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Mathematische Zeitschrift

, Volume 284, Issue 3–4, pp 643–682 | Cite as

Rigid and Schurian modules over cluster-tilted algebras of tame type

  • Robert J. MarshEmail author
  • Idun Reiten
Article

Abstract

We give an example of a cluster-tilted algebra \(\Lambda \) with quiver Q, such that the associated cluster algebra \(\mathcal {A}(Q)\) has a denominator vector which is not the dimension vector of any indecomposable \(\Lambda \)-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra \(\Lambda \), we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid \(\Lambda \)-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid \(\Lambda \)-modules in this case.

Keywords

Almost split sequences Cluster algebras Cluster categories Cluster-tilted algebras c-Vectors  d-Vectors Q-coloured quivers Tame hereditary algebras 

Mathematics Subject Classification

Primary 13F60 16G20 16G70 Secondary 18E30 

Notes

Acknowledgments

Both authors would like to thank the referee for very helpful comments and would like to thank the MSRI, Berkeley for kind hospitality during a semester on cluster algebras in Autumn 2012. RJM was Guest Professor at the Department of Mathematical Sciences, NTNU, Trondheim, Norway, during the autumn semester of 2014 and would like to thank the Department for their kind hospitality.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of MathematicsUniversity of LeedsLeedsUK
  2. 2.Department of Mathematical SciencesNTNUTrondheimNorway

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