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Mathematische Zeitschrift

, Volume 277, Issue 3–4, pp 665–690 | Cite as

Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type

  • Yuya Mizuno
Article

Abstract

We study support \(\tau \)-tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \)-tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\)-matrices of support \(\tau \)-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\)-matrices and prove that cones given by \(g\)-matrices coincide with chambers of the associated root systems.

Notes

Acknowledgments

First and foremost, the author would like to thank Osamu Iyama for his support and patient guidance. He would like to express his gratitude to Steffen Oppermann, who kindly explain results of his paper. He is grateful to Joseph Grant, Laurent Demonet and Dong Yang for answering questions and helpful comments. He thanks Kota Yamaura, Takahide Adachi and Gustavo Jasso for their help and stimulating discussions. He is very grateful to the anonymous referee for valuable comments, especially for suggesting the better terms and sentences.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityFrocho, Chikusaku, NagoyaJapan

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