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Home > Vol 27, No 1 (2008)

Ralf Schiffler

University of Massachusetts at Amherst Department of Mathematics and Statistics Amherst MA 01003-9305 USA

We give a geometric realization of cluster categories of type D _n using a polygon with n vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.

1. Assem, I., Brüstle, T., Schiffler, R., & Todorov, G. (2006). Cluster categories and duplicated algebras. J. Algebra, 305, 547-561. arXiv:math.RT/0509501.
2. Assem, I., Brüstle, T., Schiffler, R., & Todorov, G. (2006). m-cluster categories and m-replicated algebras. Preprint. arXiv:math.RT/0608727.
3. Assem, I., Simson, D., & Skowronski, A. (2006). Elements of the representation theory of associative algebras, 1: techniques of representation theory. London mathematical society student texts, Vol.
65. Cambridge: Cambridge University Press.
4. Auslander, M., Reiten, I., & Smalø, S. O. (1995). Representation theory of Artin algebras. Cambridge studies in advanced math., Vol.
36. Cambridge: Cambridge University Press.
5. Baur, K., & Marsh, R. (2006). A geometric description of m-cluster categories. Trans. Am. Math. Soc., to appear. arXiv:math.RT/0607151.
6. Buan, A., Marsh, R., Reineke, M., Reiten, I., & Todorov, G. (2006). Tilting theory and cluster combinatorics. Adv. Math., 204, 572-612. arXiv:math.RT/0402054.
7. Buan, A. B., Marsh, R., & Reiten, I. (2007). Cluster-tilted algebras. Trans. Am. Math. Soc., 359, 323-332. arXiv:math.RT/0402075.
8. Buan, A. B., Marsh, R., & Reiten, I. Cluster mutation via quiver representations. Comment. Math. Helv., to appear. arXiv:math.RT/0412077.
9. Buan, A., Marsh, R., Reiten, I., & Todorov, G. Clusters and seeds in acyclic cluster algebras. Proc. Am. Math. Soc., to appear. arXiv:math.RT/0510359.
10. Caldero, P., & Chapoton, F. (2006). Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv., 81(3), 595-616. arXiv:math.RT/0410187.
11. Caldero, P., Chapoton, F., & Schiffler, R. (2006). Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc., 358(3), 1347-1364. arXiv:math.RT/0401316.
12. Caldero, P., Chapoton, F., & Schiffler, R. (2006). Quivers with relations and cluster tilted algebras. Algebr. Represent. Theory, 9(4), 359-376. arXiv:math.RT/0411238.
13. Caldero, P., & Keller, B. From triangulated categories to cluster algebras. Invent. Math., to appear. arXiv:math.RT/0506018.
14. Caldero, P., & Keller, B. From triangulated categories to cluster algebras II. Ann. Sci. École Norm. Sup., to appear. arXiv:math.RT/0510251.
15. Fomin, S., & Zelevinsky, A. (2002). Cluster algebras I. Foundations. J. Am. Math. Soc., 15(2), 497-529 (electronic). arXiv:math.RT/0104151.
16. Fomin, S., & Zelevinsky, A. (2003). Cluster algebras II. Finite type classification. Invent. Math., 154(1), 63-121. arXiv:math.RA/0208229.
17. Fomin, S., Shapiro, M., & Thurston, D. (2006). Cluster algebras and triangulated surfaces. Part I: Cluster complexes. Preprint. arXiv:math.RA/0608367.
18. Gabriel, P. (1972). Unzerlegbare Darstellungen. Manuscripta Math., 6, 71-103.
19. Gabriel, P. (1980). Auslander-Reiten sequences and representation-finite algebras. In Representation theory, I. Proc. Workshop, Carleton Univ., Ottawa, ON,
1979. Lecture notes in math. (Vol. 831, pp. 1-71). Berlin: Springer.
20. Happel, D. (1988). Triangulated categories in the representation theory of finite dimensional algebras. London Mathematical Society, lecture notes series, Vol.
119. Cambridge: Cambridge University Press.
21. Keller, B. (2005). On triangulated orbit categories. Doc. Math., 10, 551-581 (electronic). arXiv:math.RT/0503240.
22. Keller, B., & Reiten, I. (2005). Cluster-tilted algebras are Gorenstein and stably Calabi-Yau. Preprint. arXiv:math.RT/0512471.
23. Koenig, S., & Zhu, B. (2006). From triangulated categories to abelian categories-cluster tilting in a general framework. Preprint. arXiv:math.RT/0605100.
24. Riedtmann, C. (1990). Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv., 55(2), 199-224.
25. Ringel, C. M. (1984). Tame algebras and integral quadratic forms. Lecture notes in math., Vol.
1099. Berlin: Springer.
26. Zhu, B. (2006). Equivalences between cluster categories. J. Algebra, 304, 832-850. arXiv:math.RT/0511382.