A characterization of graded symplectic structures. (English) Zbl 0729.58007
The purpose of this work is twofold. First, to give a characterization of graded symplectic structures using an intrinsic representation of graded vector fields, and second, to apply this characterization in order to build new Poisson brackets. The formalism of the theory of graded manifolds, following Kostant, is used in order to find new results when the graded sheaf is the sheaf of differentiable forms on the underlying manifold M.
As an application of of this characterization three kinds of graded symplectic forms are defined and their Poisson bracket studied.
First, for any Riemannian manifold there is an associated odd symplectic form of the graded manifold \((M,\Gamma (\Lambda T^*M)).\)
Second, if in addition we have a symplectic form on M, we obtain a canonical lifting to an even symplectic form, in such a way that the graded Poisson bracket associated to the even symplectic form is an extension of the initial Poisson bracket. When the underlying manifold is of dimension 2, we show a relation between the graded symplectic form and the Gauss curvature. All these constructions are natural.
And third, we show that the Schouten-Nijenhuis bracket is the Poisson bracket of an odd symplectic form.
As an application of of this characterization three kinds of graded symplectic forms are defined and their Poisson bracket studied.
First, for any Riemannian manifold there is an associated odd symplectic form of the graded manifold \((M,\Gamma (\Lambda T^*M)).\)
Second, if in addition we have a symplectic form on M, we obtain a canonical lifting to an even symplectic form, in such a way that the graded Poisson bracket associated to the even symplectic form is an extension of the initial Poisson bracket. When the underlying manifold is of dimension 2, we show a relation between the graded symplectic form and the Gauss curvature. All these constructions are natural.
And third, we show that the Schouten-Nijenhuis bracket is the Poisson bracket of an odd symplectic form.
Reviewer: J.Monterde (Burjasot)
MSC:
| 58A50 | Supermanifolds and graded manifolds |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |
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