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Repetitive cluster categories of type \(D_n\)

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Abstract

In this paper, we show that the repetitive cluster category of type \(D_n\), defined as the orbit category \(\mathcal {D}^b(\mathrm {mod} {\textsf {k}}D_n)/(\tau ^{-1}[1])^p\), is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, [17], which we recover when \(p=1\).

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References

  1. Baur, K., Marsh, R.: A geometric description of the\(m\)-cluster categories of type\(D_n\), Int. Math. Res. Not. IMRN, (4):Art. ID rnm011, 19, (2007)

  2. Baur, K., Marsh, R.: A geometric description of \(m\)-cluster categories. Trans. Amer. Math. Soc. 360(11), 5789–5803 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572–618 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Buan, A., Torkildsen, H.: The number of elements in the mutation class of a quiver of type\(D_n\), Electron. J. Combin. 16(1): Research Paper 49, 23, (2009)

  5. Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (\(A_n\) case). Trans. Amer. Math. Soc. 358(3), 1347–1364 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

  7. Fomin, S., Zelevinsky, A.: Cluster algebras II. Finite type classification, Inventiones Mathematicae 154(1), 63–121 (2003)

    Article  MATH  Google Scholar 

  8. Gabriel, P.: Unzerlegbare Darstellungen. Manuscripta Math 6, 71–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Mathematical Society. Lecture Notes Series 119, (Cambridge University Press), (1988)

  10. Jørgensen, P.: Quotients of cluster categories. Proc. Roy. Soc. Edinburgh Sect. A 140(1), 65–81 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Keller, B.: Calabi-Yau triangulated categories, Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep. pages 467–489. Eur. Math. Soc., Zürich, (2008)

  12. Keller, B.: Cluster algebras, quiver representations and triangulated categories, Triangulated categories, volume 375 of London Math. Soc. Lecture Note Ser. pages 76–160. Cambridge Univ. Press, Cambridge, (2010)

  13. Keller, B.: Triangulated orbit categories. Document Math. 10, 551–581 (2005)

  14. Lamberti, L.: A geometric interpretation of the triangulated structure of m-cluster categories. Communications in Algebra 42, 962–983 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lamberti, L.: Repetitive higher cluster categories of type\(A_n\), Journal of Algebra and Its Applications, 1350091

  16. Riedtmann, C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55(2), 199–224 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Schiffler, R.: A geometric model for cluster categories of type \(D_n\). J. Algebraic Combin. 27(1), 1–21 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Torkildsen, H.: Finite mutation classes of coloured quivers. Colloq. Math. 122(1), 53–58 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Zhu, B.: Cluster-tilted algebras and their intermediate coverings. Comm. Algebra 39(7), 2437–2448 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Instituto de Matemática y Estadística Rafael Laguardia, Facultad de Ingeniería - UdelaR, Montevideo, 11200, Uruguay

    Viviana Gubitosi

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  1. Viviana Gubitosi
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Correspondence to Viviana Gubitosi.

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Communicated by Vyacheslav Futorny.

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Gubitosi, V. Repetitive cluster categories of type \(D_n\). São Paulo J. Math. Sci. 16, 712–726 (2022). https://doi.org/10.1007/s40863-021-00222-7

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  • DOI: https://doi.org/10.1007/s40863-021-00222-7

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