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. show that this approach is consistent with postulates regularity structures when it comes structure group, which arises from a Hopf algebra and comodule. Our approach, where dual naturally embeds into formal power series algebra, allows interpret group Lie arising consisting derivations on algebra. These in turn are infinitesimal generators two actions pairs (non-linearities, functions space-time mod constants). also argue there exist pre-Lie morphisms between our tree-based one cases branched rough paths (Grossman-Larson, Connes-Kreimer) heat equation.