero, Chapoton and Schiﬄer in [31], [32] in the case where the quiver Q is an orientation of the graph An. The description is geometric:”