Deformed Cartan matrices and generalized preprojective algebras I: Finite type

TL;DR

This paper interprets deformed Cartan matrices of finite type using bigraded modules over generalized preprojective algebras, providing new insights into their structure and relations to quantum group representations.

Contribution

It introduces a novel interpretation of deformed Cartan matrices via bigraded modules over generalized preprojective algebras of Langlands dual type, linking algebraic and representation-theoretic concepts.

Findings

Computed first extension groups between generic kernels.

Proposed a conjecture relating extension dimensions to pole orders of R-matrices.

Connected deformed Cartan matrices to bigraded module categories.

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 Gei\ss-Leclerc-Schr\"{o}er [Invent. math. 209 (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc [J. Eur. Math. Soc. 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.