Post-Lie algebras in Regularity Structures

TL;DR

This paper constructs a Hopf algebra related to Regularity Structures as the universal envelope of a post-Lie algebra, connecting combinatorial structures like decorated trees and multi-indices.

Contribution

It introduces a new construction of the deformed Hopf algebra using post-Lie algebra theory, unifying different combinatorial approaches in singular SPDEs.

Findings

Established the Hopf algebra as the universal envelope of a post-Lie algebra.

Linked combinatorial structures to algebraic frameworks in Regularity Structures.

Provided a basis symmetric with respect to certain elements.

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.