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

Zhicheng Zhu, Adrien Busnot Laurent

Abstract

Butcher forests extend naturally into aromatic and clumped forests and play a fundamental role in the numerical analysis of volume-preserving methods. The description of numerical volume-preservation is filled with open problems and recent attempts showed progress on specific dynamics and in low-dimension. Following this trend, we introduce aromatic and clumped multi-indices, that are simpler algebraic objects that better describe the Taylor expansions in low dimension. We provide their algebraic structure of pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra, and we generalise in the aromatic context the Hopf embedding from multi-indices to the BCK Hopf algebra.

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

Abstract

Butcher forests extend naturally into aromatic and clumped forests and play a fundamental role in the numerical analysis of volume-preserving methods. The description of numerical volume-preservation is filled with open problems and recent attempts showed progress on specific dynamics and in low-dimension. Following this trend, we introduce aromatic and clumped multi-indices, that are simpler algebraic objects that better describe the Taylor expansions in low dimension. We provide their algebraic structure of pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra, and we generalise in the aromatic context the Hopf embedding from multi-indices to the BCK Hopf algebra.
Paper Structure (12 sections, 18 theorems, 111 equations, 1 figure)

This paper contains 12 sections, 18 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on aromatic and clumped Butcher forests
  3. The pre-Lie-Rinehart algebra of aromatic trees
  4. Hopf structure of aromatic and clumped forests
  5. Aromatic and clumped multi-indices
  6. The free aromatic Novikov algebra
  7. The Hopf algebras of aromatic and clumped monomials
  8. Hopf embedding between multi-indices and forests
  9. Free edges, graftings, cuts, and Hopf embeddings
  10. Free edges and extended maps
  11. Combinatorics of the graftings and cuts
  12. The aromatic LOT coproduct

Key Result

Proposition 2.5

The $\mathcal{A}\xspace$-module $\mathcal{A}\xspace T$ is a pre-Lie-Rinehart algebra when equipped with the maps

Figures (1)

  • Figure 1:

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5: Floystad20tup
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 41 more