Bibliography

Bibliography Next: Octonions Online Up: The Octonions Previous: Acknowledgements

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,ドル 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,ドル 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,ドル 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,ドル 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.

Next: Octonions Online Up: The Octonions Previous: Acknowledgements

6. Bibliography

home