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, Gei ss 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.