Title: Vinberg’s characterization of dynkin diagrams using subadditive functions with application to DTr-periodic modules
Abstract: Let R be an Artin algebra. Given two indecomposable modules M,N, let Irr(M,N) = rad(M,N)/rad2(M,N) be the bimodule of irreducible maps [5] and denote by aMN the length of Irr(M,N) as an End(N)-module, ' its length as an End(M)-module. Note that in case M is not by aMN injective, then aMN is equal to the multiplicity ol N occuring in the middle term of the Auslander-Reiten sequence starting with M, whereas ' is equal to the multiplicity of M if N is not projective, then aMN occurring in the middle term of the Auslander-Reiten sequence ending with N. The Auslander-Reiten quiver A(R) has as vertices the isomorphism classes of the indecomposable R-modules, and there is an arrow [M] + IN]