Nijnehuis Geometry III: gl-regular Nijenhuis operators

Bolsinov, Alexey; Konyaev, Andrey; Matveev, Vladimir

doi:10.48550/arxiv.2007.09506

Cited by 5 publications

(11 citation statements)

References 17 publications

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“…The (i)-part of Fact 2 is in [13], see also [16,28,29]. In view of formula (7), the two conditions from Definition 1 are nothing else but a geometric reformulation of the compatibility condition for Poisson structures of order one, which explains the name Poisson compatible. The (ii)-part is an easy corollary of [11,Theorem 3.2], see also proof of Theorem 3 below.…”

Section: Brief Description Of Main Results Structure Of the Paper And...mentioning

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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Section: Brief Description Of Main Results Structure Of the Paper And...mentioning

confidence: 99%

Section: Introduction 1forewordmentioning

confidence: 99%

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

Bolsinov¹,

Yu.²,

Matveev³

2021

Preprint

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“…This paper continues the Nijenhuis Geometry programme started in [1] and further developed in [2,3,20]. This programme was initially motivated by the fact that Niejnhuis operators (i.e.…”

Section: Introductionmentioning

confidence: 87%

“…In this case, the metric g is not geodesically compatible with L. Systems of hydrodynamic type (2) such that A is a Nijunhuis operator are well-understood in the case when L is diagonalisable; they decouple in Hopf equations and can be solved (almost) explicitly. An inclusion of singular points to this situation was done in [2].…”

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confidence: 99%

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

Bolsinov,

Konyaev,

Matveev

2020

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We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional 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 generalise a result of Sinjukov (1961) from constant curvature metrics to arbitrary Einstein metrics.

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“…Let G ⇒ M be a Lie groupoid with Lie algebroid A ⇒ M. The structure maps of G will be denoted s, t : G → M (source and target), m : G (2) → M (multiplication), u : M → G (unit), and i : G → G (inversion). We denote by G (k) the manifold of k composable arrows in G. Let T ∈ Ω 1 (G, T G).…”

Section: A Review Of Multiplicative and Im (1 1) Tensorsmentioning

confidence: 99%

Integrating Nijenhuis Structures

Pugliese¹,

Sparano²,

Vitagliano³

2022

Preprint

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A Nijenhuis operator on a manifold M is a (1, 1) tensor N whose Nijenhuistorsion vanishes. A Nijenhuis operator N on M determines a Lie algebroid structure (T M ) N on the tangent bundle T M . In this sense a Nijenhuis operator can be seen as an infinitesimal object. In this paper, we identify its global counterpart. Namely, we show that when the Lie algebroid (T M ) N is integrable, then it integrates to a Lie groupoid equipped with appropriate additional structure responsible for N , and viceversa, the Lie algebroid of a Lie groupoid equipped with such additional structure is of the type (T M ) N for some Nijenhuis operator N . We illustrate our integration result in various examples.

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Nijnehuis Geometry III: gl-regular Nijenhuis operators

Cited by 5 publications

References 17 publications

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

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

Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type

Integrating Nijenhuis Structures

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