Aromatic and clumped multi-indices: algebraic structure and Hopf embeddings
This work addresses open problems in numerical analysis for volume-preserving methods, focusing on low-dimensional dynamics, and is incremental as it builds on recent attempts in the field.
The paper tackles the problem of describing numerical volume-preservation in low-dimensional dynamics by introducing aromatic and clumped multi-indices as simpler algebraic objects for Taylor expansions, resulting in the provision of their algebraic structures including pre-Lie-Rinehart algebra, Hopf algebroid, and Hopf algebra, and generalizing the Hopf embedding in the aromatic context.
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.