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arXiv · 2603.13105

Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings

Zhicheng Zhu·Adrien Busnot Laurent
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Abstract

Butcher forests extend naturally into aromatic and clumped forests and play a fundamental role in the numerical analysis of volume-preserving methods. The design of general volume-preserving methods is a challenging open problem, and recent attempts showed progress on specific dynamics. We introduce aromatic and clumped multi-indices, that are algebraic objects that simplify the study of volume-preservation to the one-dimensional setting, while retaining much of the structure (in stark opposition to standard multi-indices). We provide their algebraic structure of pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra, apply them in numerical analysis, and generalise to the aromatic context the Hopf embedding from multi-indices to the BCK Hopf algebra.

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Zhicheng Zhu, Adrien Busnot Laurent. 2026-03-13. Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings. https://arxiv.org/abs/2603.13105

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