Preprojective algebras and c‐sortable words

By C. Amiot, Osamu Iyama, I. Reiten, G. Todorov. 2010

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/642179a85d7420d3c738d0b0696d432540ccfced

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Citations in the corpus

Affine Mirković-Vilonen polytopes (2011, background): “We thank Claire Amiot for suggesting to us that the reflection functors in [5] are related to those in [1], which allowed us to take the current literature [14, 27, 59] into account in section 5.”
Classifying tilting complexes over preprojective algebras of Dynkin type (2015, background): “If Q is a non-Dynkin quiver, the bounded derived category Db(fd ) of the category fd of finite dimensional modules are 2-Calabi–Yau (2-CY for short), that is, there is a functorial isomorphism D HomDb(fd )(X, Y ) ∼= HomDb(fd )(Y, X [2]).”
Representations of Algebras and Related Topics (2011, background): “pt An with the linear orientation), there is a triangle On Generalized Cluster Categories 35 equivalence SubΠw ≃CQ when w = c2 where c is the Coxeter element associated with the orientation of Q. See [23, 5] for further results on these categories. This construction and the constructions in Section ”
Classifying $$\tau $$τ-tilting modules over preprojective algebras of Dynkin type (2013, background): “If Q is a non-Dynkin quiver, the bounded derived category Db(fd ) of the category fd of finite dimensional modules are 2-Calabi–Yau (2-CY for short), that is, there is a functorial isomorphism D HomDb(fd )(X, Y ) ∼= HomDb(fd )(Y, X [2]).”