Insertion pre-Lie products and translation of rough paths based on multi-indices

We use the diagram-free approach to regularity structures introduced by Otto et. al. to build rough paths based on multi-indices. We identify the analogue of the insertion pre-Lie algebra of trees and use it to build the corresponding group of translations of rough paths. We make this identification precise by showing that the natural dictionary between trees and multi-indices is a pre-Lie morphism under insertion, which in turn yields a Hopf algebra morphism to the rooted tree Hopf algebra equipped with the extraction-contraction coproduct.