Cluster algebras via cluster categories with infinite-dimensional morphism spaces

Abstract

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.

32 pages, added reference

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Calabi-Yau algebras
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2
• Categorification of skew-symmetrizable cluster algebras
• Algèbres amassées et applications

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• Classical and Quantum Dilogarithm Identities
• Universal geometric cluster algebras
• Generic bases for cluster algebras and the Chamber Ansatz
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2
• Periodicities in cluster algebras and dilogarithm identities
• Structure of seeds in generalized cluster algebras
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• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
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• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• C-vectors via $\tau$-tilting theory
• A categorification of acyclic principal coefficient cluster algebras
• Density of $g$-vector cones from triangulated surfaces
• A conjecture on $C$-matrices of cluster algebras
• Synchronicity phenomenon in cluster patterns
• The classification of Zamolodchikov periodic quivers
• $t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique
• Cluster exchange groupoids and framed quadratic differentials
• Cluster algebras arising from cluster tubes
• Exact WKB analysis and cluster algebras
• Quantum Dilogarithm Identities at Root of Unity
• Dilogarithm Identities for Sine-Gordon and Reduced Sine-Gordon Y-Systems
• Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization
• Cluster algebras arising from cluster tubes I: integer vectors
• Positivity for Regular Cluster Characters in Acyclic Cluster Algebras
• Two-term tilting complexes for preprojective algebras of non-Dynkin type
• Strongly primitive species with potentials I: Mutations
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Classical and Quantum Dynamics on Orbifolds
• Acyclic cluster algebras with dense $g$-vector fans
• Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
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• Denominator vectors and dimension vectors from triangulated surfaces
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• Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map
• On maximal green sequence for quivers arising from weighted projective lines