This paper focuses on finite dimensional point derivations for the non-self-adjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to noninner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional.
"Finite dimensional point derivations for graph algebras." Illinois J. Math. 52 (2) 419 - 435, Summer 2008. https://doi.org/10.1215/ijm/1248355342