Post-Lie algebra structures for perfect Lie algebras

Abstract

We study the existence of post-Lie algebra structures on pairs of Lie algebras (Formula presented.), 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 (Formula presented.), where (Formula presented.) is perfect non-semisimple, and (Formula presented.) is (Formula presented.). We also show that there is no post-Lie algebra structure on (Formula presented.), where (Formula presented.) is perfect and (Formula presented.) is reductive with a 1-dimensional center.