RESEARCH ARTICLE

On duality preservability of Auslander-Reiten quivers of derived categories and cluster categories

  • Fang LI ,
  • Lingyu WAN
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  • Department of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou 310027, China

Received date: 30 Nov 2015

Accepted date: 07 Mar 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. Meantime, we characterize the condition of generalized path algebras as cleft extensions of path algebras.

Cite this article

Fang LI , Lingyu WAN . On duality preservability of Auslander-Reiten quivers of derived categories and cluster categories[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 957 -984 . DOI: 10.1007/s11464-016-0559-2

1
Auslander M, Reiten I, Smalo S O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1997

2
Demonet L. Skew group algebras of path algebras and preprojective algebras. 2009, arXiv: 0902.1390

3
Happel D. Triangulated Categories in the Representation of Finite Dimensional Algebras. London Math Soc Lecture Note Ser, Vol 119. Cambridge: Cambridge Univ Press, 1988

DOI

4
Hou B, Yang S. Skew group algebras of deformed preprojective algebras. J Algebra, 2011, 332(1): 209–228

DOI

5
Hubery A W. Representations of Quivers Respecting a Quiver Automorphism and a Theorem of Kac. Ph D Thesis, The University of Leeds, 2002

6
Keller B. Cluster algebras and cluster categories. Bull Iranian Math Soc, 2011, 37: 187–234

7
Li F. Characterization of left artinian algebras through pseudo path algebras. J Aust Math Soc, 2007, 83(03): 385–416

DOI

8
Li F. Modulation and natural valued quiver of an algebra. Pacific J Math,2012, 256(1): 105–128

DOI

10
Li F, Chen L L. The natural quiver of an artinian algebra. Algebr Represent Theory, 2010, 13(5): 623–636

DOI

11
Li F, Lin Z Z. Approach to artinian algebras via natural quivers. Trans Amer Math Soc, 2012, 364(3): 1395–1411

DOI

12
Reiten I, Riedtmann C. Skew group algebras in the representation theory of artin algebras. J Algebra, 1985, 92(1): 224–282

DOI

13
Zhang M. The dual quiver of the Auslander-Reiten quiver of path algebras. Algebr Represent Theory, 2012, 15(2): 203–210

DOI

14
Zhang M, Li F. Representations of skew group algebras induced from isomorphically invariant modules over path algebras. J Algebra, 2009, 321(2): 567–581

DOI

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