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

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

doi:10.48550/arxiv.2009.07802

Cited by 3 publications

(9 citation statements)

References 24 publications

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“…(B1) A generic in a certain sense maximal pencil corresponds to the well-known multi-Poisson structure discovered by M. Antonowitz and A. Fordy in [1] and studied by E. Ferapontov and M. Pavlov [17], see also [2,3,9]. We refer to it as to Antonowitz-Fordy-Frobenius pencil, AFF-pencil.…”

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

confidence: 91%

“…, g n are flat and pairwise compatible, so that they generate an n + 1-dimensional flat pencil with remarkable properties, see e.g. [17,9]. We can write this pencil as {P (L)g 0 }, where P (•) is an arbitrary polynomial of degree ≤ n (21) and L and g 0 are given by (19).…”

Section: Aff-pencilmentioning

confidence: 99%

“…The entries of the connections Γ and Γ of the diagonal metrics g ij and ḡij := gL −1 were calculated many times in the literature, see e.g. [9,Lemma 7.1], and are given by the following formulas:…”

Section: General Form Of the Metric In Diagonal Coordinatesmentioning

confidence: 99%

“…, then g = P (L)g LC has constant curvature K = − 1 4 a n+1 (see Fact 8). All together, the metrics (77) form a pencil of compatible constant curvature metrics, which can be thought of as one-dimensional extension of the AFF-pencil (26), see details in [17], [9]. In Frobenuis coordinates u 1 , .…”

Section: On the Uniqueness Of Frobenius Coordinates For A Pair Of Met...mentioning

confidence: 99%

“…. , j k such that i = next(j s ), then we first apply the flattened direct product operation to the pro-Frobenius pencils P js and then take the warped product with the extended AFF pencil F i : At the very end, when we come to the root of the whole tree, say i = 1, and obtain a pro-Frobenuis pencil P 1 , we need to perform one more operation, namely, making it flat 9 If P is an extended Frobenius pencil in dimension n, then in terms of polynomials λ = P and λ = P , this relation takes the form a 1 = ±â n+1 , cf. the second part of Condition (iii) from Section 4.1.…”

Section: Algebraic Interpretation Of Warped Productmentioning

confidence: 99%

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

Bolsinov¹,

Yu.²,

Matveev³

2021

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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: 91%

Section: Aff-pencilmentioning

confidence: 99%

Section: General Form Of the Metric In Diagonal Coordinatesmentioning

confidence: 99%

Section: On the Uniqueness Of Frobenius Coordinates For A Pair Of Met...mentioning

confidence: 99%

Section: Algebraic Interpretation Of Warped Productmentioning

confidence: 99%

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

Bolsinov¹,

Yu.²,

Matveev³

2021

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Nijenhuis geometry III: gl-regular Nijenhuis operators

2024

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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 system in which the operator takes first or second companion form, and give a local description of such operators. We apply this local description to study singular points. In particular, we obtain normal forms of gl-regular Nijenhuis operators near singular points in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces.This paper is an important step in the research programme suggested in [5,10]. Basic definitions and main resultsGiven a (1, 1) tensor field L on a manifold M n , one defines the Nijenhuis torsion of L aswhere ξ, η are arbitrary vector fields. If N L identically vanishes, then L is said to be a Nijenhuis operator.Nijenhuis geometry studies Nijenhuis operators and their properties, both local and global. A research programme and general strategy for studying such operators were suggested in [5]. This paper is devoted to the next item of our agenda (after [5], [6], [8], [16]) and is focused on Nijenhuis operators satisfying gl-regularity condition.We start with the following equivalent definitions of gl-regular operators L : R n → R n , see e.g. Wikipedia [27,28] (the same notation L will be used for the matrix corresponding to this operator, with appropriate amendments under coordinate transformations if necessary):• L is a regular element of the Lie algebra gl(n, R) in the sense that the adjoint orbit O(L) = {P LP −1 | P ∈ GL(n, R)} ⊂ gl(n, R) has maximal dimension.

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Integrating Nijenhuis Structures

Pugliese¹,

Sparano²,

Vitagliano³

2022

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

Cited by 3 publications

References 24 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

Nijenhuis geometry III: gl-regular Nijenhuis operators

Integrating Nijenhuis Structures

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