Is there a relationship between pre-Lie algebras and post-Lie algebra?

Question

You can find a short survey in this paper https://arxiv.org/pdf/1712.09415.pdf on Post-Lie algebras. I am interested in them because both of them can be constructed on rooted trees. But I don't know ''is there a relationship between pre-Lie algebras and post-Lie algebra''?

Answer 1

Yes, there are relations between both. It depends on which level you are looking for such relations. In our article

Affine actions on Lie groups and post-Lie algebra structures

we explain geometric and algebraic relations, pre-Lie algebra structures being a special case of post-Lie algebra structures. The references mention work by Vallette and Loday, who give many other viewpoits, i.e., by operad theory, rooted trees etc. The operads PreLie and PostLie arise in the context of Manin's black product, and in other topics, e.g. homology of generalized partition posets and renormalization theory.

Manin's black product ${\mathcal {P}} \bullet {\mathcal {Q}}$ of binary quadratic operads has the operad ${\mathcal {Lie}}$ as neutral element, i.e., $$ {\mathcal {P}}={\mathcal {Lie}}\bullet {\mathcal {P}}={\mathcal {P}}\bullet {\mathcal {Lie}}. $$

$\bullet$ The bi-successor and tri-successor of a quadratic operad ${\mathcal {P}}$ are given by \begin{align*} Bi ({\mathcal {P}}) & = {\mathcal {PreLie}}\bullet {\mathcal {P}}, \\ Tri ({\mathcal {P}}) & = {\mathcal {PostLie}}\bullet {\mathcal {P}}. \end{align*}