Some Remarks on {$\mathfrak{g}$}-invariant Fedosov Star Products and Quantum Momentum Mappings

In these notes we consider the usual Fedosov star product on a symplectic manifold $(M,\omega )$ emanating from the fibrewise Weyl product $\circ$, a symplectic torsion free connection $\mathrm{\nabla }$ on M, a formal series $\Omega \in \nu Z^2_{\rm\tiny dR}(M)[[\nu]]$ 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 $(\mathrm{\nabla },\mathrm{\Omega },s)$ determining the star product * on which the Lie derivative ${\text{Lie}}_X$ with respect to X is a derivation of *. Moreover, we also give additional conditions on which ${\text{Lie}}_X$ is even a quasi-inner derivation. Using these results we find necessary and sufficient criteria for a Fedosov star product to be $\mathfrak{g}$-invariant and to admit a quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present a cohomological condition on $\mathrm{\Omega }$ 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.