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.

Automated summary

We provide an interpretation of the (q, t)-deformed Cartan matrices of finite type and their inverses using bigraded modules over the generalized preprojective algebras of Langlands dual type. As an application, we calculate the first extension groups between the generic kernels introduced by Hernandez-Leclerc and propose a conjecture that their dimensions coincide with the pole orders of the normalized R-matrices between the corresponding Kirillov-Reshetikhin modules.