Preprojective algebras and c-sortable words

Claire Amiot (1) , Osamu Iyama (2) , Idun Reiten (3) , Gordana Todorov (4)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Nagoya University
3 IMF - Institutt for matematiske fag
4 neu - Departement of Mathematics [Boston]

Let $Q$ be an acyclic quiver and $\Lambda$ be the complete 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 $\mathbf{w}$ of $w$. We are especially interested in the case where the word $\mathbf{w}$ 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.

Mots clés: preprojective algebra; quiver representation; generalized cluster category; Coxeter group; 2-Calabi-Yau triangulated category; tilting theory; cluster-tilting

Domaines: Théorie des représentations [math.RT]

Claire Amiot, Osamu Iyama, Idun Reiten, Gordana Todorov. Preprojective algebras and c-sortable words. 2010. ⟨hal-00458893v3⟩