Post-Lie algebra structures for nilpotent Lie algebras

摘要

We study post-Lie algebra structures on (g, n) for nilpotent Lie algebras. First, we show that if g is nilpotent such that H-0 (g, n) = 0, then also n must be nilpotent, of bounded class. For post-Lie algebra structures x . y on pairs of 2-step nilpotent Lie algebras (g, n) we give necessary and sufficient conditions such that x circle y = 1/2 (x . y + y . x) defines a CPA-structure on g, or on n. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension n = 5 is complete. Finally, we classify all post-Lie algebra structures on (g, n) for g congruent to n congruent to n(3), where n 3 is the three-dimensional Heisenberg Lie algebra.