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Cluster algebras via cluster categories with infinite-dimensional morphism spaces

2011; Cambridge University Press; Volume: 147; Issue: 6 Linguagem: Inglês

10.1112/s0010437x11005483

1570-5846

Pierre‐Guy Plamondon,

Nonlinear Waves and Solitons

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 [Compositio Math. 143 (2007), 112–164] 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.