Deformed Cartan matrices and generalized preprojective algebras of 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-Schr{o}er~[Invent.~math.~{bf{209}} (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc~[J.~Eur.~Math.~Soc.~{bf 18} (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.