Tilting Modules and Support τ-Tilting Modules over Preprojective Algebras Associated with Symmetrizable Cartan Matrices

For any given symmetrizable Cartan matrix C with a symmetrizer D, Geiß et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τ-tilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(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 τ-tilting Π(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.