Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

Bolsinov, Alexey V.; Konyaev, Andrey Yu.; Matveev, Vladimir S.

doi:10.1007/s40879-020-00429-6

Cited by 12 publications

(18 citation statements)

References 52 publications

(176 reference statements)

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“…One can slightly generalize this example to include the points at which the operator L is not diagonalizable. In the context of the present paper, this generalization is essentially due to [3], where we discussed singularities of geodesically compatible pairs: the metric is [3, equation (37)] and the operator is [3, equation (36)]. We repeat these formulas here: (10) Much earlier, this form of g and L appeared in the study of multi-Hamiltonian structures for the coupled KdV system by Antonowicz and Fordy [1].…”

Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

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Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

2021

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We connect two a priori unrelated topics, the theory of geodesically equivalent metrics in differential geometry, and the theory of compatible infinitedimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is (n + 1)(n + 2)/2 dimensional; we describe it completely and show that it is maximal. Another has dimension n + 2 and is, in a certain sense, polynomial. We show that a nontrivial polynomial family of compatible Poisson structures of dimension n + 2 is unique and comes from a pair of geodesically equivalent metrics. In addition, we generalize a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

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Section: Basic Definitions and Main Resultsmentioning

confidence: 99%

“…It is well known that if L 1 and L 2 are both Nijenhuis, then αL 1 + βL 2 is Nijenhuis if and only if the Frolicher-Nijenhuis bracket of L 1 and L 2 vanishes (see [15] and e.g. [3]):…”

Section: Lemma 33 Consider a Pair Of Non-degenerate Nijenhuis Operators L 1 And L 2 Thenmentioning

confidence: 99%

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

2021

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“…In our paper we study compatibility of nonhomogeneous Poisson structures of type P 3 +P 1 such that the part of order 3 is Darboux-Poisson. That is, we have 4 nondegenerate Poisson structures: A g and A ḡ constructed by flat metrics g and ḡ by (6), and B h and Bh constructed by flat metrics h and h by (8). We assume that A g + B h and A ḡ + Bh are (nonhomogeneous) Poisson structures and ask the question when these structures are compatible in the sense that any of their linear combinations is a Poisson structure [27].…”

Section: Mathematical Setupmentioning

confidence: 99%

“…If (i1), (i2) and (i3) are fulfilled, we introduce the following commutative multiplication 8 ⋆ w on the direct product a × â (here we use natural inclusions a, â ⊂ a × â): 8 Here the index w stands for warped product However, we need to ensure that Conditions (i1 * ), (i3 * ) and (i3 * ) are met, that is: a in s (λ) = 0 for s = n, b in (λ) = 0, and a nn n (λ) = 4 K.…”

Section: Algebraic Interpretation Of Warped Productmentioning

confidence: 99%

“…Nijenhuis operator is a (1,1)-tensor field L = L i j on a manifold M of dimension n such that its Nijenhuis torsion vanishes. Nijenhuis geometry as initiated in [6] (where also all necessary definitions can be found) and further developped in [7,8,23] studies Nijenhuis operators and their applications. There are many topics in mathematics and mathematical physics where Nijenhuis operators appear naturally; this paper is devoted to the study of ∞-dimensional compatible Poisson brackets of type P 3 + P 1 , where the lower index i indicates the order of the homogeneous bracket P i (the necessary definitions will be given in Section 1.2).…”

Section: Introduction 1forewordmentioning

confidence: 99%

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Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

Bolsinov¹,

Yu.²,

Matveev³

2021

Preprint

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We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third order term P 3 is Darboux-Poisson and non-degenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. of linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular we show that each such Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with vertices labeled by natural numbers whose sum is the dimension of the manifold. These pencils are naturally related to certain (polynomial, in the most nondegenerate case) pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the Nijenhuis pencil.

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Integrating Nijenhuis structures

Pugliese

Sparano

Vitagliano

2023

Annali di Matematica

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A Nijenhuis operator on a manifold is a (1, 1) tensor whose Nijenhuis torsion vanishes. A Nijenhuis operator $${\mathcal {N}}$$ N determines a Lie algebroid that knows everything about $${\mathcal {N}}$$ N . In this sense, a Nijenhuis operator is an infinitesimal object. In this paper, we identify its global counterpart. Namely, we characterize Lie groupoids integrating the Lie algebroid of a Nijenhuis operator. We illustrate our integration result in various examples, including that of a linear Nijenhuis operator on a vector space or, which is equivalent, a pre-Lie algebra structure.

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Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

Abstract: We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

Cited by 12 publications

References 52 publications

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

Applications of Nijenhuis geometry II: maximal pencils of multi-Hamiltonian structures of hydrodynamic type

Applications of Nijenhuis geometry III: Frobenius pencils and compatible non-homogeneous Poisson structures

Integrating Nijenhuis structures

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