From submodule categories to preprojective algebras

Claus Michael Ringel, Pu Zhang

Published 2012-11-20, updated 2014-02-27Version 2

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 1936 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 of type 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, Schroeer). Direct connections between the submodule category S(n) and the module category of the preprojective algebra of type A_{n-1} have been established quite a long time ago by Auslander and Reiten, and recently also by Li and Zhang, but apparently this remained unnoticed. The aim of this note is to provide details on this relationship. As a byproduct we see that here we deal with ideals I in triangulated categories T such that I is generated by an idempotent and T/I is abelian.