Computing with Hamiltonian operators

Vitolo, Raffaele

doi:10.1016/j.cpc.2019.05.012

Cited by 14 publications

(17 citation statements)

References 58 publications

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“…The calculations have been done by means of computer algebra systems. In particular, Schouten brackets involving nonlocal operators have been calculated by the Reduce package CDE, and checked by the Maple package jacobi.mpl; both packages are described in [14,67] (see also [47]). Further calculations have been done in Reduce (finding first-order nonlocal operators when N = 3, and finding third-order operators in the cases N = 3, N = 4) and in Maple, also using the package Jets [8] (finding third-order operators in the case N = 5).…”

Section: Jhep08(2021)129mentioning

confidence: 99%

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WDVV equations and invariant bi-Hamiltonian formalism

Vašíček

Vitolo

2021

J. High Energ. Phys.

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The purpose of the paper is to show that, in low dimensions, the WDVV equations are bi-Hamiltonian. The invariance of the bi-Hamiltonian formalism is proved for N = 3. More examples in higher dimensions show that the result might hold in general. The invariance group of the bi-Hamiltonian pairs that we find for WDVV equations is the group of projective transformations. The significance of projective invariance of WDVV equations is discussed in detail. The computer algebra programs that were used for calculations throughout the paper are provided in a GitHub repository.

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Section: Jhep08(2021)129mentioning

confidence: 99%

“…We will omit the corresponding operator A 1 2 , as it can be easily reconstructed. The Schouten bracket [A 1 1 , A 1 2 ] turns out to be zero and thus the above operators are compatible; the computation has been performed by means of the Reduce package CDE, see [47,67].…”

Section: Jhep08(2021)129mentioning

confidence: 99%

WDVV equations and invariant bi-Hamiltonian formalism

Vašíček

Vitolo

2021

J. High Energ. Phys.

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“…However this problem can be easily fixed and the implementation of the main result on a computer algebra program seems possible. For instance, the Reduce package [17,23] already allows to use local and nonlocal variables and contains an implementation of the Schouten bracket for local operators in terms of odd variables.…”

Section: Discussionmentioning

confidence: 99%

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

Lorenzoni

Vitolo

2020

Journal of Geometry and Physics

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Poisson brackets between conserved quantities are a fundamental tool in the theory of integrable systems. The subclass of weakly nonlocal Poisson brackets occurs in many significant integrable systems. Proving that a weakly nonlocal differential operator defines a Poisson bracket can be challenging. We propose a computational approach to this problem through the identification of such operators with superfunctions on supermanifolds.

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“…The construction below is a special case of the general approach suggested in [21]. For n = 1, the construction can be found in [20], see also [10,30,36].…”

Section: Mathematical Setupmentioning

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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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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Computing with Hamiltonian operators

Cited by 14 publications

References 58 publications

WDVV equations and invariant bi-Hamiltonian formalism

WDVV equations and invariant bi-Hamiltonian formalism

Weakly nonlocal Poisson brackets, Schouten brackets and supermanifolds

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

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