Post-Lie algebra structures for perfect Lie algebras

Abstract

We study the existence of post-Lie algebra structures on pairs of Lie algebras ( g , 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 ( g , n ) , where g is perfect non-semisimple, and n is s l 3 ( C ) . We also show that there is no post-Lie algebra structure on ( g , n ) , where g is perfect and n is reductive with a 1-dimensional center.