Post-Lie algebra structures for perfect Lie algebras

Abstract

We study the existence of post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, where one of the algebras is perfect non-semisimple, and the other one is abelian, nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several nonexistence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is perfect non-semisimple, and $\mathfrak{n}$ is $\mathfrak{s}{\mathfrak{l}}_3(\mathbb{C})$. We also show that there is no post-Lie algebra structure on $(\mathfrak{g},\mathfrak{n})$, where $\mathfrak{g}$ is perfect and $\mathfrak{n}$ is reductive with a 1-dimensional center.

Keywords:

• Perfect Lie algebra
• post-Lie algebra structure

2020 Mathematics Subject Classification:

• Primary: 17B30
• 17D25

Additional information

Funding

Dietrich Burde and Mina Monadjem are supported by the Austrian Science Foundation FWF, grant P 33811. Karel Dekimpe is supported by Methusalem grant Meth/21/03 - long term structural funding of the Flemish Government.