The symmetric invariants of centralizers and Slodowy grading

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• 作者：Jean-Yves Charbonnel ; Anne Moreau
• 关键词：Symmetric invariant ; Centralizer ; Polynomial algebra ; Slodowy grading ; 17B35 ; 17B20 ; 13A50 ; 14L24
• 刊名：Mathematische Zeitschrift
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：282
• 期：1-2
• 页码：273-339
• 全文大小：1,125 KB
• 参考文献：1.Arakawa, T., Premet, A.: Quantization of Fomenko-Mishchenko algebras via affine W-algebras (preprint)
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• 作者单位：Jean-Yves Charbonnel (1)
  Anne Moreau (2)
  
  1. Groupes, Représentations et géométrie, Institut de Mathématiques de Jussieu - Paris Rive Gauche, UMR 7586, Université Paris Diderot - CNRS, Bâtiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France
  2. Laboratoire de Mathématiques et Applications de Poitiers (LMA), Boulevard Marie et Pierre Curie, Téléport 2, BP 30179, 86962, Futuroscope Chasseneuil Cedex, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

Let \(\mathfrak {g}\) be a finite-dimensional simple Lie algebra of rank \(\ell \) over an algebraically closed field \(\Bbbk \) of characteristic zero, and let e be a nilpotent element of \(\mathfrak {g}\). Denote by \(\mathfrak {g}^{e}\) the centralizer of e in \(\mathfrak {g}\) and by \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) the algebra of symmetric invariants of \(\mathfrak {g}^{e}\). We say that e is good if the nullvariety of some \(\ell \) homogenous elements of \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) in \(({\mathfrak g}^{e})^{*}\) has codimension \(\ell \). If e is good then \( \mathrm{S}({\mathfrak g}^{e}) ^{{\mathfrak g}^{e}} \) is a polynomial algebra. The main result of this paper stipulates that if for some homogenous generators of \( \mathrm{S}({\mathfrak g}) ^{{\mathfrak g}} \), the initial homogenous components of their restrictions to \(e+\mathfrak {g}^{f}\) are algebraically independent, with (e, h, f) an \(\mathfrak {sl}_2\)-triple of \(\mathfrak {g}\), then e is good. As applications, we pursue the investigations of Panyushev et al. (J. Algebra 313:343–391, 2007) and we produce (new) examples of nilpotent elements that satisfy the above polynomiality condition, in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type \(\mathbf{D}_{7}\). Keywords Symmetric invariant Centralizer Polynomial algebra Slodowy grading