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Preprojective algebras and c-sortable words
Claire Amiot1, Osamu Iyama2, Idun Reiten3, Gordana Todorov4

Let $Q$ be an acyclic quiver and $\Lambda$ be the completion of the preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in \cite{Bua2} a finite dimensional algebra $\Lambda_w=\Lambda/I_w$. In this paper we look at filtrations of $\Lambda_w$ associated to any reduced expression $\ww$ of $w$. We are specially interested in the case where the word $\ww$ is $c$-sortable where $c$ is a Coxeter element. In this situation, the consecutive quotients of this filtration can be related to tilting $kQ$-modules with finite torsionfree class. This nice description allows us to construct a triangle equivalence between the 2-Calabi-Yau triangulated category $\underline{\Sub}\Lambda_w$ and the generalized cluster category associated with an Auslander algebra.
1 :  Mathematisches Institut Bonn
2 :  Nagoya University
3 :  IMF - Institutt for matematiske fag
4 :  neu - Departement of Mathematics [Boston]
preprojective algebra – quiver representation – generalized cluster category – Coxeter group – 2-Calabi-Yau triangulated category – tilting theory – cluster-tilting