We follow definitions given by V. Lunts and A. Rosenberg and study the algebra of differential operators \(D(\mathbb{k}Q)\) on the quiver algebra \(\mathbb{k}Q\) over a finite quiver Q and investigate their properties. We completely describe \(D(\mathbb{k}Q)\) by finding a set \(\cal{S}\) of differential operators which generate \(D(\mathbb{k}Q)\) both as a left and as a right \(\mathbb{k}Q\) -module. The key is the simultaneous introduction of the set \(\cal{S}\) in a natural algebraic inductive way and as combinatorially defined maps. Using this, we are able to describe the algebra structure of \(D(\mathbb{k}Q)\) by formulas which express every product Θ · Γ for \(\Theta. \Gamma \in \cal{S}\) as a linear combination of elements of \(\cal{S}\) with non-negative integer coefficients. We also find generalized Leibnitz-type formulas for these operators, which suggest that \(D(\mathbb{k}Q)\) behaves like a representative algebra, and suggest a possible comultiplication and quantum group structure of \(D(\mathbb{k}Q)\) . We end with some questions and concrete examples.
