Post-Lie algebra structures and generalized derivations of semisimple Lie algebras

TL;DR

This paper investigates the existence and construction of post-Lie algebra structures on pairs of Lie algebras, especially when one is semisimple, and classifies related generalized derivations.

Contribution

It provides new existence results, constructions, and classifications of post-Lie algebra structures involving semisimple Lie algebras and generalized derivations.

Findings

No post-Lie structures for semisimple g and solvable n.

Constructed canonical structures for semisimple n and certain solvable g.

Classified post-Lie structures induced by generalized derivations.

Abstract

We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple n and solvable, unimodular g. We also determine the generalized $(\al,\be,\ga)$-derivations of n in the semisimple case. As an application we classify post-Lie algebra structures induced by generalized derivations.