Article

# Preprojective algebras and c-sortable words

02/2010;
Source: arXiv

ABSTRACT 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.

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ABSTRACT: Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When $\Cc_A$ is $\Hom$-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{\"o}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category $\Cc_{(Q,W)}$ associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra $\Jj(Q,W)$. Comment: 46 pages, small typos as it will appear in Annales de l'Institut Fourier
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Compositio Mathematica 06/2009; 145(04):1035 - 1079. · 1.19 Impact Factor

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### Keywords

acyclic quiver

expression $\mathbf{w}$

field $k$

filtration

filtrations

Reiten

Scott

tilting $kQ$-modules