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. Contents 1. Introduction 2. Preliminaries on the Nijenhuis and Haantjes geometry 3. The generalized Nijenhuis tensors and block-diagonalization 3.1. Eigen-distributions and spectral properties of generalized Nijenhuis operators 3.2. Block-diagonalization 4. Generalized Haantjes algebras 4.1. Definitions 4.2. Cyclic generalized Haantjes algebras 5. Generalized Nijenhuis torsions and simultaneous block-diagonalization 6. Block-diagonalization of generalized Haantjes algebras 6.1. A level-three generalized Haantjes algebra 6.2. Block-diagonalization 7. A level-four generalized Haantjes algebra 7.1. Spectral analysis 7.2. Block-diagonalization Acknowledgement References
We show that the theory of classical Hamiltonian systems admitting separation variables can be formulated in the context of (ω, H ) structures. They are essentially symplectic manifolds endowed with a Haantjes algebra H , namely an algebra of (1,1) tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation.We shall prove that a multiseparable system admits as many ωH structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.
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