Preprojective algebras and c-sortable words

摘要

Let Q be an acyclic quiver and Λ 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 ['Cluster structures for 2-Calabi-Yau categories and unipotent groups', Compos. Math. 145 (2009) 1035-1079] have introduced and studied a finite-dimensional algebra Λw=Λ/Iw. In this paper, we look at filtrations of Λw associated to any reduced expression w of w. We are especially interested in the case where the word 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. 2011 London Mathematical Society2011