W- algebras and Duflo Isomorphism

2012/10/29 by Panagiotis Batakidis, Batakidis, Panagiotis, Nikolaos Papalexiou +1
Mathematics · #17B20 #53D55 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:17B20 #msc:53D55

paper · pdf · doi:10.48550/arxiv.1210.7759

expanded version, 19 pages, 3 figures

arxiv created 2016/07/27 · arxiv updated 2017/02/14

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. We then consider the linear Poisson case X=\mathfrakg^∗ for a semi simple Lie algebra \mathfrakg. As an application we provide an isomorphism between the Cattaneo-Felder-Torossian reduction algebra H⁰(\mathfrakg,\mathfrakm,χ) and the W- algebra (U(\mathfrakg)/U(\mathfrakg)\mathfrakm_χ)^\mathfrakm. We also show that in the W- algebra setting, (S(\mathfrakg)/S(\mathfrakg)\mathfrakm_χ)^\mathfrakm is polynomial. Finally, we compute generators of H⁰(\mathfrakg,\mathfrakm,χ) as a deformation of (S(\mathfrakg)/S(\mathfrakg)\mathfrakm_χ)^\mathfrakm.

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