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 rule Dχ(r) = χ(g)r for every r ∈ R in the homogeneous component of “degree ” g ∈ H. Obvi-ously, Dχ is diagonalizable. Conversely, if D is a diagonalizable derivation of the k-algebra S and H is the additive subgroup of k generated by the eigenvalues of D then, for the inclusion map ι: H → k+, we have D = Dι. In our first paper [FGGM], we proved that the span of all diagonalizable derivations of R comprise a Lie ideal of Der(R) whenever k has character-istic zero or is algebraically closed of positive characteristic. This result turned out to be a powerful tool in describing what we called spanned-by-split derivations (i.e., those which are sums of diagonalizable derivations) in several classes of algebras. In what follows, we shall use the notation SPDer(R) for the subspace of Der(R) consisting of spanned-by-split derivations. This paper describes some of the examples which motivated our original paper. Consider R presented as kΓ/I, a path algebra modulo relations, graded by the fundamental group pi1(Γ, I). In section one, we review an