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Cluster algebras via cluster categories with infinite-dimensional morphism spaces

Part of: Abelian categories

Published online by Cambridge University Press:  28 September 2011

Pierre-Guy Plamondon
Pierre-Guy Plamondon*
Affiliation:
Institut de Mathématiques de Jussieu, Université Paris Diderot – Paris 7, UMR 7586 du CNRS, Case 7012, Bâtiment Chevaleret, 75205 Paris Cedex 13, France (email: plamondon@math.jussieu.fr)
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Abstract

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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 [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.

Keywords

triangulated categoriescluster categoriescluster algebras

MSC classification

Secondary: 18E30: Derived categories, triangulated categories
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1921 - 1954
Copyright
Copyright © Foundation Compositio Mathematica 2011

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