Equivariant decomposition of polynomial vector fields

TL;DR

This paper introduces a systematic rational method to compute structure constants for Lie algebra bases related to polynomial vector fields, enabling explicit normal form calculations in 2D and 3D cases.

Contribution

It develops a new rational approach for calculating structure constants, facilitating the explicit normalization of vector fields with nilpotent linear parts.

Findings

Successfully computed normal forms for 2D and 3D vector fields

Provided explicit formulas for structure constants in the new basis

Demonstrated the method on Euler family vector fields

Abstract

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the linear nilpotent. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the present paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch-Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D case.