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The Symmetric Invariants of Centralizers and Slodowy Grading II


Let 𝔤 be a finite-dimensional simple Lie algebra of rank over an algebraically closed field 𝕜 of characteristic zero. We identify 𝔤 with 𝔤 through the Killing form of 𝔤. Let (e, h, f) be an 𝔰𝔩2-triple of 𝔤. Denote by 𝔤e the centralizer of e in 𝔤 and by \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) the algebra of symmetric invariants of 𝔤e. We say that e is good if the nullvariety of some homogeneous elements of \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) in (𝔤e) has codimension . In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then \(\mathrm {S}(\mathfrak {g}^{e})^{\mathfrak {g}^{e}}\) is a polynomial algebra. In this paper, we prove that the converse of the main result of Charbonnel and Moreau (Math. Zeitsch. 282, n° 1-2, 273–339 2016) is true. Namely, we prove that e is good if and only if for some homogeneous generating sequence q 1, … , q l of S(𝔤)𝔤, the initial homogeneous components of their restrictions to e + 𝔤f are algebraically independent over 𝕜.

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The second author is partially supported by the ANR Project GeoLie Grant number ANR-15-CE40-0012. We thank the referee for valuable comments and suggestions.

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Correspondence to Anne Moreau.

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Presented by Vyjayanthi Chari.

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Charbonnel, JY., Moreau, A. The Symmetric Invariants of Centralizers and Slodowy Grading II. Algebr Represent Theor 20, 1341–1363 (2017).

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  • Symmetric invariant
  • Centralizer
  • Polynomial algebra
  • Slodowy grading

Mathematics Subject Classification (2010)

  • 17B35
  • 17B20
  • 13A50
  • 14L24