Higher Haantjes Brackets and Integrability

TL;DR

This paper introduces a new class of brackets extending the Fr"olicher--Nijenhuis bracket, with applications to operator integrability and block-diagonalization, generalizing Haantjes' theorem.

Contribution

It develops an infinite class of brackets that generalize existing torsions, providing new criteria for operator integrability without spectral analysis.

Findings

Vanishing higher Nijenhuis torsion implies integrability of eigen-distributions.

The new brackets unify and extend previous torsion concepts.

Operators with zero higher torsion can be locally block-diagonalized.

Abstract

We propose a new, infinite class of brackets generalizing the Fr\"olicher--Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.