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International Mathematics Research Notices (2007) Vol. 2007 : article ID rnm011, 19 pages, doi:10.1093/imrn/rnm011 published on May 24, 2007
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Copyright © The Author 2007. Published by Oxford University Press.

A Geometric Description of the m-cluster Categories of Type Dn

Karin Baur1, and Robert J. Marsh2

1 Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England, UK
2 Department of PureMathematics, University of Leeds, Leeds LS2 9JT, England, UK

Correspondence: Correspondence to be sent to: Karin Baur, Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England, UK. e-mail: k.baur{at}mcs.le.ac.uk

We show that the m-cluster category of type Dn is equivalent to a certain geometrically-defined category of arcs in a punctured regular nm – m + 1-gon. This generalizes a result of Schiffler for m = 1. We use the notion of the mth power of a translation quiver to realize the m-cluster category in terms of the cluster category.

Communicated by Andrei Zelevinsky



References

  1. Keller B. On triangulated orbit categories. Documenta Mathematica (2005) 10:551–581.[Web of Science]
  2. Assem I., Brüstle T., Schiffler R., Todorov G. Defining anm-cluster categories and m-replicated algebras. (2006) Preprint arxiv:math.RT/0608727.
  3. Baur K., Marsh R. J. A geometric description of m-cluster categories. Transactions of the American Mathematical Society (2006) arxiv:math.RT/0607151.
  4. Thomas H. Defining an m-cluster category. (2006) Preprint arxiv:math.RT/0607173.
  5. Wralsen A. PhD thesis, Norges Teknisk Naturvitenskapelige Universtitet, in preparation.
  6. Zhu B. Generalized cluster complexes via quiver representations. (2006) Preprint arxiv:math.RT/0607155.
  7. Fomin S., Reading N. Generalized cluster complexes and Coxeter combinatorics. International Mathematics Research Notices (2005) 44:2709–2757.
  8. Tzanaki E. Polygon dissections and some generalizations of cluster complexes. Journal of Combinatorial Theory Series A (2006) 113(no. 6):1189–1198.[CrossRef][Web of Science]
  9. Buan A. B., Marsh R. J., Reineke M., Reiten I., Todorov G. Tilting theory and cluster combinatorics. Advances in Mathematics (2006) 204:572–618.[Web of Science]
  10. Caldero P., Chapoton F., Schiffler R. Quivers with Relations arising from Clusters (An case). Transactions of the American Mathematical Society (2006) 358(no. 3):1347–1364.[CrossRef][Web of Science]
  11. Keller B., Reiten I. Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. (2005) Preprint arxiv:math.RT/0512471.
  12. Fomin S., Zelevinsky A. Cluster algebras I: foundations. Journal of the American Mathematical Society (2002) 15(no. 2):497–529.[CrossRef][Web of Science]
  13. Schiffler R. A geometric model for cluster categories of type Dn. (2006) Preprint arxiv:math.RT/0608264.
  14. Fomin S., Shapiro M., Thurston D. Cluster algebras and triangulated surfaces. Part I: Cluster complexes. (2006) Preprint arxiv:math.RA/0608367.
  15. Fock V. V., Goncharov A. B. Moduli spaces of local systems and higher Teichmüller theory. Publications Mathématiques De l'Institut Des Hautes Études Scientifiques (2006) 103:1–211.
  16. Fock V. V., Goncharov A. B. Cluster ensembles, quantization and the dilogarithm. (2003) Preprint arxiv:math.AG/0311245.
  17. Gekhtman M., Shapiro M., Vainshtein A. Cluster algebras and Poisson geometry. Moscow Mathematical Journal (2003) 3(no. 3):899–934.
  18. Gekhtman M., Shapiro M., Vainshtein A. Cluster algebras and Weil-Petersson forms. Duke Mathematical Journal (2005) 127:291–311.[CrossRef][Web of Science]
  19. Happel D. (1988) Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 119. Triangulated categories in the representation theory of finite-dimensional algebras.
  20. Ringel C. M. Tame Algebras and Integral Quadratic Forms (1984) Berlin: Springer. Lecture Notes in Mathematics 1099.

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This Article
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Right arrow Alert me when this article is cited
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