An operadic approach to substitution in Lie–Butcher series

Abstract

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.