From submodule categories to preprojective algebras

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• 作者：Claus Michael Ringel ; Pu Zhang
• 关键词：Primary 16G20 ; 16E65 ; Secondary 16D90 ; 16E05 ; 16G10 ; 16G50 ; 16G70 ; 18G25
• 刊名：Mathematische Zeitschrift
• 出版年：2014
• 出版时间：October 2014
• 年：2014
• 卷：278
• 期：1-2
• 页码：55-73
• 全文大小：1,255 KB
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• 作者单位：Claus Michael Ringel (1) (2)
  Pu Zhang (1)
  
  1. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China
  2. King Abdulaziz University, P O Box 80200, Jeddah, Saudi Arabia
  
• ISSN：1432-1823

文摘

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, Schr?er). 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.