An operadic approach to substitution in Lie-Butcher series

TL;DR

This paper introduces an operadic framework to describe substitution in Lie-Butcher series, extending known algebraic structures for B-series and applying them to Lie-Butcher series.

Contribution

It develops a new operadic approach to characterize substitution in Lie-Butcher series using bialgebraic structures derived from the post-Lie operad.

Findings

Derived a bialgebra $\

,

describes substitution in Lie-Butcher series using operadic methods.

Abstract

The paper follows an operadic approach to provide a bialgebraic description of substitution for Lie-Butcher series. We first show how the well-known bialgebraic description for substitution in Butcher's $B$-series can be obtained from the pre-Lie operad. We then apply the same construction to the post-Lie operad to arrive at a bialgebra $\mathcal{Q}$. By considering a module over the post-Lie operad, we get a cointeraction between $\mathcal{Q}$ and the Hopf algebra $\mathcal{H}_N$ that describes composition for Lie-Butcher series. We use this coaction to describe substitution for Lie-Butcher series.