In these notes we consider the usual Fedosov star product on a symplectic
manifold (M,ω) emanating from the fibrewise Weyl product ∘, a
symplectic torsion free connection ∇ on M, a formal series Ω∈νZdR2(M)[[ν]] of closed two-forms on M and a certain formal
series s of symmetric contravariant tensor fields on M. For a given symplectic
vector field X on M we derive necessary and sufficient conditions for the
triple (∇,Ω,s) determining the star product * on which the Lie
derivative \Lie_X with respect to X is a derivation of *. Moreover, we also
give additional conditions on which \Lie_X is even a quasi-inner derivation.
Using these results we find necessary and sufficient criteria for a Fedosov
star product to be g-invariant and to admit a quantum Hamiltonian.
Finally, supposing the existence of a quantum Hamiltonian, we present a
cohomological condition on Ω that is equivalent to the existence of a
quantum momentum mapping. In particular, our results show that the existence of
a classical momentum mapping in general does not imply the existence of a
quantum momentum mapping.Comment: 15 pages, one corollary and one definition added to Section 4, typos
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