An operadic approach to substitution in Lie-Butcher series

03/19/2021

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by   Ludwig Rahm, et al.

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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 𝒬. By considering a module over the post-Lie operad, we get a cointeraction between 𝒬 and the Hopf algebra ℋ_N that describes composition for Lie-Butcher series. We use this coaction to describe substitution for Lie-Butcher series.