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.

arXiv comment32 pages, added reference

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• Calabi-Yau algebras
• Quivers with relations arising from clusters (A_n case)
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• Categorification of skew-symmetrizable cluster algebras
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• Bases for cluster algebras from surfaces
• Triangular bases in quantum cluster algebras and monoidal categorification conjectures
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r
• Cluster categories for marked surfaces: punctured case
• Classical and Quantum Dilogarithm Identities
• Universal geometric cluster algebras
• Generic bases for cluster algebras and the Chamber Ansatz
• Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2
• Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
• Combinatorial frameworks for cluster algebras
• Periodicities in cluster algebras and dilogarithm identities
• Structure of seeds in generalized cluster algebras
• Colored BPS Pyramid Partition Functions, Quivers and Cluster Transformations
• Cluster characters II: A multiplication formula
• Quiver mutation and combinatorial DT-invariants
• On Generalized Cluster Categories
• C-vectors via $\tau$-tilting theory
• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• 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
• Quantum Dilogarithm Identities at Root of Unity
• Exact WKB analysis and cluster algebras
• Cluster algebras arising from cluster tubes
• 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
• Classical and Quantum Dynamics on Orbifolds
• Two-term tilting complexes for preprojective algebras of non-Dynkin type
• Positivity for Regular Cluster Characters in Acyclic Cluster Algebras
• Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams
• Acyclic cluster algebras with dense $g$-vector fans
• Strongly primitive species with potentials I: Mutations
• Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras
• Continuously triangulating the continuous cluster category
• Ideal mutations in triangulated categories and generalized Auslander-Reiten theory
• On maximal green sequence for quivers arising from weighted projective lines
• Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map
• Denominator vectors and dimension vectors from triangulated surfaces
• Categorical dynamical systems arising from sign-stable mutation loops
• Quivers with analytic potentials
• Mutations of reflections and existence of pseudo-acyclic orderings for type $A_n$