Post-Lie algebras in Regularity Structures

Yvain Bruned, Foivos Katsetsiadis

Published 2022-07-31Version 1

In this work, we construct the deformed Butcher-Connes-Kreimer Hopf algebra appearing in 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, used in arXiv:1610.0846(8) and multi-indices, used in arXiv:2103.0418(7). Our construction is inspired from arXiv:2103.0418(7) where the Hopf algebra was obtained as the universal envelope of a Lie algebra and the authors showed 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.