Post-Lie algebra structures and generalized derivations of semisimple Lie algebras

We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple $\mathfrak{g}$ and solvable $\mathfrak{n}$ we show that there exist no post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$. For semisimple $\mathfrak{n}$ and certain solvable $\mathfrak{g}$ we construct natural post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple $\mathfrak{n}$ and solvable, unimodular $\mathfrak{g}$. We also determine the generalized $(\alpha,\beta,\gamma)$-derivations of $\mathfrak{n}$ in the semisimple case. As an application we classify certain post-Lie algebra structures related to generalized derivations.