Cluster algebras via cluster categories with infinite-dimensional morphism spaces

Pierre-Guy Plamondon
2011 Compositio Mathematica  
AbstractWe 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'sCluster 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
more » ... it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of theE-invariant and show that an arbitrary decorated representation with vanishingE-invariant is characterized by itsg-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.
doi:10.1112/s0010437x11005483 fatcat:mfwpfj7d2fabfhnysy7f46vlhi