An operadic approach to substitution in Lie-Butcher series

Ludwig Rahm

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arXiv · 2103.10893

An operadic approach to substitution in Lie-Butcher series

Ludwig Rahm

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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.

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Ludwig Rahm. 2021-03-19. An operadic approach to substitution in Lie-Butcher series. https://arxiv.org/abs/2103.10893

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