W- algebras and Duflo Isomorphism

Abstract

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 $(S(\mathfrak{g})/S(\mathfrak{g}){\mathfrak{m}}_{\chi }{)}^{\mathfrak{m}}$.

expanded version, 19 pages, 3 figures

References in corpus (2)

• Biquantization of symmetric pairs and the quantum shift
• Equivalence theorems and an explicit formula for the biquantization character