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Cluster algebras via cluster categories with infinite-dimensional morphism spaces

  • Pierre-Guy Plamondon (a1)
Abstract
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.

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[Ami09]C. Amiot , Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier 59 (2009), 25252590.

[BFZ05]A. Berenstein , S. Fomin and A. Zelevinsky , Cluster algebras III: upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 152.

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[CCS06]P. Caldero , F. Chapoton and R. Schiffler , Quivers with relations arising from clusters (An case), Trans. Amer. Math. Soc. 358 (2006), 13471364.

[CK08]P. Caldero and B. Keller , From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169211.

[DWZ08]H. Derksen , J. Weyman and A. Zelevinsky , Quivers with potentials and their representations I: mutations, Selecta Math. (N.S.) 14 (2008), 59119.

[DWZ10]H. Derksen , J. Weyman and A. Zelevinsky , Quivers with potentials and their representations II: applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749790.

[Dup11]G. Dupont , Generic variables in acyclic cluster algebras, J. Pure Appl. Algebra 215 (2011), 628641.

[FZ02]S. Fomin and A. Zelevinsky , Cluster algebras I: foundations, J. Amer. Math. Soc. 15 (2002), 497529.

[FK10]C. Fu and B. Keller , On cluster algebras with coefficients and 2-Calabi–Yau categories, Trans. Amer. Math. Soc. 362 (2010), 859895.

[GLS08]C. Geiss , B. Leclerc and J. Schröer , Preprojective algebras and cluster algebras, in Trends in representation theory of algebras and related topics, EMS Series of Congress Reports, vol. 1 (European Mathematical Society, Zürich, 2008), 253283.

[KY11]B. Keller and D. Yang , Derived equivalences from mutations of quivers with potential, Adv. Math. 226 (2011), 21182168.

[Pal08]Y. Palu , Cluster characters for 2-Calabi–Yau triangulated categories, Ann. Inst. Fourier 58 (2008), 22212248.

[Pla11]P.-G. Plamondon , Cluster characters for cluster categories with infinite-dimensional morphism spaces, Adv. Math. 227 (2011), 139.

[Sch10]R. Schiffler , On cluster algebras arising from unpunctured surfaces II, Adv. Math. 223 (2010), 18851923.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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