Deformed symmetrizable generalized Cartan matrices and generalized preprojective algebras

Motivated from studies of the representation theory of quantum loop algebras, Geiss-Leclerc-Schröer introduced the notion of the generalized preprojective algebra associated with a symmetrizable generalized Cartan matrix and its symmetrizer. 

We study a several parameter deformation of a symmetrizable generalized Cartan matrix as a numerical aspect of the graded module category of the generalized preprojective algebra. In particular, we will interpret some numerical formula about this matrix in terms of braid group symmetries of our graded module category. This is a joint work with Ryo Fujita (RIMS).