The structure group for quasi-linear equations via universal enveloping algebras

We consider the approach of replacing trees by multi-indices as an index set of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and Weber to tackle quasi-linear singular SPDEs. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group $\mathsf{G}$. In particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra $\mathsf{T}^+$ and a comodule $\Delta\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$. In fact, this approach, where the dual $\mathsf{T}^*$ of the abstract model space $\mathsf{T}$ naturally embeds into a formal power series algebra, allows to interpret $\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (nonlinearities, functions of space-time mod constants). These actions are shift of space-time and tilt by space-time polynomials. The Hopf algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The coordinates are determined by an underlying pre-Lie algebra structure of the derived algebra of $\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness and the restrictive definition of $\mathsf{T}$, allow for this purely algebraic construction of $\mathsf{G}$. We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the generalized parabolic Anderson model.