## Natural transformations of symplectic structures into Poisson’s and Jacobi’s brackets.(English)Zbl 0966.53052

The Poisson bracket $$\{f,g\}$$ of a pair of smooth functions $$f$$ and $$g$$ on a symplectic manifold $$(M,\omega)$$ is a mapping $$A_M(\omega):C^\infty M\times C^\infty M\to C^\infty M$$. Denote by $$S(M)$$ the set of all symplectic structures on $$M$$ and by $$P(M)$$ the set of all Poisson brackets on $$M$$. Then the family of maps $$A_M:S(M)\to P(M)$$ can be interpreted as a natural transformation of symplectic structures into Poisson brackets. It is proved that all natural transformations of such a type are of the form $$A_M(\omega)(f,g)=c\cdot \{f,g\}$$, where $$c\in \mathbb R$$ and $$\{,\}$$ is the standard Poisson bracket on $$(M,\omega)$$. The same is proved for natural transformations of symplectic structures into Jacobi brackets.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids
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