Deformed Cartan Matrices and Generalized Preprojective Algebras I: Finite Type

We give an interpretation of the (q, t)-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schroer [33]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc [40] and propose a conjecture that their dimensions coincide with the pole orders of the normalized R-matrices between the corresponding Kirillov-Reshetikhin modules.