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We study post-Lie algebra structures on pairs of Lie algebras$(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups.We prove existence results for such structures depending on the interplay ofthe algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider theclasses of simple, semisimple, reductive, perfect, solvable, nilpotent, abelianand unimodular Lie algebras. Furthermore we consider commutative post-Liealgebra structures on perfect Lie algebras. Using Lie algebra cohomology weprove that such structures are trivial in several cases. We classifycommutative structures on low-dimensional Lie algebras, and study the case ofnilpotent Lie algebras.

Post-Lie algebra structures on pairs of Lie algebras