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W-algebras and Duflo isomorphism

P. Batakidis

Department of Mathematics, Penn State University, McAllister Building, State College, PA 16802, USA

E-mail Address: batakidis@psu.edu

E-mail Address: batakidis@gmail.com

Corresponding author.

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N. Papalexiou

Department of Mathematics, University of the Aegean, Karlovasi 83200, Samos, Greece

E-mail Address: papalexi@aegean.gr

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https://doi.org/10.1142/S021949881850041XCited by:1 (Source: Crossref)

Abstract

We prove that when Kontsevich’s deformation quantization is applied on weight homogeneous Poisson structures, the operators in the ∗-product formula are weight homogeneous. In the linear Poisson case for a semisimple Lie algebra $𝔤$ the Poisson manifold $X$ is $𝔤^{*}$ . As an application we provide an isomorphism between the Cattaneo–Felder–Torossian reduction algebra $H^{0} (𝔪^{⊥}, χ, 𝔮)$ and the $W$ -algebra ${(U (𝔤) / U (𝔤) 𝔪_{χ})}^{𝔪}$ . We also show that in the $W$ -algebra setting, ${(S (𝔤) / S (𝔤) 𝔪_{χ})}^{𝔪}$ is polynomial. Finally, we compute generators of $H^{0} (𝔪^{⊥}, χ, 𝔮)$ as a deformation of ${(S (𝔤) / S (𝔤) 𝔪_{χ})}^{𝔪}$ .

Communicated by I. M. Musson

Keywords:

AMSC: 53D55, 20G42, 17B35, 17B20

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