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 [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.

arXiv comment41 pages, v3: changed the title, minor corrections and added explanations, updated references

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Cited by in corpus (2)

• $t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras
• Deformed Cartan matrices and generalized preprojective algebras II: General type