Frontiers of Mathematics in China >
A relation between tilting graphs and cluster-tilting graphs of hereditary algebras
Received date: 24 Jul 2013
Accepted date: 27 Aug 2014
Published date: 12 Feb 2015
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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.
Fang LI , Yichao YANG . A relation between tilting graphs and cluster-tilting graphs of hereditary algebras[J]. Frontiers of Mathematics in China, 2015 , 10(2) : 275 -291 . DOI: 10.1007/s11464-015-0426-6
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
|
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