The Symmetric Invariants of Centralizers and Slodowy Grading II
The Symmetric Invariants of Centralizers and Slodowy Grading II
复制标题
扶正器的对称不变量和缓慢分级 II
DOI：
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发表时间：
2016
影响因子：
0.6
通讯作者：
Anne Moreau
中科院分区：
文献类型：
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作者：
Jean;Anne Moreau
Let ? be a finite-dimensional simple Lie algebra of rank ℓ over an algebraically closed field ? of characteristic zero. We identify ? with ?∗ through the Killing form of ?. Let (e, h, f) be an ??2-triple of ?. Denote by ?e the centralizer of e in ? and by S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} the algebra of symmetric invariants of ?e. We say that e is good if the nullvariety of some ℓ homogeneous elements of S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} in (?e)∗ has codimension ℓ. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} 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 q1, … , ql of S(?)?, the initial homogeneous components of their restrictions to e + ?f are algebraically independent over ?.
Let ? be a finite-dimensional simple Lie algebra of rank ℓ over an algebraically closed field ? of characteristic zero. We identify ? with ?∗ through the Killing form of ?. Let (e, h, f) be an ??2-triple of ?. Denote by ?e the centralizer of e in ? and by S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} the algebra of symmetric invariants of ?e. We say that e is good if the nullvariety of some ℓ homogeneous elements of S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} in (?e)∗ has codimension ℓ. In our previous work (Charbonnel and Moreau. Math. Zeitsch. 282, n° 1-2, 273–339 2016), we showed that if e is good then S(?e)?edocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$mathrm {S}(mathfrak {g}^{e})^{mathfrak {g}^{e}}$end{document} 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 q1, … , ql of S(?)?, the initial homogeneous components of their restrictions to e + ?f are algebraically independent over ?.