Clusters and seeds in acyclic cluster algebras

By Aslak Bakke Buan, Robert J. Marsh, I. Reiten, G. Todorov. 2005

97 citations recorded by the source corpus.

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

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

View Paper: https://www.semanticscholar.org/paper/0d57944ff25c9df3465ded8ff75baeefba6a95f1

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Citations in the corpus

On the quiver Grassmannian in the acyclic case (2006, result): “We deduce another proof of the denominator property in the acyclic case, [8], [ 2 ], [14].”
Cluster categories (2010, methodology): “Already from these properties of α one gets the following, which was proved independently in [25] and [34] for acyclic cluster algebras.”
Acyclic Calabi–Yau categories (2006, background): “If k is a field and Q a quiver without oriented cycles, the associated cluster category CQ is the ‘largest’ 2-Calabi–Yau category under the derived category of representations of Q over k.”
On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, unknown): “…to Fomin-Zelevinsky’s cluster algebras [19], [20], [8], [21] developed by Marsh-Reineke-Zelevinsky [35], Buan-Marsh-Reineke-Reiten-Todorov [6], [7], Caldero-Chapoton and Caldero-Keller [9], [10], [11], Geiss-Leclerc-Schr¨oer [25], [26], [23], [2] and many others; cf. the surveys [4], [31],…”
Representations of Algebras and Related Topics (2011, background): “e in general 2-CY categories (see aslo Remark 3.15). The category CQ provides a ﬁrst and very important example which permits to categorify the acyclic cluster algebras. We refer to [24], [30], [26], [28], [33], [34] for results on the categoriﬁcation of acyclic cluster algebras. Detailed overviews ”
Coloured quiver mutation for higher cluster categories (2008, background): “Each cluster in a fixed cluster algebra comes together with a finite quiver, and in the categorical model this quiver is in fact the Gabriel quiver of the corresponding tilting object [ BMRT ].”
Calabi–Yau triangulated categories (2008, result): “The theorem is proved in [16] on the basis of the previous results obtained by many authors notably Buan-Marsh-Reiten-Todorov [13], Buan-Marsh-Reiten [4], Buan-Marsh-Reineke-Reiten-Todorov [3], Caldero-Chapoton [14], Marsh-ReinekeZelevinsky [46], .”
From triangulated categories to cluster algebras (2005, background): “We prove that the Laurent expansion of a cluster variable has positive coefficients, when the seed is associated to an orientation of a given Dynkin diagram.”
A categorification of non-crossing partitions (2013, background): “ween non-crossing partitions and representations of quivers has already been pointed out by Ingalls and Thomas [32]; it arises from the categoriﬁcation of cluster algebras [19] via cluster categories [9, 39]. We also introduce the notion of a (mono-)morphism between generalised Cartan lattices, and ”