Post-Lie algebras of derivations and regularity structures

Abstract

Given a commutative algebra $\mathcal{A}$, we exhibit a canonical structure of post-Lie algebra on the space $\mathcal{A}\otimes \mathrm{Der}(\mathcal{A})$, where $\mathrm{Der}(\mathcal{A})$ is the space of derivations on $\mathcal{A}$, in order to use the machinery given by Ebrahimi-Fard–Lundervold–Munthe-Kaas (2015) and Oudom–Guin (2025), 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.