# Poisson brackets of the exterior forms

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Let $(M,\omega)$ be a symplectic manifold. Poisson brackets of the exterior forms are defined by the following formulas:

$$\{ \alpha, \{ \beta , \gamma \} \}= \{ \{ \alpha, \beta \},\gamma \}+ \{ \beta ,\{ \alpha , \gamma \} \}$$

$$\{ \alpha ,\beta \}= (-1)^{deg(\alpha) deg(\beta )+1} \{ \beta , \alpha \}$$

$$\{ f \alpha,\beta \}= \alpha \wedge \nabla_{X_f} \beta + f \{ \alpha, \beta \}$$

with $\alpha,\beta,\gamma \in \Lambda^* (TM)$ and $f \in {\cal C}^{\infty}(M)$, $X_f=(df)^*$. $\nabla$ is a symplectic connection.

Can we quantize the structure?

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