Poisson Bracket,Deformed Bracket and Gauge Group Actions in Kontsevich Deformation Quantization

We express the difference between the Poisson bracket and a deformed bracket for Kontsevich deformation quantization on any Poisson manifold by means of the second derivative of the formality quasi-isomorphism. The counterpart in star products of the action of formal diffeomorphisms on Poisson formal bivector fields is also investigated.