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Quivers with Relations and Cluster Tilted Algebras

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Abstract

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.

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References

  1. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. To appear in Adv. Math.

  2. Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. To appear in Trans. Amer. Math. Soc.

  3. Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representations. Arxiv. Math. RT/0412077

  4. Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. To appear in Comment. Math. Helv.

  5. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\({A}_{n}\) case). Trans. Amer. Math. Soc. 358, 1347–1364 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Arxiv. Math. RT/0506018

  7. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15(2), 497–529 (2002) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154, 63–121 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Fomin, S., Zelevinsky, A.: \(Y\)-systems and generalized associahedra. Ann. Math. (2) 158(3), 977–1018 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Keller, B.: Triangulated orbit categories (2003) Arxiv. Math. RT/0503240

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Authors and Affiliations

  1. Institut Girard Desargues, Villeurbanne Cedex, France

    Philippe Caldero & Frédéric Chapoton

  2. Carleton University, Ottawa, Ontario, Canada

    Ralf Schiffler

Authors
  1. Philippe Caldero
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  2. Frédéric Chapoton
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  3. Ralf Schiffler
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Correspondence to Philippe Caldero.

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Caldero, P., Chapoton, F. & Schiffler, R. Quivers with Relations and Cluster Tilted Algebras. Algebr Represent Theor 9, 359–376 (2006). https://doi.org/10.1007/s10468-006-9018-1

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  • DOI: https://doi.org/10.1007/s10468-006-9018-1

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