Poisson-Nijenhuis structures and the Vinogradov bracket

Beltrán, José; Monterde, J.

doi:10.1007/bf02108287

Cited by 9 publications

(7 citation statements)

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“…We thus obtain a formulation of the compatibility of a Poisson and a Nijenhuis structure that is the simplest that we have found in the literature [1] [17], and had eluded previous attempts to characterize PN structures.…”

Section: 2mentioning

confidence: 98%

The lie bialgebroid of a Poisson-Nijenhuis manifold

Kosmann–Schwarzbach

1996

Lett Math Phys

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Abstract. We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures.Résumé. Nousétudions une nouvelle classe de bigébroïdes de Lie, associés aux structures de Poisson-Nijenhuis.Introduction. Nijenhuis operators have been introduced in the theory of integrable systems in the work of Magri, Gelfand and Dorfman (see the book [4]), and, under the name of hereditary operators, in that of Fuchssteiner and Fokas. Poisson-Nijenhuis structures were defined by Magri and Morosi in 1984 [15] in their study of completely integrable systems. There is a compatibility condition between the Poisson structure and the Nijenhuis structure that is expressed by the vanishing of a rather complicated tensorial expression. In this letter, we shall prove that this condition can be expressed in a very simple way, using the notion of a Lie bialgebroid [14] [7] [12]. A Lie bialgebroid is a pair of vector bundles in duality, each of which is a Lie algebroid, such that the differential defined by one of them on the exterior algebra of its dual is a derivation of the Schouten bracket. Here we show that a Poisson structure and a Nijenhuis structure constitute a Poisson-Nijenhuis structure if and only if the following condition is satisfied: the cotangent and tangent bundles are a Lie bialgebroid when equipped respectively with the bracket of 1-forms defined by the Poisson structure, and with the deformed bracket of vector fields defined by the Nijenhuis structure.Let me add three "historical" remarks. This result was first conjectured by Magri during a conversation that we held at the time of the Semestre Symplectique at the Centre Emile Borel. Secondly, the Lie bracket of differential 1-forms on a Poisson manifold, defining the Lie-algebroid structure of its cotangent bundle, was defined by Fuchssteiner in an article of 1982 [6] which is not often cited, though it is certainly one of the first papers to mention this important definition. Thirdly, as A. Weinstein has shown [18] [19], Sophus Lie's book [11] contains a comprehensive theory of Poisson manifolds under the name of function groups, including, among many results, a proof of the contravariant form of the Jacobi identity, a proof of the duality between Lie algebra structures and linear Poisson structures on vector spaces, the notions of distinguished functions (Casimir functions) and polar groups (dual pairs), and the existence of canonical coordinates. Moreover, Carathéodory, in his book [2], proves explicitly the tensorial character of the Poisson bivector and gives a rather complete account of this theory, based on a short article by Lie [10] that appeared even earlier than the famous "Theorie der Transformationsgruppen", Part II, of 1890.

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Section: 2mentioning

confidence: 98%

The lie bialgebroid of a Poisson-Nijenhuis manifold

Kosmann–Schwarzbach

1996

Lett Math Phys

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“…A proof of Theorem 6 is also in [75], based on the fact that C(π, N ) = 0 implies ι π (d N df ) = − 1 2 H π I 1 (f ) which fact is a corollary of a theorem of Beltrán and Monterde [8].…”

Section: Remarkmentioning

confidence: 99%

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

Kosmann–Schwarzbach¹

2008

SIGMA

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“…Poisson-Nijenhuis structures [16] play an important role both in classical differential geometry (see, for example [1,10]) and in geometry of partial differential equations, see [11,14]. In the latter case existence of a Poisson-Nijenhuis structure virtually amounts to complete integrability of the equation under consideration.…”

Section: Introductionmentioning

confidence: 99%

“…We show that an invariant (with respect to the flow determined by the equation) Nijenhuis tensors are recursion operators for the symmetries, while invariant Poisson bi-vectors amount to Hamiltonian structures. We also define the ℓ-and ℓ * -coverings and reduce construction of recursion operators and Hamiltonian structures to solution of equation (1). The Schouten and Frölicher-Nijenhuis brackets as well as the compatibility conditions are reformulated in terms of the Jacobi brackets of the corresponding solutions and explicit formulas for these brackets are obtained.…”

Section: Introductionmentioning

confidence: 99%