Post-Lie algebra structures for nilpotent Lie algebras

TL;DR

This paper investigates post-Lie algebra structures on nilpotent Lie algebras, establishing conditions for their properties, classifying structures on specific cases, and exploring their relation to CPA-structures and LR-structures.

Contribution

It provides new necessary and sufficient conditions for post-Lie structures on nilpotent Lie algebras, including classifications and properties of structures on Heisenberg algebras.

Findings

Nilpotent Lie algebra structures imply nilpotency of associated algebras.

Conditions for CPA-structures derived from post-Lie algebra structures.

Complete LR-structures on high-dimensional Heisenberg algebras.

Abstract

We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be nilpotent, of bounded class. For post-Lie algebra structures $x\cdot y$ on pairs of $2$-step nilpotent Lie algebras $(\mathfrak{g},\mathfrak{n})$ we give necessary and sufficient conditions such that $x\circ y=\frac{1}{2}(x\cdot y+y\cdot x)$ defines a CPA-structure on $\mathfrak{g}$, or on $\mathfrak{n}$. As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of dimension $n\ge 5$ is complete. Finally we classify all post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for $\mathfrak{g}\cong \mathfrak{n}\cong \mathfrak{n}_3$, where $\mathfrak{n}_3$ is the $3$-dimensional Heisenberg Lie algebra.