On cluster algebras with coefficients and 2-Calabi-Yau categories

By Changjian Fu, B. Keller. 2007

146 citations recorded by the source corpus.

Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in 'Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/8fffbc8d2480629690cd412ca4a6cfcae9988c6e

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Citations in the corpus

Ordered Exchange Graphs (2013, methodology): “(4.1) More precisely, it follows from Keller’s Morita theorem for triangulated categories ([59, Theorem 4.3]) that there is a non-positive dg algebra B together with a triangle equivalence D(B)→ D(A) taking B toM .”
Cluster categories (2010, background): “There are several interesting generalizations beyond the acyclic case, and we refer to ([37],[52],[56],[57],[58],[93],[96],[97]).”
Positivity for cluster algebras (2013, methodology): “The most general results are the following: • A formula involving the Euler-Poincaré characteristic of quiver Grassmannians obtained in [15], [8] using categorification and generalizing results in [5], [6].”
Acyclic Cluster Algebras Revisited (2012, methodology): “These papers all use heavy machinery of some kind: [12, 22] use general 2-Calabi-Yau triangulated categories, [9, 10] use representations of quivers with potentials, and [20] uses Donaldson-Thomas theory.”
Categorification of Skew-symmetrizable Cluster Algebras (2009, methodology): “For example, this allowed Fu and Keller [ 21 ] to prove the previous conjecture for a family of cluster algebras containing acyclic cluster algebras.”
On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories (2007, result): “In [ 9 ], the results of this paper have been used to prove these conjectures for certain classes of cluster algebras.”