dorsal/arxiv
View SchemaBoundary values as Hamiltonian variables. II. Graded Structures
| Authors | Vladimir O. Soloviev |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9501017 |
| URL | https://arxiv.org/abs/q-alg/9501017 |
Abstract
It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect to divergences. Bilinear operations, such as action of vector fields onto functionals, commutator of vector fields, interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator is proposed and skew-adjoint operators are constructed with the help of boundary terms. Fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis bracket for Poisson bivector with itself. The simple procedure for testing this condition proposed by P. Olver is applicable with a minimal modification. It is demonstrated that the second structure of the Korteweg--de Vries equation is not Hamiltonian with respect to the new brackets.
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"abstract": "It is shown that the new Poisson brackets proposed in Part I of this work (J.\nMath. Phys. 34, 5747(hep-th/9305133)) arise naturally in an extension of the\nformal variational calculus incorporating divergences. The linear spaces of\nlocal functionals, evolutionary vector fields, functional forms, multi-vectors\nand differential operators become graded with respect to divergences. Bilinear\noperations, such as action of vector fields onto functionals, commutator of\nvector fields, interior product of forms and vectors and the Schouten-Nijenhuis\nbracket are compatible with the grading. A definition of the adjoint graded\noperator is proposed and skew-adjoint operators are constructed with the help\nof boundary terms. Fulfilment of the Jacobi identity for the new Poisson\nbrackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis\nbracket for Poisson bivector with itself. The simple procedure for testing this\ncondition proposed by P. Olver is applicable with a minimal modification. It is\ndemonstrated that the second structure of the Korteweg--de Vries equation is\nnot Hamiltonian with respect to the new brackets.",
"arxiv_id": "q-alg/9501017",
"authors": [
"Vladimir O. Soloviev"
],
"categories": [
"q-alg",
"dg-ga",
"hep-th",
"math.DG",
"math.QA"
],
"title": "Boundary values as Hamiltonian variables. II. Graded Structures",
"url": "https://arxiv.org/abs/q-alg/9501017"
},
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