Let (M,ω) be a symplectic manifold and (E,∇) a fiber bundle with connection. Let W be a 2-form with endomorphisms. I define Poisson brackets of endomorphisms:
{e,f}=W(∇e,∇f)
2W(X∗⊗e,∇f)=e.∇Xf+∇Xf.e
With e,f two endomorphisms and X a tangent vector.
Are the Poisson brackets well defined?