6. Bibliography

1

John F. Adams, Lectures on Exceptional Lie Groups, eds. Zafer Mahmud and Mamoru Mimira, University of Chicago Press, Chicago, 1996.

2

Michael Atiyah and Friedrich Hirzebruch, Bott periodicity and the parallelizability of the spheres. Proc. Cambridge Philos. Soc. 57 (1961), 223-226.

3

Helena Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, available as arXiv:math/9802116.

4

Chris H. Barton and Anthony Sudbery, Magic squares of Lie algebras. Available as arXiv:math/0001083.

5

Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp. 313-316.

6

F. van der Blij, History of the octaves, Simon Stevin 34 (1961), 106-125.

7

F. van der Blij and Tonny A. Springer, Octaves and triality, Nieuw Arch. v. Wiskunde 8 (1960), 158-169.

8

Armand Borel, Le plan projectif des octaves et les sphéres commes espaces homogènes, Compt. Rend. Acad. Sci. 230 (1950), 1378-1380.

9

Raoul Bott and John Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958) 87-89.

10

Robert B. Brown, Groups of type $\E _7$, Jour. Reine Angew. Math. 236 (1969), 79-102.

11

Élie Cartan, Sur la structure des groupes de tranformations finis et continus, Thèse, Paris, Nony, 1894.

12

Élie Cartan, Les groupes réels simples finis et continus, Ann. Sci. École Norm. Sup. 31 (1914), 262-255.

13

Élie Cartan, Nombres complexes, in Encyclopédie des sciences mathématiques, 1, ed. J. Molk, 1908, 329-468.

14

Élie Cartan, Le principe de dualité et la théorie des groupes simple et semi-simples, Bull. Sci. Math. 49 (1925), 361-374.

15

Arthur Cayley, On Jacobi's elliptic functions, in reply to the Rev. B. Brownwin; and on quaternions, Philos. Mag. 26 (1845), 208-211.

16

Arthur Cayley, On Jacobi's elliptic functions, in reply to the Rev. B. Bronwin; and on quaternions (appendix only), in The Collected Mathematical Papers, Johnson Reprint Co., New York, 1963, p. 127.

17

Sultan Catto, Carlos J. Moreno and Chia-Hsiung Tze, Octonionic Structures in Physics, to appear.

18

Claude Chevalley and Richard D. Schafer, The exceptional simple Lie algebras $\F _4$ and $\E _6$, Proc. Nat. Acad. Sci. USA 36 (1950), 137-141.

19

Yvonne Choquet-Bruhat and Cécile DeWitt-Morette, Analysis, Manifolds and Physics, part II, Elsevier, Amsterdam, 2000, pp. 263-274.

20

William K. Clifford, Applications of Grassmann's extensive algebra, Amer. Jour. Math. 1 (1878), 350-358.

21

Frederick R. Cohen, On Whitehead squares, Cayley-Dickson algebras and rational functions, Bol. Soc. Mat. Mexicana 37 (1992), 55-62.

22

E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and the superstring, Comm. Math. Phys. 122 (1989), 393-410. Available online courtesy of Project Euclid.

23

Harold Scott MacDonald Coxeter, Integral Cayley numbers, Duke Math. Jour. 13 (1946), 561-578.

24

Michael J. Crowe, A History of Vector Analysis, University of Notre Dame Press, Notre Dame, 1967.

25

C. W. Curtis, The four and eight square problem and division algebras, in Studies in Modern Algebra, ed. A. Albert, Prentice-Hall, Englewood Cliffs, New Jersey, 1963, pp. 100-125.

26

Pierre Deligne et al, eds., Quantum Fields and Strings: A Course for Mathematicians, 2 volumes, Amer. Math. Soc., Providence, Rhode Island, 1999.

27

Leonard E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. Math. 20 (1919), 155-171.

28

Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer, Dordrecht, 1994.

29

M. J. Duff, ed., The World in Eleven Dimensions: Supergravity, Supermembranes and M-Theory, Institute of Physics Publishing, Bristol, 1999.

30

Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.

31

J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nucl. Phys. B298 (1988), 92-108.

32

John R. Faulkner, A construction of Lie algebras from a class of ternary algebras, Trans. Amer. Math. Soc. 155 (1971), 397-408.

33

John R. Faulkner and Joseph C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), 1-35.

34

Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries, Spinor Construction of Vertex Operator Algebras, Triality, and $\E _8^{(1)}$, Contemp. Math. 121, Amer. Math. Soc., Providence, Rhode Island, 1991.

35

Hans Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, mimeographed notes, 1951. Also available in Geom. Dedicata 19 (1985), 7-63.

36

Hans Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195-200.

37

Hans Freudenthal, Beziehungen der $e_7$ und $\e _8$ zur Oktavenebene, I, II, Indag. Math. 16 (1954), 218-230, 363-368. III, IV, Indag. Math. 17 (1955), 151-157, 277-285. V - IX, Indag. Math. 21 (1959), 165-201, 447-474. X, XI, Indag. Math. 25 (1963) 457-487.

38

Hans Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145-190.

39

Hans Freudenthal, Bericht über die Theorie der Rosenfeldschen elliptischen Ebenen, in Raumtheorie, Wege Der Forschung, CCLXX, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978, pp. 283-286.

40

Lynn E. Garner, An Outline of Projective Geometry, North Holland, New York, 1981.

41

Robert Perceval Graves, Life of Sir William Rowan Hamilton, 3 volumes, Arno Press, New York, 1975.

42

Michael B. Green, John H. Schwarz and Edward Witten, Superstring Theory, volume 1, Cambridge University Press, Cambridge, 1987, pp. 344-349.

43

B. Grossman, T. E. Kephart, and James D. Stasheff, Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map, Comm. Math. Phys. 96 (1984), 431-437.

44

Murat Günaydin, Generalized conformal and superconformal group actions and Jordan algebras, Mod. Phys. Lett. A8 (1993), 1407-1416. Also available as arXiv:hep-th/9301050.

45

Murat Günaydin, Kilian Koepsell, and Hermann Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups. Available as arXiv:hep-th/0008063.

46

Murat Günaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Comm. Math. Phys. 61 (1978), 69-85.

47

Feza Gürsey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebras in Particle Physics, World Scientific, Singapore, 1996.

48

William Rowan Hamilton, Four and eight square theorems, in Appendix 3 of The Mathematical Papers of Sir William Rowan Hamilton, vol. 3, eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967, pp. 648-656.

49

Thomas L. Hankins, Sir William Rowan Hamilton, John Hopkins University Press, Baltimore, 1980.

50

F. Reese Harvey, Spinors and Calibrations, Academic Press, San Diego, 1990.

51

Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898) 309-316.

52

Dale Husemoller, Fibre Bundles, Springer, Berlin, 1994.

53

Pascual Jordan, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569-575.

54

Pascual Jordan, Über eine nicht-desarguessche ebene projektive Geometrie, Abh. Math. Sem. Hamburg 16 (1949), 74-76.

55

Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29-64.

56

Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford U. Press, Oxford, 2000.

57

I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers - an Elementary Introduction to Algebras, Springer, Berlin, 1989.

58

Michel Kervaire, Non-parallelizability of the $n$ sphere for $n > 7$, Proc. Nat. Acad. Sci. USA 44 (1958), 280-283.

59

Wilhelm Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen I, Math. Ann. 31 (1888), 252-290. II, 33 (1889) 1-48. III, 34 (1889), 57-122. IV 36 (1890), 161-189.

60

T. Kugo and P.-K. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B221 (1983), 357-380.

61

J. M. Landsberg and L. Manivel: The projective geometry of Freudenthal's magic square. Available as arXiv:math/9908039.

62

Jaak Lohmus, Eugene Paal, and Leo Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, Florida, 1994.

63

Corinne A. Manogue and Tevian Dray, Octonionic Möbius transformations, Mod. Phys. Lett. A14 (1999), 1243-1256.

64

Corinne A. Manogue and Jörg Schray, Finite Lorentz transformations, automorphisms, and division algebras, Jour. Math. Phys. 34 (1993), 3746-3767.

65

Corinne A. Manogue and Jörg Schray, Octonionic representations of Clifford algebras and triality, Found. Phys. 26 (1996), 17-70.

66

Kevin McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), 612-627. Available online courtesy of AMS and Project Euclid.

67

K. Meyberg, Eine Theorie der Freudenthalschen Tripelsysteme, I, II, Ned. Akad. Wetenschap. 71 (1968), 162-190.

68

R. Guillermo Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, available as arXiv:q-alg/9710013.

69

Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207-222.

70

A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras III, Springer, Berlin, 1991, pp. 167-178.

71

Susumu Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, 1995.

72

Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, 2 volumes, Cambridge U. Press, Cambridge, 1985-86.

73

Ian R. Porteous, Topological Geometry, Cambridge U. Press, 1981.

74

Boris A. Rosenfeld, Geometrical interpretation of the compact simple Lie groups of the class $E$ (Russian), Dokl. Akad. Nauk. SSSR (1956) 106, 600-603.

75

Boris A. Rosenfeld, Geometry of Lie Groups, Kluwer, Dordrecht, 1997.

76

Richard D. Schafer, On algebras formed by the Cayley-Dickson process, Amer. Jour. of Math. 76 (1954) 435-446.

77

Richard D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995.

78

Jörg Schray, Octonions and Supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University, 1994.

79

G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Class. Quant. Grav. 4 (1987), 227-236.

80

Tonny A. Springer, The projective octave plane, I-II, Indag. Math. 22 (1960), 74-101.

81

Tonny A. Springer, Characterization of a class of cubic forms, Indag. Math. 24 (1962), 259-265.

82

Tonny A. Springer, On the geometric algebra of the octave planes, Indag. Math. 24 (1962), 451-468.

83

Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, Berlin, 2000.

84

Frederick W. Stevenson, Projective Planes, W. H. Freeman and Company, San Francisco, 1972.

85

Anthony Sudbery, Octonionic description of exceptional Lie superalgebras, Jour. Math. Phys. 24 (1983), 1986-1988.

86

Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984), 939-955.

87

Jacques Tits, Le plan projectif des octaves et les groupes de Lie exceptionnels, Bull. Acad. Roy. Belg. Sci. 39 (1953), 309-329.

88

Jacques Tits, Le plan projectif des octaves et les groupes exceptionnels $\E _6$ et $\E _7$, Bull. Acad. Roy. Belg. Sci. 40 (1954), 29-40.

89

Jacques Tits, Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Indag. Math. 28 (1966) 223-237.

90

Jacques Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386, Springer, Berlin, 1974.

91

V. S. Varadarajan, Geometry of Quantum Theory, Springer-Verlag, Berlin, 1985.

92

E. B. Vinberg, A construction of exceptional simple Lie groups (Russian), Tr. Semin. Vektorn. Tensorn. Anal. 13 (1966), 7-9.

93

Max Zorn, Theorie der alternativen Ringe, Abh. Math. Sem. Univ. Hamburg 8 (1930), 123-147.

94

Max Zorn, Alternativkörper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), 395-402.

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