Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

The cluster category of a canonical algebra

Author(s): M. Barot; D. Kussin; H. Lenzing
Journal: Trans. Amer. Math. Soc. 362 (2010), 4313-4330.
MSC (2000): Primary 16G20, 18E30
Posted: March 5, 2010
MathSciNet review: 2608408
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study the cluster category of a canonical algebra in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line $\mathbb{X}$ . As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of $\mathbb{X}$ is non-negative, or equivalently, if is of tame (domestic or tubular) representation type.

References:

1.: M. Auslander, M. I. Platzeck and I. Reiten, Coxeter functors without diagrams. Trans. Amer. Math. Soc. 250 (1979), 1-46. MR 530043 (80c:16027)
2.: M. Barot, D. Kussin and H. Lenzing, The Grothendieck group of a cluster category. J. Pure Appl. Algebra 212 (2008), 33-46. MR 2355032 (2008j:18010)
3.: A. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145 (2009), no. 4, 1035-1079. MR 2521253
4.: A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics. Adv. Math. 204 (2006), 572-618. MR 2249625 (2007f:16033)
5.: S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations. J. Amer. Math. Soc. 15 (2002), no. 2, 497-529. MR 1887642 (2003f:16050)
6.: W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras. Springer Lecture Notes in Math. 1273, 265-297, Springer-Verlag, Berlin, 1987. MR 915180 (89b:14049)
7.: -, Perpendicular categories with applications to representations and sheaves. J. Algebra 144 (1991), 273-343. MR 1140607 (93b:16011)
8.: D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. Cambridge University Press, Cambridge, 1988. MR 935124 (89e:16035)
9.: D. Happel and L. Unger, On the set of tilting objects in hereditary categories. Fields Institute Communications 45 (2005), 141-159. MR 2146246 (2006h:18006)
10.: T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projektiven gewichteten Kurven. Dissertation, Universität Paderborn, 1996.
11.: -, Rank additivity for quasi-tilted algebras of canonical type. Colloq. Math. 75 (1998), 183-193. MR 1490688 (99b:16016)
12.: O. Iyama and Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172 (2008), 117-168. MR 2385669 (2008k:16028)
13.: B. Keller, On triangulated orbit categories. Documenta Math. 10 (2005), 551-581. MR 2184464 (2007c:18006)
14.: D. Kussin, Noncommutative curves of genus zero: related to finite dimensional algebras. Mem. Amer. Math. Soc. 201 (2009), no. 942, x+128 pp. MR 2548114
15.: -, Parameter curves for the regular representations of tame bimodules. J. Algebra 320 (2008), 2567-2582. MR 2437515 (2010c:16012)
16.: H. Lenzing, Representations of finite dimensional algebras and singularity theory. Trends in ring theory (Miskolc, 1996) (V. Dlab et al., ed.), CMS Conf. Proc., vol. 22, Amer. Math. Soc., Providence, RI, 1998, pp. 71-97. MR 1491919 (99d:16014)
17.: H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. CMS Conf. Proc. 14 (1992), 313-337. MR 1265294
18.: -, Tilting sheaves and concealed-canonical algebras. CMS Conf. Proc. 18 (1996), 455-473. MR 1388067 (97f:16026)
19.: -, The automorphism group of the derived category for a weighted projective line. Comm. Algebra 28 (2000), 1685-1700. MR 1747349 (2001a:16021)
20.: H. Lenzing and J.A. de la Peña, Concealed-canonical algebras and separating tubular families. Proc. London Math. Soc. (3) 78 (1999), 513-540. MR 1674837 (2000c:16018)
21.: H. Lenzing and I. Reiten, Additive functions for quivers with relations. Colloq. Math. 82 (1999), 85-103. MR 1736037 (2000m:16021)
22.: H. Meltzer, Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Mem. Amer. Math. Soc. 171 (2004), no. 808, viii+139 pp. MR 2074151 (2005k:14033)
23.: J. Rickard, Derived equivalences as derived functors. J. London Math. Soc. 43 (1991), 37-48. MR 1099084 (92b:16043)
24.: C. M. Ringel, Tame algebras and integral quadratic forms. Lecture Notes in Mathematics 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
25.: -, The canonical algebras. Topics in Algebra, Part I. Banach Center Publ., no. 26, PWN, 1990, with an appendix by William Crawley-Boevey, pp. 407-432. MR 1171247 (93e:16022)
26.: J.-L. Verdier, Catégories dérivées (etat 0). (French) in: (P. Deligne ed.) Séminarie de Géométrie Algébrique du Bois-Marie SGA 4 $\frac{1}{2}$ . Lecture Notes in Mathematics 569, Springer-Verlag, Berlin, 1977. MR 0463174 (57:3132)
27.: B. Zhu, Equivalences between cluster categories. J. Algebra 304 (2006), 832-850. MR 2264281 (2007h:18017)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|