Diagonalizable derivations of finitedimensional algebras

When an algebra is graded by a group, any additive char-acter of the group induces a diagonalizable derivation of the ring. This construction is studied in detail for the case of a path algebra modulo relations and its fundamental group. We describe an injection of the character group into the first co-homology group following Assem-de la Peña. Rather general conditions are determined, in this context, which guarantee that a diagonalizable derivation is induced from the funda-mental group. This paper is the second installment in a series devoted to diagonalizable derivations. Suppose that R is a finite-dimensional algebra over the field k. We will denote by Der(R) the space of k-algebra derivations from R to itself. Diagonalizable derivations arise naturally whenever R is graded by a group H. Indeed, every additive character χ ∈ Hom(H, k+) can be assigned a derivation Dχ ∈ Der(R) according to the rul