Skip to main content
SpringerLink
Log in

Platonic solids, Kleinian singularities, and Lie groups

  • Conference paper
  • First Online:
Algebraic Geometry

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1008))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 29.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 39.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A'Campo, N.: Le groupe de monodromie du deploiement des singularités isoleés de courbes planes I; Math. Ann. 213, 1–32 (1975), II, Proc. Int. Cong. Math. Vancouver, Vol I, 395–404 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnol'd, V.I.: On matrices depending on parameters; Russian Math. Surveys 26, 29–43 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnol'd, V.I.: Normal forms for functions near degenerate critical points, the Weyl groups of Ak, Dk, Ek and Lagrangian singularities; Functional Anal. Appl. 6, 254–272 (1972)

    Article  Google Scholar 

  4. Arnol'd, V.I.: Critical points of smooth functions; Proc. Int. Cong. Math. Vancouver, Vol I, 19–39 (1974)

    Google Scholar 

  5. Arnol'd, V.I.: Critical points of functions on a manifold with boundary, the simple Lie groups Bk, Ck, F4 and singularities of evolutes; Russian Math. Surveys 33, 5, 99–116 (1978)

    Article  MATH  Google Scholar 

  6. Artin, M.: On isolated rational singularities of surfaces; Amer. J. Math. 88, 129–136 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  7. Artin, M.: Lectures on deformations of singularities; Tata Institute, Bombay, 1976

    Google Scholar 

  8. Bernstein, I.N., Gel'fand, I.M., Gel'fand, S.I.: Schubert cells and the cohomology of the spaces G/P; Russian Math. Surveys 28, 3, 1–26 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Borel, A.: Linear Algebraic Groups; Benjamin, New York 1969

    MATH  Google Scholar 

  10. Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten; Inventiones math. 2, 1–14 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. Brieskorn, E.: Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann. 166, 76–102 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brieskorn, E.: Die Auflösung der rationalen Singularitäten holomorpher Abbildungen; Math. Ann. 178, 255–270 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brieskorn, E.: Singular elements of semisimple algebraic groups; Actes Cong. Int. Math. Nice 1970, t.2, 279–284

    Google Scholar 

  14. Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen; Manuscripta math. 2, 103–161 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brieskorn, E.: The Development of Geometry and Topology, Notes of introductory lectures given at the University of La Habana, 1973; Mat. z. Berufspraxis Math. 17, 109–203 (1976)

    Google Scholar 

  16. Cayley, A.: A memoir on cubic surfaces; Phil. Trans. Roy. Soc. London 159, 231–326 (1869)

    Article  MATH  Google Scholar 

  17. Coxeter, H.S.M., Moser, W.O.J.: Generators and Relations for Discrete Groups, 3rd ed.; Springer, Berlin-Heidelberg-New York, 1975

    MATH  Google Scholar 

  18. Demazure, M.: Désingularisation des variétés de Schubert generalisées, Ann. Sc. l'E.N.S. 7, 53–88 (1974)

    MathSciNet  MATH  Google Scholar 

  19. Drucker, D., Frohardt, D.: Irreducible root systems and finite linear groups of degree two; Bull. London Math. Soc. 14 (2), 142–148 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Durfee, A.H.: Fifteen characterizations of rational double points and simple critical points; L'Enseignement mathematique, T. XXV 1–2, 131–163 (1979)

    MathSciNet  MATH  Google Scholar 

  21. Du Val, P.: On isolated singularities of surfaces that do not affect the conditions of adjunction I, II, III; Proc. Cambridge Phil. Soc. 30, 483–491 (1934)

    Article  MATH  Google Scholar 

  22. Gabriel, P.: Unzerlegbare Darstellungen I, Manuscripta math. 6, 71–103 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gonzalez-Sprinberg, G., Verdier, J.L.: Points doubles rationnels et représentations de groupes; C.R. Acad. Sc. Paris 293, 111–113 (1981)

    MathSciNet  MATH  Google Scholar 

  24. Gonzalez-Sprinberg, G., Verdier, J.L.: Construction géométrique de la correspondence de McKay; Ann. Sc. E.N.S., to appear

    Google Scholar 

  25. Grauert, H.: Über die Deformationen isolierter Singularitäten analytischer Mengen; Inventiones math. 15, 171–198 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Happel, D., Preiser, U., Ringel, C.M.: Binary polyhedral groups and Euclidean diagrams; Wanuscripta math. 31, 317–329 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hirzebruch, F.: The topology of normal singularities of an algebraic surface; Sém. Bourbaki No. 250, 1962–63

    Google Scholar 

  28. Humphreys: Linear Algebraic Groups, Springer Verlag, Berlin-Heidelberg-New York, 1975

    Book  MATH  Google Scholar 

  29. Husein-Zade: Intersection matrices for certain singularities of functions of two variables; Funct. Anal. Appl. 8, 11–15 (1974)

    MathSciNet  Google Scholar 

  30. Kac, V.G.: Infinite root systems, representations of graphs, and invariant theory; Inventiones math. 56, 57–92 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kas, A., Schlessinger, M.: On the versal deformation of a complex space with an isolated singularity; Math. Ann. 196, 23–29 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kepler, J.: Mysterium Cosmographicum; Tübingen 1596

    Google Scholar 

  33. Kirby, D.: The structure of an isolated multiple point of a surface I, II, III; Proc. London Math. Soc. (3) 6, 597–609 (1956), 7, 1–28 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  34. Klein, F.: Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade; Teubner, Leipzig 1884

    MATH  Google Scholar 

  35. Knörrer, H.: Group representations and resolution of rational double points, preprint Bonn University

    Google Scholar 

  36. Kodaira, K.: On compact complex analytic surfaces II; Annals of Math. 77, 563–626 (1963)

    Article  MATH  Google Scholar 

  37. Kostant, B.: Lie group representations on polynomial rings; Amer. J. Math. 85, 327–404 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  38. Lipman, J.: Rational singularities, with applications to algebraic surfaces and unique factorization; Publ. Math. IHES 36, 195–279 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  39. Looijenga, E.: On the semi-universal deformation of a simple elliptic singularity II, Topology 17, 23–40 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  40. Looijenga, E.: Rational surfaces with an anticanonical cycle; Annals of Math. 114, 267–322 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mather, J.: Stability of C∞-mappings I–VI, Annals of Math. 87, 89–104 (1968), 89, 254–291 (1969), Publ. Math. IHES 35, 127–156 (1968), 37, 223–248 (1969), Advances in Math. 4, 301–336 (1970), Lecture Notes in Math. 192, 207–253 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  42. McKay, J.: Graphs, singularities, and finite groups; Proc. Symp. Pure Math. 37, 183–186 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  43. Milnor, J.: Singular points of complex hypersurfaces, Annals of Math. Study 61, 1968

    Google Scholar 

  44. Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity; Publ. Math. IHES 9, 5–22 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  45. Pham, F.: Formules de Picard-Lefschetz généralisées et ramification des integrales; Bull. Soc. Math. France 93, 333–367 (1965)

    MathSciNet  MATH  Google Scholar 

  46. Pinkham, H.: Singularités rationelles des surfaces, Séminaire sur les singularités des surfaces, Lecture Notes in Math. 777, 147–178 (1980)

    Article  MathSciNet  Google Scholar 

  47. Plato: Timaios

    Google Scholar 

  48. Schläfli, L.: On the distribution of surfaces of the third order into species; Phil. Trans. Roy. Soc. London 153, 193–241 (1863)

    Article  Google Scholar 

  49. Schwarz, H.A.: Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Funktion ihres vierten Elementes darstellt; J. Reine Angew. Math. 75, 292–335 (1873)

    Article  MathSciNet  Google Scholar 

  50. Sekiguchi, J., Shimizu, Y.: Simple singularities and infinitesimally symmetric spaces, Proc. Japan Acad. 57(A), 42–46 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  51. Sekiguchi, J.: The nilpotent subvariety of the vector space associated to a symmetric pair; preprint, Tokyo Metropolitan University, 1983

    Google Scholar 

  52. Serre, J.P.: Représentations lineaires des groupes finis, Hermann, Paris 1967

    MATH  Google Scholar 

  53. Slodowy, P.: Simple singularities and simple algebraic groups; Lecture Notes in Math. 815, 1980

    Google Scholar 

  54. Slodowy, P.: Four lectures on simple groups and singularities; Communications of the Mathematical Institute 11, Rijksuniversiteit Utrecht, 1980

    Google Scholar 

  55. Slodowy, P.: Chevalley groups over ℂ((t)) and deformations of simply elliptic singularities. RIMS Kokyuroku 415, 19–38 (1981), Kyoto University

    Google Scholar 

  56. Slodowy, P.: Sur les groupes finis attachés aux singularités simples, Séminaire Lê, Université de Paris VII, 26 mars '81

    Google Scholar 

  57. Slodowy, P.: A character approach to Looijenga's invariant theory for generalized root systems, preprint, Bonn University 1982

    Google Scholar 

  58. Springer, T.A.: Invariant theory; Lecture Notes in Math. 585, 1977

    Google Scholar 

  59. Steinberg, R.: Conjugacy classes in algebraic groups; Lecture Notes in Math 366, 1974

    Google Scholar 

  60. Steinberg, R.: Kleinian singularities and unipotent elements; Proc. Symp. Pure Math. 37, 265–270 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  61. Steinberg, R.: Subgroups of SU2 and Dynkin diagrams; Preprint UCLA, 1981

    Google Scholar 

  62. Thom, R.: Les singularités des applications differentiables, Ann. Inst. Fourier 6, 43–87 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  63. Thom, R.: Stabilité Structurelle et Morphogénèse, Benjamin, Reading Mass., 1972

    MATH  Google Scholar 

  64. Tjurina, G.N.: Locally semiuniversal flat deformations of isolated singularities of complex spaces, Math. USSR Izvestija 3(5), 967–999 (1969)

    Article  MathSciNet  Google Scholar 

  65. Tjurina, G.N.: Resolutions of singularities of flat deformations of rational double points, Funct. Anal. Appl. 4(1), 68–73 (1970)

    Article  MathSciNet  Google Scholar 

  66. Waterhouse, W.C.: The discovery of the regular solids; Arch. Hist. Ex. Sci. 9, 212–221 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  67. Weyl, H.: Symmetry; Princeton Univ. Press, Princeton N.J. 1952

    Book  MATH  Google Scholar 

  68. Whitney, H.: On singularities of mappings of Euclidean spaces I, Ann. of Math. 62, 374–410 (1955).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, Wegelerstraße 10, D-5300, Bonn, W. Germany

    P. Slodowy

Authors
  1. P. Slodowy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

I. Dolgachev

Rights and permissions

Reprints and permissions

Copyright information

© 1983 Springer-Verlag

About this paper

Cite this paper

Slodowy, P. (1983). Platonic solids, Kleinian singularities, and Lie groups. In: Dolgachev, I. (eds) Algebraic Geometry. Lecture Notes in Mathematics, vol 1008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0065703

Download citation

  • DOI: https://doi.org/10.1007/BFb0065703

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12337-8

  • Online ISBN: 978-3-540-40971-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation