Post-Lie algebra structures on pairs of Lie algebras

TL;DR

This paper investigates the existence and classification of post-Lie algebra structures on pairs of Lie algebras, exploring their properties across various algebraic classes and using cohomology to identify trivial cases.

Contribution

It provides new existence results, classifications, and triviality conditions for post-Lie algebra structures on different classes of Lie algebras.

Findings

Existence results depend on algebraic interplay.

Classified commutative structures on low-dimensional Lie algebras.

Proved triviality of certain structures using Lie algebra cohomology.

Abstract

We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we prove that such structures are trivial in several cases. We classify commutative structures on low-dimensional Lie algebras, and study the case of nilpotent Lie algebras.