Algebraic Integrability Conditions for Killing Tensors on Constant Sectional Curvature Manifolds

TL;DR

This paper translates the geometric Nijenhuis integrability conditions for Killing tensors into algebraic equations using representation theory, enabling the construction of new integrable Killing tensors on constant curvature manifolds.

Contribution

It introduces an algebraic approach to integrability conditions for Killing tensors, leading to explicit algebraic equations and new integrable tensor families.

Findings

Derived algebraic integrability conditions as degree two and three equations.

Constructed a new family of integrable Killing tensors.

Established an isomorphism linking geometric and algebraic tensor spaces.

Abstract

We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integrability conditions for the corresponding algebraic curvature tensor, resulting in two simple algebraic equations of degree two and three. As a first application of this we construct a new family of integrable Killing tensors.