  We study the category of graded finite-dimensional representations of the
polynomial current algebra associated to a simple Lie algebra. We prove that
the category has enough injectives and compute the graded character of the
injective envelopes of the simple objects as well as extensions between simple
objects. The simple objects in the category are parametized by the affine
weight lattice. We show that with respect to a suitable refinement of the
standard ordering on affine the weight lattice the category is highest weight.
We compute the Ext quiver of the algebra of endomorphisms of the injective
cogenerator of the subcategory associated to a interval closed finite subset of
the weight lattice. Finally, we prove that there is a large number of
interesting quivers of finite, affine and tame type that arise from our study.
We also prove that the path algebra of star shaped quivers are the Ext algebra
of a suitable subcategory.
