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Algebras and Representation Theory

, Volume 22, Issue 5, pp 1239–1260 | Cite as

Tilting Modules and Support τ-Tilting Modules over Preprojective Algebras Associated with Symmetrizable Cartan Matrices

  • Changjian FuEmail author
  • Shengfei Geng
Article
  • 55 Downloads

Abstract

For any given symmetrizable Cartan matrix C with a symmetrizer D, Geiß et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τ-tilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(C, D) and the corresponding Weyl group W(C) provided that C has no component of Dynkin type. When C is of Dynkin type, we also establish a bijection between the set of all basic support τ-tilting Π(C, D)-modules and the corresponding Weyl group W(C). These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035–1079 2009) and Mizuno (Math. Zeit. 277(3), 665–690 2014) over classical preprojective algebras.

Keywords

Symmetrizable Cartan matrix Preprojective algebras Locally free modules Generalized simple modules Cofinite tilting ideals Support τ-tilting modules 

Mathematics Subject Classification (2010)

16G10 16G20 

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Notes

Acknowledgments

Partially supported by the China Scholarship Council and the National Natural Science Foundation of China (No. 11471224).

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSiChuan UniversityChengduPeople’s Republic of China

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