Classic Groups: Clifford Algebras, Projective Quadrics, and Spin Groups

  • Pierre Anglè
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 50)

Keywords

Unitary Group Clifford Algebra Simple Algebra Spin Representation Quaternion Algebra 
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Bibliography

  1. Albert A., Structures of Algebras, American Mathematical Society, vol XXIV, New York, 1939.Google Scholar
  2. Anglès P., Construction de revêtements du groupe conforme d’un espace vectoriel muni d’une métrique de type (p,q), Annales de l’I.H.P., section A, vol XXXIII no 1, pp. 33–51, 1980.Google Scholar
  3. Anglès P., Géométrie spinorielle conforme orthogonale triviale et groupes de spinorialité conformes, Report HTKK Mat A 195, pp. 1–36, Helsinki University of Technology, 1982.Google Scholar
  4. Anglès P., Construction de revêtements du groupe symplectique réel C Sp(2r, R) Géométrie conforme symplectique réelle. Définition des structures spinorielles conformes symplectiques réelles, Simon Stevin (Gand-Belgique), vol 60 no 1, pp. 57–82, Mars 1986.MATHGoogle Scholar
  5. Anglès P., Algèbres de Clifford Cr,s des espaces quadratiques pseudo-euclidiens standards Er,s et structures correspondantes sur les espaces de spineurs associés. Plongements naturels de quadriques projectives Q(Er,s) associés aux espaces Er,s. Nato ASI Séries vol 183, 79–91, Clifford algebras édité par JSR Chisholm et A.K. Common D. Reidel Publishing Company, 1986.Google Scholar
  6. Anglès P., Real conformal spin structures, Scientiarum Mathematicarum Hungarica, vol 23, pp. 115–139, Budapest, Hongary, 1988.MATHGoogle Scholar
  7. Anglès P. and R. L. Clerc, Operateurs de creation et d’annihilation et algèbres de Clifford, Ann. Fondation Louis de Broglie, vol. 28, no 1, pp. 1–26, 2003.Google Scholar
  8. Artin E., Geometric Algebra, Interscience, 1954; or in French, Algèbre géomètrique, Gauthier Villars, Paris, 1972.Google Scholar
  9. Atiyah M. F., R. Bott, and A. Shapiro, Clifford Modules, Topology, vol 3, pp. 3–38, 1964.CrossRefMathSciNetGoogle Scholar
  10. Berger M., Géométrie Différentielle, Armand Colin, Paris, 1972.MATHGoogle Scholar
  11. Berger M., Géométrie, vol. 1–5, Cedic Nathan, Paris, 1977.Google Scholar
  12. Blaine Lawson H. and M. L. Michelson, Spin Geometry, Princeton University Press, 1989.Google Scholar
  13. Bourbaki N., Algèbre, Chapitre 9: Formes sesquilineaires et quadratiques, Hermann, Paris, 1959.Google Scholar
  14. Bourbaki N., Algèbre, Chapitres 1 à 3, Hermann, Paris, 1970.MATHGoogle Scholar
  15. Bourbaki N., Algèbre de Lie, Chapitres 1 à 3, Hermann, Paris, 1970.Google Scholar
  16. Cartan E., Annales de l’ E.N.S., 31, pp. 263–355, 1914.Google Scholar
  17. Cartan E., Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. de France, 41, pp. 1–53, 1931.Google Scholar
  18. Cartan E., Leçons sur la théorie des spineurs, Hermann, Paris, 1938.Google Scholar
  19. Cartan E., The theory of Spinors, Hermann, Paris, 1966.MATHGoogle Scholar
  20. Chevalley C., Theory of Lie groups, Princeton University Press, 1946.Google Scholar
  21. Chevalley C., The Algebraic theory of Spinors, Columbia University Press,NewYork, 1954.Google Scholar
  22. Crumeyrolle A., Structures spinorielles, Ann. I.H.P., Sect. A, vol. XI, no 1, pp. 19–55, 1964.Google Scholar
  23. Crumeyrolle A., Groupes de spinorialité, Ann. I.H.P., Sect. A, vol. XIV, no 4, pp. 309–323, 1971.MathSciNetGoogle Scholar
  24. Crumeyrolle A., Dérivations, formes, opérateurs usuels sur les champs spinoriels, Ann. I.H.P., Sect. A, vol. XVI, no 3, pp. 171–202, 1972.MathSciNetGoogle Scholar
  25. Crumeyrolle A., Algèbres de Clifford et spineurs, Université Toulouse III, 1974.Google Scholar
  26. Crumeyrolle A., Fibrations spinorielles et twisteurs généralisés, Periodica Math. Hungarica, vol. 6–2, pp. 143–171, 1975.CrossRefMathSciNetGoogle Scholar
  27. Crumeyrolle A., Bilinéarité et géométrie affine attachées aux espaces de spineurs complexes Minkowskiens ou autres, Ann. I.H.P., Sect. A, vol. XXXIV, no 3, p. 351–371, 1981.MathSciNetGoogle Scholar
  28. Deheuvels R., Formes Quadratiques et groupes classiques, Presses Universitaires de France, Paris, 1981.MATHGoogle Scholar
  29. Deheuvels R., Groupes conformes et algèbres de Clifford, Rend. Sem. Mat. Univers. Politech. Torino, vol. 43, 2, p. 205–226, 1985.MathSciNetGoogle Scholar
  30. Deheuvels R., Tenseurs et spineurs, Presses Universitaires de France, Paris, 1993.MATHGoogle Scholar
  31. Deligne P., Kazhdan D., Etingof P., Morgan J.W., Freed D.S., Morrison D.R., Jeffrey L.C., Witten E., Quantum fields and strings: a course for mathematicians, vol. I and vol. II, ed. American Math. Soc., Institute for advanced study, 2000.Google Scholar
  32. Dieudonné J., Les determinants sur un corps non commutatif, Bull. Soc. Math. de France, 71, pp. 27–45, 1943.MATHGoogle Scholar
  33. Dieudonné J., On the automorphisms of the classical groups, Memoirs of Am. Math. Soc., no 2, pp. 1–95, 1951.Google Scholar
  34. Dieudonné J., On the structure of unitary groups, Trans. Am. Math. Soc., 72, 1952.Google Scholar
  35. Dieudonné J., La géomètrie des groupes classiques, Springer Verlag, Berlin, Heidelberg, New York, 1971.MATHGoogle Scholar
  36. Dieudonné J., Sur les groupes classiques, Hermann, Paris, 1973.Google Scholar
  37. Helgason S., Differential Geometry and Symmetric Spaces, Academic Press, New York and London, 1962.MATHGoogle Scholar
  38. Husemoller D., Fibre bundles, McGraw Inc., 1966.Google Scholar
  39. Kahan T., Théorie des groupes en physique classique et quantique, Tome 1, Dunod, Paris, 1960.Google Scholar
  40. Karoubi M., Algèbres de Clifford et K-théorie, Annales Scientifiques de l’E.N.S., série, tome 1, pp. 14–270, 1968.MathSciNetGoogle Scholar
  41. Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, Berlin, 1972.MATHGoogle Scholar
  42. Lam T.Y., The algebraic theory of quadratic forms, W.A. Benjamin Inc., 1973.Google Scholar
  43. LichnerowiczA., Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France, 92, pp. 11–100, 1964.Google Scholar
  44. Lichnerowicz A., Cours du Collège de France, ronéotypé non publié, 1963–1964.Google Scholar
  45. Lounesto P., Spinor valued regular functions in hypercomplex analysis, Thesis, Report HTKK-Math-A 154, Helsinki University of Technology, 1–79, 1979.Google Scholar
  46. Lounesto P., Latvamaa E., Conformal transformations and Clifford algebras, Proc. Amer. Math. Soc., 79, pp. 533–538, 1980.Google Scholar
  47. Lounesto P., Clifford algebras and spinors, Second edition, Cambridge University Press, 2001.Google Scholar
  48. Maia M.D., Conformal spinors in general relativity, Journal of Math. Physic., vol. 15, no 4, pp. 420–425, 1974.CrossRefMathSciNetGoogle Scholar
  49. O’Meara O.T., Introduction to quadratic forms, Springer-Verlag, Berlin, Göttingen, Heidelberg, 1973.MATHGoogle Scholar
  50. Porteous I.R., Topological geometry, 2ν δ gdition, Cambridge University Press, 1981.Google Scholar
  51. Postnikov M., Leçons de géométries: Groupes et algèbres de Lie, Trad. Française, Ed. Mir, Moscou, 1985.Google Scholar
  52. Riescz M., Clifford numbers and spinors, Lectures series no 38, University of Maryland, 1958.Google Scholar
  53. Satake I., Algebraic structures of symmetric domains, Iwanami Shoten publishers and Princeton University Press, 1981.Google Scholar
  54. Serre J.P., Applications algébriques de la cohomologie des groupes, II. Théorie des algèbres simples, Séminaire H. Cartan, E.N.S., 2 exposés 6.01, 6.09, 7.01, 7.11, 1950–1951.Google Scholar
  55. Steenrod N., The topology of fibre bundles, Princeton University Press, New Jersey, 1951.MATHGoogle Scholar
  56. Sternberg S., Lectures on differential geometry, P. Hall, New York, 1965.Google Scholar
  57. Sudbery A., Division algebras, pseudo-orthogonal groups and spinors, J. Phys. A. Math. Gen. 17, pp. 939–955, 1984.CrossRefMathSciNetMATHGoogle Scholar
  58. Wall C.T.C., Graded algebras anti-involutions, simple groups and symmetric spaces, Bull. Am. Math. Soc., 74, pp. 198–202, 1968.MATHMathSciNetCrossRefGoogle Scholar
  59. Weil A., Algebras with involutions and the classical groups, Collected papers, vol. II, pp. 413–447, 1951–1964; reprinted by permission of the editors of Journal of Ind. Math. Soc., Springer Verlag, New York, 1980.Google Scholar
  60. Wolf J. A., Spaces of constant curvature, Publish or Perish, Inc. Boston, 1974.MATHGoogle Scholar
  61. Wybourne B.G., Classical Groups for Physicists, JohnWiley and sons, Inc. NewYork, 1974.MATHGoogle Scholar

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© Birkhäuser Boston 2008

Authors and Affiliations

  • Pierre Anglè
    • 1
  1. 1.Laboratoire Emile Picard Institut de Mathématiques de ToulouseUniversité Paul SabatierFrance

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