  In this paper, first we study infinitesimal deformations of a Lie algebra
with a representation and introduce the notion of a Nijenhuis pair, which gives
a trivial deformation of a Lie algebra with a representation. Then we introduce
the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a
representation, which is a generalization of the $r$-$n$ structure
($r$-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam and
Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives
rise to a hierarchy of Kupershmidt operators. Finally, we define a
Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a
Lie algebra with respect to the adjoint representation, and study the relation
between Rota-Baxter-Nijenhuis structures and $r$-matrix-Nijenhuis structures.
