Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings
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| Format: | Preprint |
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2026
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| _version_ | 1866914450111463424 |
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| author | Zhu, Zhicheng Laurent, Adrien Busnot |
| author_facet | Zhu, Zhicheng Laurent, Adrien Busnot |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13105 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings Zhu, Zhicheng Laurent, Adrien Busnot Combinatorics Numerical Analysis Rings and Algebras 16T05, 37M15, 05C05, 16T30, 17A30 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. |
| title | Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings |
| topic | Combinatorics Numerical Analysis Rings and Algebras 16T05, 37M15, 05C05, 16T30, 17A30 |
| url | https://arxiv.org/abs/2603.13105 |