Mathematische Zeitschrift

, Volume 278, Issue 1–2, pp 55–73 | Cite as

From submodule categories to preprojective algebras

Article

Abstract

Let \(S(n)\) be the category of invariant subspaces of nilpotent operators with nilpotency index at most \(n\). Such submodule categories have been studied already in 1934 by Birkhoff, they have attracted a lot of attention in recent years, for example in connection with some weighted projective lines (Kussin, Lenzing, Meltzer). On the other hand, we consider the preprojective algebra \(\Pi _n\) of type \(\mathbb {A}_n\); the preprojective algebras were introduced by Gelfand and Ponomarev, they are now of great interest, for example they form an important tool to study quantum groups (Lusztig) or cluster algebras (Geiss, Leclerc, Schröer). We are going to discuss the connection between the submodule category \(\mathcal {S}(n)\) and the module category \(\hbox {mod}\;\Pi _{n-1}\) of the preprojective algebra \(\Pi _{n-1}\). Dense functors \(\mathcal {S}(n) \rightarrow \hbox {mod}\;\Pi _{n-1}\) are known to exist: one has been constructed quite a long time ago by Auslander and Reiten, recently another one by Li and Zhang. We will show that these two functors are full, dense, objective functors with index \(2n\), thus \(\hbox {mod}\;\Pi _{n-1}\) is obtained from \(\mathcal {S}(n)\) by factoring out an ideal which is generated by \(2n\) indecomposable objects. As a byproduct we also obtain new examples of ideals in triangulated categories, namely ideals \(\mathcal {I}\) in a triangulated category \(\mathcal {T}\) which are generated by an idempotent such that the factor category \(\mathcal {T}/\mathcal {I}\) is an abelian category.

Mathematics Subject Classification (2010)

Primary 16G20 16E65 Secondary 16D90 16E05 16G10 16G50 16G70 18G25 

Notes

Acknowledgments

The authors are indebted to the referee for a careful reading of the paper and for many valuable comments concerning possible improvements of the paper, in particular, for suggesting to add Sect. 9. This work was supported by the NSF of China (11271251) and Specialized Research Fund for the Doctoral Program of Higher Education (20120073110058).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.King Abdulaziz UniversityJeddahSaudi Arabia

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