Cluster algebras via cluster categories with infinite-dimensional morphism spaces

Details Cluster algebras via cluster categories with infinite-dimensional morphism spaces Cluster algebras via cluster categories with infinite-dimensional morphism spaces

2010-04-06

arxiv.org/abs/1004.0830v2

Mathematics

Representation Theory

32 pages, added reference

Scientific paper

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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