W- algebras and Duflo Isomorphism

TL;DR

This paper explores the structure of W-algebras via deformation quantization, establishing isomorphisms with reduction algebras and analyzing their polynomial nature and generators.

Contribution

It demonstrates an isomorphism between certain reduction algebras and W-algebras, and analyzes their algebraic properties in the context of semi-simple Lie algebras.

Findings

Operators in the star product are weight homogeneous for weight homogeneous Poisson structures.

An isomorphism between the reduction algebra and the W-algebra is established.

The W-algebra quotient is shown to be polynomial, and generators are computed as deformations.

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