Post-Lie algebras in Regularity Structures

2022/07/31 by Yvain Bruned, Bruned, Yvain, Foivos Katsetsiadis +1 · 3 citations
Materials Science · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Porphyrin and Phthalocyanine Chemistry #Probability (math.PR) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2208.00514

openalex publication_date 2022/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
• Geometric post-Lie deformations of post-Lie algebras and regularity structures
• Construction of pre-and post-Lie algebras for stochastic PDEs

Related

• Post-Lie deformations of pre-Lie algebras and their applications in Regularity Structures
• On post-Lie structures for free Lie algebras
• Hopf and pre-Lie algebras in regularity structures
• Post-Lie algebras of derivations and regularity structures
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra