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

Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation