  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.
