From triangulated categories to cluster algebras II

By Philippe Caldero, B. Keller. 2005

299 citations recorded by the source corpus.

The cluster category is a triangulated category introduced for its combinatorial similarities with cluster algebras. We prove that a cluster algebra \(\mathcal{A}\) of finite type can be realized as a Hall algebra, called exceptional Hall algebra, of the cluster category. This realization provides a natural basis for \(\mathcal{A}\). We prove new results and formulate conjectures on ‘good basis’ properties, positivity, denominator theorems and toric degenerations.

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

View Paper: https://www.semanticscholar.org/paper/e854172bc9a3c1a11eb2cd0c0280299a02b9e8f0

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Citations in the corpus

Quivers with potentials and their representations II: Applications to cluster algebras (2009, methodology): “It was shown in [7, 8], 2 that the F -polynomial of M has nonnegative coeﬃcients in this case.”
Cluster characters for cluster categories with infinite-dimensional morphism spaces (2010, background): “Keller in [8] that, for cluster algebras associated with acyclic quivers, the Caldero-Chapoton map induces a bijection between the set of isomorphism classes of indecomposable rigid objects and the set of cluster variables.”
Representations of Algebras and Related Topics (2011, background): “ 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 of this subj”
Cluster algebras and derived categories (2012, background): “j (i.e. formal compositions of ≥ 0 arrows). Let α1, ..., αn be the simple roots of the root system corresponding to the underlying graph of Q. The following theorem is a consequence of the results of [12]. Cluster algebras and derived categories 23 • e1 e e2 2 −e1 −e −e2 • α1 α2 α1 +α2 −α −α2 Figure”
Mutation in triangulated categories and rigid Cohen–Macaulay modules (2006, methodology): “c and Schr¨oer [35] applied mutation to the study of rigid modules over preprojective algebras and the coordinate rings of maximal unipotent subgroups of semisimple Lie groups. We refer the reader to [21, 22, 26, 49, 51, 13, 69, 63, 62] for more recent developments in the study of cluster categories”