Deformation quantization of symplectic vector fields

In this paper, we study deformation quantization of symplectic vector fields à la Fedosov. We provide an explicit construction to quantize each symplectic vector field as a derivation of the deformed star algebra. This recovers a result of Müller-Bahns and Neumaier with an explicit formula of higher total degree terms. Moreover, we show that this quantization yields a non-abelian $2$-cocycle on the Lie algebra of symplectic vector fields with values in the deformed star algebra. For a Fedosov deformation quantization with vanishing first characteristic class, this non-abelian $2$-cocycle can provide a formal deformation of the crossed-product algebra generated by the action of symplectic vector fields. In this sense, we can quantize any Lie algebra action by symplectic vector fields.