Post-Lie algebras in Regularity Structures

Abstract

In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra coming from the theory of Regularity Structures as the universal envelope of a post-Lie algebra. We show that this can be done using either of the two combinatorial structures that have been proposed in the context of singular SPDEs: decorated trees and multi-indices. Our construction is inspired from multi-indices where the Hopf algebra was obtained as the universal envelope of a Lie algebra, and it has been proved that one can find a basis that is symmetric with respect to certain elements. We show that this Lie algebra comes from an underlying post-Lie structure.

Paracontrolled calculus and regularity structures II

“…The proofs of them are given in Section 4.2. The following fact is also proved in Proposition 4.1 of [10].…”

“…Here is how to build a large family of K-admissible interpretation maps from a naive one. Recall from [6] and Bruned & Nadeem's work [10] the following definition.…”