Tilting theory of preprojective algebras and $c$-sortable elements

Abstract

For a finite acyclic quiver $Q$ and the corresponding preprojective algebra $\Pi$, we study the factor algebra ${\Pi }_w$ associated with a element $w$ in the Coxeter group introduced by Buan-Iyama-Reiten-Scott. The algebra ${\Pi }_w$ has a natural $\mathbb{Z}$-grading, and we prove that ${\underline{\mathsf{Sub}}}^{\mathbb{Z}}{\Pi }_w$ has a tilting object $M$. Moreover, we show that the endomorphism algebra of $M$ is isomorphic to the stable Auslander algebra of a certain torsion free class of $\mathsf{mod}kQ$.

26 pages, new theorem is added (Theorem 1.2, v2)

References in corpus (1)

• Cluster structures for 2-Calabi-Yau categories and unipotent groups

Cited by in corpus (3)

• Tilting and cluster tilting for preprojective algebras and Coxeter groups
• Homological dimension formulas for trivial extension algebras
• Singularity categories of derived categories of hereditary algebras are derived categories