Cartan matrices and Calabi-Yau completions

Abstract: Quantum Cartan matrices and their inverses are of interest in the representation theory of quantum affine algebras. We show how to categorify them for simply laced Dynkin diagrams using bigraded Calabi-Yau completions (aka bigraded Ginzburg algebras). In particular, we will recover a formula due to Hernandez-Leclerc (2015) and Fujita (2019) expressing the inverse quantum Cartan matrix in terms of the (ordinary) derived category and thus in terms of the combinatorics of the repetition quiver. We will end by discussing the problem of obtaining analogous results in the non simply laced case.

combinatoricscategory theoryrings and algebrasrepresentation theory

Audience: researchers in the topic

( slides )

Sherbrooke Meeting on Representation Theory of Algebras, Corona Edition (fully online)

Series comments: Please contact Thomas Brüstle or Juan Carlos Bustamante if you are interested to participate.