Classical Yang-Baxter equations and Nijenhuis operators for Lie algebras

In this paper the conditions that when a Lie algebra is Nijenhuis are investigated. Furthermore all the Nijenhuis operators on $\mathfrak{sl}_2$ under the standard Cartan-Weyl basis are given. On the other hand, the relations between the classical Yang-Baxter equation and Nijenhuis operators $N$ on a Lie algebra and $P$ on a Lie coalgebra are derived by means of the bialgebraic theory for Nijenhuis Lie algebras.