A relation between tilting graphs and cluster-tilting graphs of hereditary algebras

Abstract

We give the condition of isomorphisms between tilting graphs and cluster-tilting graphs of hereditary algebras. As a conclusion, it is proved that a graph is a skeleton graph of Stasheff polytope if and only if it is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra. At last, when comparing such uniformity, the geometric realizations of simplicial complexes associated with tilting modules and clustertilting objects are discussed respectively.

Keywords

Tilting graph / cluster-tilting graph / cluster category / Stasheff polytope / linear quiver

References

[1] Assem I, Simson D, Skowroński A. Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory. London Math Soc Student Texts, 65. Cambridge: Cambridge Univ Press, 2006

[2] Auslander M, Reiten I. Representation theory of Artin algebras. III. Almost split sequences. Comm Algebra, 1975, 3(3): 239-294

[3] Buan A B, Marsh R J. Cluster-tilting theory. In: Trends in Representation Theory of Algebras and Related Topics. Contemporary Mathematics, 406. Providence: Amer Math Soc, 2006, 1-30

[4] Buan A B, Marsh R J, Reineke M, Reiten I, Todorov G. Tilting theory and cluster combinatorics. Adv Math, 2006, 204(2): 572-618

[5] Fomin S, Shapiro M, Thurston D. Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math, 2008, 201(1): 83-146

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

[7] Fomin S, Zelevinsky A. Y-systems and generalized associahedra. Ann Math, 2003, 158(3): 977-1018

[8] Happel D. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Note Series, 119. Cambridge: Cambridge Univ Press, 1988

[9] Happel D, Unger L. On the quiver of tilting modules. J Algebra, 2005, 284(2): 857-868

[10] Hügel L A, Happel D, Krause H. Handbook of Tilting Theory. London Math Soc Lecture Note Series, 332. Cambridge: Cambridge Univ Press, 2007

[11] Iyama O, Yoshino Y. Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent Math, 2008, 172(1): 117-168

[12] Kase R. The number of arrows in the quiver of tilting modules over a path algebra of type A and D. Research Institute for Math Science Kôkyūroku, 2012, 1795: 154-162

[13] Keller B. On triangulated orbit categories. Doc Math, 2005, 10: 551-581

[14] Riedtmann C, Schofield A. On a simplicial complex associated with tilting modules. Comm Math Helv, 1991, 66(1): 70-78

[15] Unger L, Ungruhe M. On the genus of the graph of tilting modules. Beiträge Algebra Geom, 2004, 45(2): 415-427

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg