Tilting modules and support $\tau$-tilting modules over preprojective algebras associated with symmetrizable Cartan matrices

Abstract

For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Gei\ss-Leclerc-Schr\"{o}er \cite{GLS14} introduced a generalized preprojective algebra $\Pi(C, D)$. We study tilting modules and support $\tau$-tilting modules for the generalized preprojective algebra $\Pi(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $\Pi(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $\tau$-tilting $\Pi(C,D)$-modules and the corresponding Weyl group $W(C)$. These results generalize the classification results of \cite{BIRS} and \cite{M} over classical preprojective algebras.