Affine actions on Lie groups and post-Lie algebra structures

We introduce post-Lie algebra structures on pairs of Lie algebras (g,n) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras g and n. One result is, for example, that if there exists a post-Lie algebra structure on (g,n), where g is nilpotent, then n must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.