On post-Lie structures for free Lie algebras

We study post-Lie structures on free Lie algebras, the Grossman-Larson product on their enveloping algebras, and provide an abstract formula for its dual coproduct. This might be of interest for the general theory of post-Hopf algebras. Using a magmatic approach, we explore post-Lie algebras connected to multiple zeta values and their $q$-analogues. For multiple zeta values, this framework yields an algebraic interpretation of the Goncharov coproduct. Assuming that the Bernoulli numbers satisfy the so called threshold shuffle identities, we present a post-Lie structure, whose induced Lie bracket we expect to restrict to the dual of indecomposables of multiple $q$-zeta values. Our post-Lie algebras align with Ecalle's theory of bimoulds: we explicitly identify the ari bracket with a post-Lie structure on a free Lie algebra, and conjecture a correspondence for the uri bracket.