W- algebras and Duflo Isomorphism

We prove that when Kontsevich's deformation quantization is applied on weight homogeneous Poisson structures, the operators in the $\ast-$ product formula are weight homogeneous. We then consider the linear Poisson case $X=\mathfrak{g}^\ast$ for a semi simple Lie algebra $\mathfrak{g}$. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra $H^0(\mathfrak{g},\mathfrak{m},\chi)$ and the $W-$ algebra $(U(\mathfrak{g})/U(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$. We also show that in the $W-$ algebra setting, $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$ is polynomial. Finally, we compute generators of $H^0(\mathfrak{g},\mathfrak{m},\chi)$ as a deformation of $(S(\mathfrak{g})/S(\mathfrak{g})\mathfrak{m}_\chi)^\mathfrak{m}$.