Post-Lie algebras of derivations and regularity structures

Given a commutative algebra $\mathcal{A}$, we exhibit a canonical structure of post-Lie algebra on the space $\mathcal{A}\otimes {\rm Der}(\mathcal{A})$ where ${\rm Der}(\mathcal{A})$ is the space of derivations on $\mathcal{A}$, in order to use the machinery given by Oudom-Guin (2008) and Ebrahimi-Fard--Lundervold--Munthe-Kaas (2015), and to define a Hopf algebra structure on the associated enveloping algebra with a natural action on $\mathcal{A}$. We apply these results to the setting of Linares-Otto-Tempelmayr (2023), giving a simpler and more efficient construction of their action and extending the recent work by Bruned-Katsetsiadis (2023). This approach gives an optimal setting to perform explicit computations in the associated structure group.