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Quasihomogene affine Varietäten für SL(2,ℂ)

Part of the Lecture Notes in Mathematics book series (LNM,volume 1146)

Abstract

This paper contains some contributions to a refined study of linear actions of the group G=Sl(2,ℂ) on a vectorspace V. More precisely, it deals with the study of the sophisticated geometry of orbitclosures. The case where V=Rn is the irreducible G-module of binary n-forms is of particular importance, since any finite-dimensional G-module is a direct sum of such irreducible G-modules. Hadžiev 1966 [Ha1] started investigating orbitclosures of binary n-forms according to their dimension and their decomposition into G-orbits. His results were generalized to orbitclosures in arbitrary finite-dimensional G-modules by Popov 1973 [P1] and [P2], who also began the classification of G-orbitclosures up to G-isomorphism. An intrinsic definition of orbitclosures required for such a classification is the following: An algebraic variety with a regular action of the group G, briefly a G-variety, is called quasihomogeneous if it contains a dense orbit. Popov 1973 [P1] classified all normal affine quasihomogeneous G-varieties. In addition, Luna and Vust 1983 [LV] classified the normal (not necessarily affine) quasihomogeneous G-varieties. These classifications use discrete parameters, which even in the most complicated affine case are rather simple, namely a natural and a rational number. This paper deals with the investigation of non-normal affine quasi-homogeneous G-varieties. In particular, numerical invariants for G-isomorphism are obtained, which are of an essentially higher complexity than in the normal case. Moreover, it is shown that the classification involves continuous parameters and modulispaces of arbitrary high dimension can occur.

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Literaturverzeichnis

  1. Bartels, D., On nonnormality of affine quasihomogeneous Sl(2,ℂ)-varieties, Séminaire d'Algèbre P. Dubreil et M.P. Malliavin, Lect. Notes in Math. 924, Springer Verlag (1982), 384–399.

    Google Scholar 

  2. Birkes, D., Orbits of linear algebraic groups, Ann. of Math. 93 (1971), 459–475.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Grace, J.H., Young, A., The algebra of invariants, Cambridge Univ. Press, (1903).

    Google Scholar 

  4. Grosshans, F., Observable groups and Hilbert's fourteenth problem, Amer. J. Math. 95 (1973), 229–253.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Hadžiev, D., Description of the closed orbits and closures of nonclosed orbits in irreducible representations of the Lorentz group, Dokl. Akad. Nauk Uzbek SSR 12 (1966), 3–6.

    MathSciNet  Google Scholar 

  6. Hadžiev, D., Some questions in the theory of vectorinvariants, Math. USSR Sbornik 1 (1967), 383–396.

    CrossRef  MATH  Google Scholar 

  7. Hilbert, D., Über die vollen Invariantensysteme, Math. Ann. 42 (1893), 313–373.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Humphreys, J.E., Linear algebraic groups, Springer GTM 21 (1975).

    Google Scholar 

  9. Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer GTM 9 (1972).

    Google Scholar 

  10. Jordan, C., Memoire sur les covariants des formes binaires, Journal de Math. 2 (3) (1876), 177–233 und 5 (3) (1879), 345–378.

    MATH  Google Scholar 

  11. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Toroidal embeddings I, Lect. Notes in Math. 339 (1973).

    Google Scholar 

  12. Kraft, H., Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, Vieweg Verlag (1984).

    Google Scholar 

  13. Luna, D., Vust, Th., Plongements d'espace homogène, Comm. Math. Helvetici 58 (1983), 186–245.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Mumford, D., Geometric invariant theory, Erg. der Math. 34, Springer Verlag (1970).

    Google Scholar 

  15. Popov, V.L., Quasihomogeneous affine algebraic varieties of the group Sl(2), Math. USSR Izv. 7 no 4 (1973), 793–831.

    CrossRef  MATH  Google Scholar 

  16. Popov, V.L., Structure of the closure of orbits in spaces of finite-dimensional linear Sl(2)-representations, Math. Notes 16 no 6 (1974), 1159–1162.

    CrossRef  MATH  Google Scholar 

  17. Schur, I., Vorlesung über Invariantentheorie, Grundl. Math. Wiss. 143, Springer Verlag (1968).

    Google Scholar 

  18. Springer, T.A., Invariant theory, Lect. Notes in Math. 585, Springer Verlag (1977).

    Google Scholar 

  19. Vust, Th., Sur la theorie des invariants des groupes classique, Ann. Inst. Fourier 26 2 (1976), 1–31.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1985 Springer-Verlag

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Bartels, D. (1985). Quasihomogene affine Varietäten für SL(2,ℂ). In: Malliavin, MP. (eds) Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin. Lecture Notes in Mathematics, vol 1146. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074535

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  • DOI: https://doi.org/10.1007/BFb0074535

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  • Print ISBN: 978-3-540-15686-4

  • Online ISBN: 978-3-540-39628-4

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