  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.
