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 $D$, Geiß~ et al. (2016) 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 Buan et al. (Compos. Math. 145(4), 1035-1079, 2009) and Mizuno (Math. Zeit. 277(3), 665-690, 2014) over classical preprojective algebras.

19 pages. typos fixed, to appear in ART

References in corpus (1)

• Kac-Moody groups and cluster algebras