From triangulated categories to cluster algebras

By Philippe Caldero, B. Keller. 2005

347 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, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f3f7d49171dfa4125d8f477282ae549a6d7eb75c

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Citations in the corpus

Quivers with potentials and their representations I: Mutations (2007, neutral): “low in Section 10). 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. Most of the results in these papers are f”
Quiver varieties and cluster algebras (2009, methodology): “unctions of Euler numbers of quiver Grassmannian varieties. This important result was ﬁrst proved by Caldero-Chapoton in type ADE [7]. Later it was generalized to any acyclic quiver by Caldero-Keller [8] using various results in the cluster category theory (see [37] for the reference). We recall the”
Generic bases for cluster algebras from the cluster category (2011, background): “or any quiver with potential in [36]. An indecomposable reachable object of CQ,W is an object which can be obtained from some Γi by a sequence of mutations (at non-frozen vertices). Theorem 3.8 ([5], [6], [43]). The cluster character induces a surjection from the set of isomorphism classes of indeco”
An approach to non-simply laced cluster algebras (2005, methodology): “rem becomes a generalization to the entire ﬁnite type of theorem 1.9 of [FZ03]. 11 Another result of the representation theoretical approach that can be extended is the ﬁrst multiplication formula of [CK05a]. Let M be a G-invariant kQ-module, we decompose M into G-indecomposable kQ-modules M = L i M”
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories (2008, neutral): “he object E is isomorphic to N (we do not ﬁx an isomorphism). Notice that this subset is a cone (i.e. stable under multiplication by non zero scalars) in the vector space Ext1(L,M). Proposition 5.13 ([31]). The subset Ext1(L,M)N is constructible in Ext1(L,M). In particular, it is an algebraic variet”
Cluster algebras and representation theory (2010, methodology): “1 is the analogue for modΛ of a formula of Caldero and Keller [6] for the cluster categories introduced by Buan, Marsh, Reineke, Reiten and Todorov [4] to model cluster algebras with an acyclic seed.”
Positivity for cluster algebras from surfaces (2009, methodology): “ion to quiver representations and the introduction of the cluster category [BMRRT] (see also [CCS1] in type A). More speciﬁcally, there is a geometric interpretation (found in [CC] and generalized in [CK]) of coeﬃcients in Laurent expansions as Euler-Poincar´e characteristics of appropriate Grassman”
Representations of Algebras and Related Topics (2011, background): “eneral 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 thi”
Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories (2008, neutral): “In their study [6] of the connections between cluster algebras (see [22]) and quiver representations, P.”
On cluster algebras with coefficients and 2-Calabi-Yau categories (2007, background): “…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], [39], [40].”
Bases for cluster algebras from surfaces (2011, background): “struction of bases for cluster algebras is a problem that has gained a lot of attention recently. Caldero and Keller showed that for cluster algebras of ﬁnite type, the cluster monomials form a basis [CK]. For cluster algebras which are not of ﬁnite type, the cluster monomials do not span the cluste”