Cluster algebras as Hall algebras of quiver representations

By Philippe Caldero, Frédéric Chapoton. 2004

477 citations recorded by the source corpus.

Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.

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

View Paper: https://www.semanticscholar.org/paper/b61bdd667c6ef4b6d263d196fd42a5a14988b1fc

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Citations in the corpus

Ordered Exchange Graphs (2013, methodology): “In parallel, instead of objects in C(Q,W ), Derksen, Weyman and Zelevinsky constructed in [28] the Caldero– Chapoton map for decorated representations over the Jacobian algebra J and gave the first complete proof of some of Fomin and Zelevinsky’s conjectures in the skewsymmetric case; Nagao construc”
Cluster algebras IV: Coefficients (2006, unknown): “We denote xi = xi,t, x′i = xi,t′ , yi = yi,t, y ′ i = yi,t′ , bij = b t ij , b ′ ij = b t′ ij (3.9) (so as to match the notation of Definition 2.4) and ŷj = ŷj,t, ŷ′j = ŷj,t′ .”
Quiver Grassmannians and degenerate flag varieties (2011, background): “It was observed in [4] that these varieties play an important role in cluster algebra theory [19]; namely, the cluster variables can be described in terms of the Euler characteristic of quiver Grassmannians.”
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].”
Categorification of Skew-symmetrizable Cluster Algebras (2009, methodology): “In the skew-symmetric case, according to the work of Caldero–Chapoton [ 7 ], of Caldero–Keller [9, 10], of Palu [55], of Dehy–Keller [15] and of Fu–Keller [21] one can assign to every object X of C a Laurent polynomial in the cluster variables of an initial seed of the cluster algebra A categorified”
Quivers with Relations and Cluster Tilted Algebras (2004, background): “On the other hand, the first and second author obtained in [ 4 ] a formula for the numerators of the Laurent polynomial expansions of the cluster variables.”
Cluster structures from 2-Calabi–Yau categories with loops (2008, background): “... 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 clu”