Planar Regularity Structures

Rahm, Ludwig

doi:10.48550/arxiv.2212.04856

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2023

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“…Indeed, the special case of a pre-Lie algebra has already been used by Loday and Ronco in [36] to formalise the idea of a combinatorial Hopf algebra by means of a categorical equivalence. Lastly, our use of the post-Lie structure has already had some impact by allowing one to construct planar Regularity Structures in [41] for SPDEs in homogeneous spaces as well as to revisit, in [7], the Chapoton-Foissy isomorphism given in [27,19]. Also, in [33], the authors were inspired by our construction to study a class of post-Lie algebras comprised of derivations.…”

Section: Introductionmentioning

confidence: 99%

Post-Lie algebras in Regularity Structures

Bruned,

Katsetsiadis

2023

Forum of Mathematics, Sigma

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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.

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