Software

Computational tools developed as part of my research on integrable systems and Haantjes geometry. All software is free for academic and research use.

Nijenhuis and Haantjes Torsion Package for Maple

Version: 1.0 (February 2026)

Language: Maple

License: Free for academic and research use

Overview

This Maple package computes the Nijenhuis and Haantjes torsions of (1,1)-tensor fields. These geometric invariants are fundamental tools in:

Separation of variables for integrable systems
Classification of Hamiltonian systems
Study of Haantjes algebras and symplectic-Haantjes manifolds
Differential geometry of almost-complex structures

Download & Documentation

Download Maple Package (.mpl) GitHub Repository View Full Documentation View Quick Start Guide

Quick Start

Installation

Copy the functions from NijenhuisHaantjesTorsion.mpl into your Maple workbook.

Basic Usage

# Define your tensor L as an n×n matrix
L := Matrix([[f11, f12, f13],
             [f21, f22, f23],
             [f31, f32, f33]]);

# Define coordinates
coords := [x, y, z];

# Compute Haantjes torsion
H := HaantjesTorsion(L, coords);

# Check if tensor is Haantjes
IsHaantjes(L, coords);   # Returns true/false

Main Functions

`NijenhuisTorsion(L, coords)`	Full Nijenhuis torsion tensor
`HaantjesTorsion(L, coords)`	Full Haantjes torsion tensor
`IsNijenhuis(L, coords)`	Check if all Nijenhuis components vanish
`IsHaantjes(L, coords)`	Check if all Haantjes components vanish

Options

# Disable simplification for faster computation
H := HaantjesTorsion(L, coords, simplify_each=false);

# Enable verbose output
H := HaantjesTorsion(L, coords, verbose=true);

Mathematical Background

For a (1,1)-tensor field L on a manifold with coordinates x = (x¹,...,xⁿ), the Nijenhuis torsion is the (1,2)-tensor defined by:

(T_L)ⁱ_jk = Σ_α [ (∂Lⁱ_k/∂x^α)L^α_j - (∂Lⁱ_j/∂x^α)L^α_k + (∂L^α_j/∂x^k - ∂L^α_k/∂x^j)Lⁱ_α ]

The Haantjes torsion H_L is a derived quantity that provides a weaker (but often more useful) condition for local diagonalizability.

Key property: T_L = 0 implies H_L = 0, but not conversely.

References

D. Reyes, P. Tempesta, G. Tondo, "Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry", Commun. Nonlinear Sci. Numer. Simul. 104 (2022) 106021. DOI
P. Tempesta, G. Tondo, "Higher Haantjes Brackets and Integrability", Commun. Math. Phys. 389 (2022) 1647–1671. DOI
J. Haantjes, "On X_m-forming sets of eigenvectors", Nederl. Akad. Wetensch. Proc. Ser. A 58 (1955) 158–162.
A. Nijenhuis, "X_n-1-forming sets of eigenvectors", Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951) 200–212.

Contact

For questions, bug reports, or collaboration inquiries: ondrej.kubu@icmat.es

Acknowledgments

This code was developed as part of research on integrable systems and Haantjes geometry. OK's postdoctoral fellowship is financed by project FOSTERING ICMAT'S STRATEGIC SCIENTIFIC LINES (reference 202450E223) of ICMAT-CSIC, Spain, supported by the Severo Ochoa Programme for Centres of Excellence in R&D (CEX2019-000904-S).