Generalized Nijenhuis Torsions and block-diagonalization of operator fields

TL;DR

This paper introduces generalized Nijenhuis torsions to analyze the normal forms of operator fields, proving local block-diagonalization under certain conditions and defining a new algebraic structure called generalized Haantjes algebra.

Contribution

It extends classical torsion concepts to higher levels, providing a new framework for simultaneous block-diagonalization of commuting operator fields and introducing generalized Haantjes algebras.

Findings

Vanishing generalized Nijenhuis torsion ensures local block-diagonalization.

Introduction of generalized Haantjes algebra as a natural extension.

Provides tools for studying normal forms of operator fields.

Abstract

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We propose a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level $l$ vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras.