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

We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). 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 stochastic heat equation.