Compositio Mathematica

Research Article

Cluster algebras via cluster categories with infinite-dimensional morphism spaces

Pierre-Guy Plamondona1

a1 Institut de Mathématiques de Jussieu, Université Paris Diderot – Paris 7, UMR 7586 du CNRS, Case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France (email:


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.

(Received May 12 2010)

(Accepted February 07 2011)

(Online publication September 28 2011)

2010 Mathematics Subject Classification

  • 18E30


  • triangulated categories;
  • cluster categories;
  • cluster algebras


The author was financially supported by an NSERC scholarship.