Deformed Cartan Matrices and Generalized Preprojective Algebras I: Finite Type

Abstract

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–Schröer [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.