Affine actions on Lie groups and post-Lie algebra structures

Abstract

We introduce post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g}\text{,}\mathfrak{n})$ 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 $\mathfrak{g}$ and $\mathfrak{n}$. One result is, for example, that if there exists a post-Lie algebra structure on $(\mathfrak{g}\text{,}\mathfrak{n})$, where $\mathfrak{g}$ is nilpotent, then $\mathfrak{n}$ must be solvable. Furthermore special cases and examples are given. This includes a classification of all complex, two-dimensional post-Lie algebras.