相关论文: Higher Haantjes Brackets and Integrability
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…
We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility…
We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. This structure, expressed in terms of suitable operators whose Haantjes…
A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds), as a natural…
In this work, we introduce the notion of polarization of generalized Nijenhuis torsions and establish several algebraic identities. We prove that these polarizations are relevant in the characterization of Haantjes $C^{\infty}$(M)-modules…
We characterize a class of integrable Hamiltonian hydrodynamic chains, based on the necessary condition for the integrability provided by the vanishing of the Haantjes tensor. We prove that the vanishing of the first few components of the…
We briefly recall the history of the Nijenhuis torsion of (1,1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplectic-Haantjes structures of Tempesta and Tondo…
This work is the first, and main, of the series of papers in progress dedicated to Nienhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis tensor. It serves as an introduction to Nijenhuis Geometry that should be…
For a Banach--Lie group $G$ and an embedded Lie subgroup $K$ we consider the homogeneous Banach manifold $\mathcal M=G/K$. In this context we establish the most general conditions for a bounded operator $N$ acting on $Lie(G)$ to define a…
The generalized Zernike family $H_{(N)} = p_1^2 + p_2^2 + \sum_{n=1}^N \gamma_n\,(q_1 p_1 + q_2 p_2)^n$ is a parametric family of two-dimensional superintegrable Hamiltonians, admitting $N$ integrals of motion of degree $N$ in the momenta.…
We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate…
A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian…
In commutative differential geometry the Fr\"olicher-Nijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In \cit!{3} the Fr\"olicher-Nijenhuis bracket was developped for universal differential forms of…
We carry over to a quite general noncommutative setting some of the basic tools of differential geometry, using from the very beginning the setting of convenient vector spaces developed by Froelicher and Kriegl, which allows to carry all of…
A Nijenhuis operator on a manifold $M$ is a $(1,1)$ tensor $\mathcal N$ whose Nijenhuis-torsion vanishes. A Nijenhuis operator $\mathcal N$ on $M$ determines a Lie algebroid structure $(TM)_{\mathcal N}$ on the tangent bundle $TM$. In this…
We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which satisfy the Nijenhuis integrability…
We extend the Heine-Stieltjes Theorem to concern all (non-degenerate) differential operators preserving the property of having only real zeros. This solves a conjecture of B. Shapiro. The new methods developed are used to describe intricate…
We extend the characterization of the integrability of an almost complex structure $J$ on differentiable manifolds via the vanishing of the Fr\"olicher-Nijenhuis bracket $[J, J] ^{FN}$ to an analogous characterization of torsion-free…
In the infinite-dimensional Banach setting, we consider general smooth Banach fibrations $\tau:M\to M_0$ and `$(1,1)$-tensors' $N:TM\to TM$ that are projectable (in the obvious sense) onto Nijenhuis operators $N_0:TM_0\to TM_0$ on $M_0$. We…
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to…