Post-Lie algebra structures for nilpotent Lie algebras

Abstract

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.