# Deformed Cartan matrices and generalized preprojective algebras II: General type

## TL;DR

This paper introduces a new class of deformed symmetrizable generalized Cartan matrices with multiple parameters, providing a categorical interpretation and a combinatorial formula for their inverses, extending previous work in symmetric, finite, and affine cases.

## Contribution

It defines deformed Cartan matrices with multiple parameters and connects them to graded modules over generalized preprojective algebras, offering a new perspective and formulas for their inverses.

## Key findings

- Categorical interpretation via graded modules over generalized preprojective algebras.
- A combinatorial formula for the inverses of deformed Cartan matrices.
- Equivalence with mass-deformed Cartan matrices in symmetric, finite, and affine cases.

## Abstract

We propose a definition of deformed symmetrizable generalized Cartan matrices with several deformation parameters, which admit a categorical interpretation by graded modules over the generalized preprojective algebras in the sense of Gei\ss-Leclerc-Schr\"oer. Using the categorical interpretation, we deduce a combinatorial formula for the inverses of our deformed Cartan matrices in terms of braid group actions. Under a certain condition, which is satisfied in all the symmetric cases or in all the finite and affine cases, our definition coincides with that of the mass-deformed Cartan matrices introduced by Kimura-Pestun in their study of quiver $\mathcal{W}$-algebras.