Affine actions on Lie groups and post-Lie algebra structures

TL;DR

This paper introduces post-Lie algebra structures on pairs of Lie algebras, explores their existence and properties, especially in relation to nilpotent and solvable Lie groups, and classifies all complex two-dimensional cases.

Contribution

It defines post-Lie algebra structures, links them to NIL-affine actions, and provides existence criteria and classifications, including a complete classification of two-dimensional complex cases.

Findings

Post-Lie structures exist only if the second Lie algebra is solvable when the first is nilpotent.

Post-Lie structures naturally arise in NIL-affine actions on nilpotent Lie groups.

Complete classification of all complex, two-dimensional post-Lie algebras.

Abstract

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