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Cluster algebras via cluster categories with infinite-dimensional morphism spaces
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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.
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From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes
Introduces a piecewise-linear flow on cluster complexes whose leaves generalize green mutation, proving these complexes are spheres for Dynkin quivers and contractible for Euclidean quivers.
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