Natural transformations of symplectic structures into Poisson's and Jacobi's brackets (Q2716335)

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.