Vector fields and admissible embeddings for quiver moduli

We introduce a double framing construction for moduli spaces of quiver representations. It allows us to reduce certain sheaf cohomology computations involving the universal representation, to computations involving line bundles, making them amenable to methods from geometric invariant theory. We will use this to show that in many good situations the vector fields on the moduli space are isomorphic as a vector space to the first Hochschild cohomology of the path algebra. We also show that considering the universal representation as a Fourier-Mukai kernel in the appropriate sense gives an admissible embedding of derived categories.