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An operadic approach to substitution in Lie-Butcher series (2103.10893v1)
Published 19 Mar 2021 in math.CO, cs.NA, math.NA, and math.RA
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.