Preprojective algebras and c-sortable words

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.

There was a mistake in section 4 of the previous version. This section is removed in this new version. Some results still true of section 4 are now in subsection 3.3. Accepted for publication in Proc. of the London Math. Soc

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• Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras
• Deformed Cartan matrices and generalized preprojective algebras II: General type
• Calabi-Yau structures on Drinfeld quotients and Amiot's conjecture
• $c$-Birkhoff polytopes