  The representations of a quiver Q over a field k have been studied for a long
time. It seems to be worthwhile to consider also representations of Q over
arbitrary finite-dimensional k-algebras A. Here we draw the attention to the
case when A = k[\epsilon] is the algebra of dual numbers, thus to the
\Lambda-modules with \Lambda = kQ[\epsilon]. The algebra \Lambda{} is a
1-Gorenstein algebra, therefore the torsionless \Lambda-modules are known to be
of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay
modules). They form a Frobenius category \Cal L, thus the corresponding stable
category is a triangulated category. As we show, the category \Cal L is the
category of perfect differential kQ-modules and the stable category of \Cal L
is triangle equivalent to the orbit category of the derived category D^b(mod
kQ) modulo the shift. The homology functor H from mod \Lambda{} to \mod kQ
yields a bijection between the indecomposables in the stable category of \Cal L
and those in mod kQ. Our main interest lies in the inverse, it is given by the
minimal \Cal L-approximation. Also, we determine the kernel of the restriction
of the functor H to \Cal L and we describe the Auslander-Reiten quiver of \Cal
L.
