  We show that the category of representations of the Euclidean group of
orientation-preserving isometries of two-dimensional Euclidean space is
equivalent to the category of representations of the preprojective algebra of
infinite type A. We also consider the moduli space of representations of the
Euclidean group along with a set of generators. We show that these moduli
spaces are quiver varieties of the type considered by Nakajima. Using these
identifications, we prove various results about the representation theory of
the Euclidean group. In particular, we prove it is of wild representation type
but that if we impose certain restrictions on weight decompositions, we obtain
only a finite number of indecomposable representations.
