Combinatorial model for the cluster categories of type \(E\). (English) Zbl 1316.05130
Summary: In this paper, we give a geometric-combinatorial description of the cluster categories of type \(E\). In particular, we give an explicit geometric description of all cluster-tilting objects in the cluster category of type \(E_6\). The model we propose here arises from combining two polygons, and it generalizes the description of the cluster category of type \(A\) and \(D\).
MSC:
| 05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
cluster categories; type \(E\); triangulated Calabi-Yau categories; polygon dissection; cluster algebra; Auslander-Reiten quiverReferences:
| [1] | Bongartz, K.: Critical simply connected algebras. Manuscr. Math. 46, 117-136 (1984) · Zbl 0537.16024 · doi:10.1007/BF01185198 |
| [2] | Brüstle, T., Zhang, J.: On the cluster category of a marked surface without punctures. Algebra Number Theory 5, 529-566 (2011) · Zbl 1250.16013 · doi:10.2140/ant.2011.5.529 |
| [3] | Buan, A.B., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204, 572-618 (2006) · Zbl 1127.16011 · doi:10.1016/j.aim.2005.06.003 |
| [4] | Buan, A.B., Marsh, R.J., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. 83, 143-177 (2008) · Zbl 1193.16016 · doi:10.4171/CMH/121 |
| [5] | Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters \[(A_n\] An case). Trans. Am. Math. Soc. 358, 1347-1364 (2006) · Zbl 1137.16020 · doi:10.1090/S0002-9947-05-03753-0 |
| [6] | Caldero, P., Keller, B.: From triangulated categories to cluster algebras. Invent. Math. 172, 169-211 (2008) · Zbl 1141.18012 · doi:10.1007/s00222-008-0111-4 |
| [7] | Demonet, L.: Categorification of skew-symmetrizable cluster algebras. Algebr. Represent. Theory 14(6), 1087-1162 (2011) · Zbl 1236.13019 · doi:10.1007/s10468-010-9228-4 |
| [8] | Fomin, S., Zelevinsky, A.: \[YY\]—systems and generalized associahedra. Ann. Math. 158(2), 977-1018 (2003) · Zbl 1057.52003 · doi:10.4007/annals.2003.158.977 |
| [9] | Fomin, S. Pylyavskyy, P.: Tensor graphs and cluster algebras, eprint arXiv:1210.1888 (2012) · Zbl 1386.13062 |
| [10] | Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras, vol. 119 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (1988) · Zbl 0635.16017 |
| [11] | Jørgensen, Peter: Quotients of cluster categories. Proc. R. Soc. Edinb. Sect. A 140(1), 65-81 (2010) · Zbl 1201.16020 · doi:10.1017/S0308210508000425 |
| [12] | Keller, B.: On triangulated orbit categories. Doc. Math. 10, 551-581 (2005) · Zbl 1086.18006 |
| [13] | Keller, B.: Cluster algebras, quiver representations and triangulated categories, In: Triangulated Categories, London Mathematical Society. Lecture Note Series, vol. 375, pp. 76-160. Cambridge University Press, Cambridge (2010) · Zbl 1215.16012 |
| [14] | Lamberti, L.: Repetitive higher cluster categories of type \[A_n\] An. J. Algebra Appl. 13(2), 1350091 (2014) · Zbl 1292.16010 · doi:10.1142/S0219498813500916 |
| [15] | Lamberti, L.: A geometric interpretation of the triangulated structure of m-cluster categories. Commun. Algebra 42(3), 962-983 (2014) · Zbl 1293.16011 · doi:10.1080/00927872.2012.718397 |
| [16] | Miyachi, J.-I., Yekutieli, A.: Derived Picard groups of finite-dimensional hereditary algebras. Compos. Math. 129, 341-368 (2001) · Zbl 0999.16012 · doi:10.1023/A:1012579131516 |
| [17] | Riedtmann, C.: Algebren, Darstellungsköcher, Überlagerungen und zurück. Comment. Math. Helv. 55, 199-224 (1980) · Zbl 0444.16018 · doi:10.1007/BF02566682 |
| [18] | Schiffler, R.: A geometric model for cluster categories of type \[D_n\] Dn. J. Algebr. Combin. 27, 1-21 (2008) · Zbl 1165.16008 · doi:10.1007/s10801-007-0071-6 |
| [19] | Scott, J.S.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. 92(3), 345-380 (2006) · Zbl 1088.22009 · doi:10.1112/S0024611505015571 |
| [20] | Torkildsen, H.A.: A geometric realization of the m-cluster category of type A, eprintarXiv:1208.2138 (2012) · Zbl 1338.16020 |
| [21] | Zhu, B.: Cluster-tilted algebras and their intermediate coverings. Commun. Algebra 39, 2437-2448 (2011) · Zbl 1250.16015 · doi:10.1080/00927872.2010.489082 |
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