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.