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The octonions

Author: John C. Baez
Journal: Bull. Amer. Math. Soc. 39 (2002), 145-205
MSC (2000): Primary 17-02, 17A35, 17C40, 17C90, 22E70
Published electronically: December 21, 2001
Erratum: Bull. Amer. Math. Soc. 42 (2005), 213-213.
MathSciNet review: 1886087
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Abstract: The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry.

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  • J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20–104. MR 141119, DOI
  • J. F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. With a foreword by J. Peter May; Edited by Zafer Mahmud and Mamoru Mimura. MR 1428422
  • M. F. Atiyah and F. Hirzebruch, Bott periodicity and the parallelizability of the spheres, Proc. Cambridge Philos. Soc. 57 (1961), 223–226. MR 126282, DOI
  • Helena Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), no. 1, 188–224. MR 1713433, DOI
  • BS Chris H. Barton and Anthony Sudbery, Magic squares of Lie algebras, preprint available as math.RA/0001083.
  • Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
  • F. van der Blij, History of the octaves, Simon Stevin 34 (1960/61), 106–125. MR 130283
  • F. van der Blij and T. A. Springer, Octaves and triality, Nieuw Arch. Wisk. (3) 8 (1960), 158–169. MR 123622
  • Boos Dominik Boos, Ein tensorkategorieller Zugang zum Satz von Hurwitz, Diplomarbeit, ETH Zurich, March 1998. Borel Armand Borel, Le plan projectif des octaves et les sphéres commes espaces homogènes, Compt. Rend. Acad. Sci. 230 (1950), 1378–1380.
  • R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89. MR 102804, DOI
  • Robert B. Brown, Groups of type $E_{7}$, J. Reine Angew. Math. 236 (1969), 79–102. MR 248185, DOI
  • Cartan0 Élie Cartan, Sur la structure des groupes de tranformations finis et continus, Thèse, Paris, Nony, 1894. Cartan Élie Cartan, Les groupes réels simples finis et continus, Ann. Sci. École Norm. Sup. 31 (1914), 255–262. Cartan2 Élie Cartan, Nombres complexes, in Encyclopédie des sciences mathématiques, 1, ed. J. Molk, 1908, 329–468. Cartan3 Élie Cartan, Le principe de dualité et la théorie des groupes simple et semi-simples, Bull. Sci. Math. 49 (1925), 361–374. Cayley Arthur Cayley, On Jacobi’s elliptic functions, in reply to the Rev. B. Bronwin; and on quaternions, Philos. Mag. 26 (1845), 208–211. Cayley2 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. CMT Sultan Catto, Carlos J. Moreno and Chia-Hsiung Tze, Octonionic Structures in Physics, to appear. CS Claude Chevalley and Richard D. Schafer, The exceptional simple Lie algebras $\mathrm {F}_4$ and $\mathrm {E}_6$, Proc. Nat. Acad. Sci. USA 36 (1950), 137–141.
  • Yvonne Choquet-Bruhat and Cécile DeWitt-Morette, Analysis, manifolds and physics. Part II, North-Holland Publishing Co., Amsterdam, 1989. 92 applications. MR 1016603
  • Clifford William K. Clifford, Applications of Grassmann’s extensive algebra, Amer. Jour. Math. 1 (1878), 350–358.
  • F. R. Cohen, On the Whitehead square, Cayley-Dickson algebras, and rational functions, Bol. Soc. Mat. Mexicana (2) 37 (1992), no. 1-2, 55–62. Papers in honor of José Adem (Spanish). MR 1317562
  • E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and the superstring, Comm. Math. Phys. 122 (1989), no. 3, 393–410. MR 998659
  • Coxeter Harold Scott MacDonald Coxeter, Integral Cayley numbers, Duke Math. Jour. 13 (1946), 561–578.
  • Michael J. Crowe, A history of vector analysis. The evolution of the idea of a vectorial system, University of Notre Dame Press, Notre Dame, Ind.-London, 1967. MR 0229496
  • Curtis 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.
  • Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (eds.), Quantum fields and strings: a course for mathematicians. Vol. 1, 2, American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. MR 1701618
  • Dickson Leonard E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. Math. 20 (1919), 155–171.
  • Geoffrey M. Dixon, Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics, Mathematics and its Applications, vol. 290, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1307379
  • M. J. Duff (ed.), The world in eleven dimensions: supergravity, supermembranes and M-theory, Studies in High Energy Physics Cosmology and Gravitation, IOP Publishing, Bristol, 1999. MR 1735754
  • Noam D. Elkies and Benedict H. Gross, The exceptional cone and the Leech lattice, Internat. Math. Res. Notices 14 (1996), 665–698. MR 1411589, DOI
  • Emch Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.
  • J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nuclear Phys. B 298 (1988), no. 1, 92–108. MR 929226, DOI
  • John R. Faulkner, A construction of Lie algebras from a class of ternary algebras, Trans. Amer. Math. Soc. 155 (1971), 397–408. MR 294424, DOI
  • J. R. Faulkner and J. C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), no. 1, 1–35. MR 444729, DOI
  • Alex J. Feingold, Igor B. Frenkel, and John F. X. Ries, Spinor construction of vertex operator algebras, triality, and $E^{(1)}_8$, Contemporary Mathematics, vol. 121, American Mathematical Society, Providence, RI, 1991. MR 1123265
  • F. D. Veldkamp, In honor of Hans Freudenthal on his eightieth birthday, Geom. Dedicata 19 (1985), no. 1, 2–5. MR 797150, DOI
  • Freudenthal Hans Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195–200. Freudenthal2 Hans Freudenthal, Beziehungen der $\mathfrak {e}_7$ und $\mathfrak {e}_8$ 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. Freudenthal3 Hans Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145–190. Freudenthal5 Hans Freudenthal, Bericht über die Theorie der Rosenfeldschen elliptischen Ebenen, in Raumtheorie, Wege Der Forschung, CCLXX, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978, pp. 283–286.
  • Lynn E. Garner, An outline of projective geometry, North-Holland Publishing Co., New York-Amsterdam, 1981. MR 600860
  • Graves Robert Perceval Graves, Life of Sir William Rowan Hamilton, 3 volumes, Arno Press, New York, 1975.
  • Michael B. Green, John H. Schwarz, and Edward Witten, Superstring theory. Vol. 1, 2nd ed., Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1988. Introduction. MR 952374
  • Bernard Grossman, Thomas W. Kephart, and James D. Stasheff, Solutions to Yang-Mills field equations in eight dimensions and the last Hopf map, Comm. Math. Phys. 96 (1984), no. 4, 431–437. MR 775040
  • Murat Günaydin, Generalized conformal and superconformal group actions and Jordan algebras, Modern Phys. Lett. A 8 (1993), no. 15, 1407–1416. MR 1222801, DOI
  • GKN Murat Günaydin, Kilian Koepsell, and Hermann Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Comm. Math. Phys. 221 (2001), 57–76
  • M. Günaydin, C. Piron, and H. Ruegg, Moufang plane and octonionic quantum mechanics, Comm. Math. Phys. 61 (1978), no. 1, 69–85. MR 503050
  • Feza Gürsey and Chia-Hsiung Tze, On the role of division, Jordan and related algebras in particle physics, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1626607
  • Hamilton William Rowan Hamilton, Four and eight square theorems, in Appendix 3 of The Mathematical Papers of William Rowan Hamilton, vol. 3, eds. H. Halberstam and R. E. Ingram, Cambridge University Press, Cambridge, 1967, pp. 648–656.
  • Thomas L. Hankins, Sir William Rowan Hamilton, Johns Hopkins University Press, Baltimore, Md., 1980. MR 618834
  • F. Reese Harvey, Spinors and calibrations, Perspectives in Mathematics, vol. 9, Academic Press, Inc., Boston, MA, 1990. MR 1045637
  • Hurwitz Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898), 309–316.
  • Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482
  • Jordan Pascual Jordan, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569–575. Jordan2 Pascual Jordan, Über eine nicht-desarguessche ebene projektive Geometrie, Abh. Math. Sem. Hamburg 16 (1949), 74–76. JNW Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29–64.
  • Dominic D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 1787733
  • I. L. Kantor and A. S. Solodovnikov, Hypercomplex numbers, Springer-Verlag, New York, 1989. An elementary introduction to algebras; Translated from the Russian by A. Shenitzer. MR 996029
  • Kervaire Michel Kervaire, Non-parallelizability of the $n$ sphere for $n > 7$, Proc. Nat. Acad. Sci. USA 44 (1958), 280–283. Killing 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.
  • Taichiro Kugo and Paul Townsend, Supersymmetry and the division algebras, Nuclear Phys. B 221 (1983), no. 2, 357–380. MR 709052, DOI
  • Greg Kuperberg, Spiders for rank $2$ Lie algebras, Comm. Math. Phys. 180 (1996), no. 1, 109–151. MR 1403861
  • LM J. M. Landsberg and L. Manivel, The projective geometry of Freudenthal’s magic square, J. Algebra 239 (2001), 477–512.
  • Jaak Lõhmus, Eugene Paal, and Leo Sorgsepp, Nonassociative algebras in physics, Hadronic Press Monographs in Mathematics, Hadronic Press, Inc., Palm Harbor, FL, 1994. MR 1320709
  • Corinne A. Manogue and Tevian Dray, Octonionic Möbius transformations, Modern Phys. Lett. A 14 (1999), no. 19, 1243–1255. MR 1703958, DOI
  • Corinne A. Manogue and Jörg Schray, Finite Lorentz transformations, automorphisms, and division algebras, J. Math. Phys. 34 (1993), no. 8, 3746–3767. MR 1230549, DOI
  • Jörg Schray and Corinne A. Manogue, Octonionic representations of Clifford algebras and triality, Found. Phys. 26 (1996), no. 1, 17–70. MR 1380124, DOI
  • Kevin McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), no. 4, 612–627. MR 466235, DOI
  • Kurt Meyberg, Eine Theorie der Freudenthalschen Tripelsysteme. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 162–174, 175–190 (German). MR 0225838
  • Moreno R. Guillermo Moreno, The zero divisors of the Cayley–Dickson algebras over the real numbers, preprint available at q-alg/9710013. Moufang Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207–222.
  • Susumu Okubo, Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics, vol. 2, Cambridge University Press, Cambridge, 1995. MR 1356224
  • È. B. Vinberg (ed.), Lie groups and Lie algebras, III, Encyclopaedia of Mathematical Sciences, vol. 41, Springer-Verlag, Berlin, 1994. Structure of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 41 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990 [ MR1056485 (91b:22001)]; Translation by V. Minachin [V. V. Minakhin]; Translation edited by A. L. Onishchik and È. B. Vinberg. MR 1349140
  • PR Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, 2 volumes, Cambridge U. Press, Cambridge, 1985-86. ;
  • Ian R. Porteous, Topological geometry, 2nd ed., Cambridge University Press, Cambridge-New York, 1981. MR 606198
  • Rosenfeld1 Boris A. Rosenfeld, Geometrical interpretation of the compact simple Lie groups of the class $\mathrm {E}$ (Russian), Dokl. Akad. Nauk. SSSR 106 (1956), 600-603.
  • Boris Rosenfeld, Geometry of Lie groups, Mathematics and its Applications, vol. 393, Kluwer Academic Publishers Group, Dordrecht, 1997. MR 1443207
  • Markus Rost, On the dimension of a composition algebra, Doc. Math. 1 (1996), No. 10, 209–214. MR 1397790
  • Helmut Salzmann, Dieter Betten, Theo Grundhöfer, Hermann Hähl, Rainer Löwen, and Markus Stroppel, Compact projective planes, De Gruyter Expositions in Mathematics, vol. 21, Walter de Gruyter & Co., Berlin, 1995. With an introduction to octonion geometry. MR 1384300
  • Schafer1 Richard D. Schafer, On algebras formed by the Cayley–Dickson process, Amer. Jour. of Math. 76 (1954), 435–446.
  • Richard D. Schafer, An introduction to nonassociative algebras, Dover Publications, Inc., New York, 1995. Corrected reprint of the 1966 original. MR 1375235
  • Schray Jörg Schray, Octonions and Supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University, 1994.
  • G. Sierra, An application of the theories of Jordan algebras and Freudenthal triple systems to particles and strings, Classical Quantum Gravity 4 (1987), no. 2, 227–236. MR 879887
  • T. A. Springer, The projective octave plane. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22 (1960), 74–101. MR 0126196
  • T. A. Springer, Characterization of a class of cubic forms, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 259–265. MR 0138661
  • T. A. Springer, On the geometric algebra of the octave planes, Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24 (1962), 451–468. MR 0142045
  • Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974
  • Fredrick W. Stevenson, Projective planes, W. H. Freeman and Co., San Francisco, Calif., 1972. MR 0344995
  • A. Sudbery, Octonionic description of exceptional Lie superalgebras, J. Math. Phys. 24 (1983), no. 8, 1986–1988. MR 713528, DOI
  • A. Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, J. Phys. A 17 (1984), no. 5, 939–955. MR 743176
  • Tits Jacques Tits, Le plan projectif des octaves et les groupes de Lie exceptionnels, Bull. Acad. Roy. Belg. Sci. 39 (1953), 309–329. Tits2 Jacques Tits, Le plan projectif des octaves et les groupes exceptionnels $\mathrm {E}_6$ et $\mathrm {E}_7$, Bull. Acad. Roy. Belg. Sci. 40 (1954), 29–40.
  • J. Tits, Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles. I. Construction, Nederl. Akad. Wetensch. Proc. Ser. A 69 = Indag. Math. 28 (1966), 223–237 (French). MR 0219578
  • Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
  • V. S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985. MR 805158
  • Vinberg E. B. Vinberg, A construction of exceptional simple Lie groups (Russian), Tr. Semin. Vektorn. Tensorn. Anal. 13 (1966), 7–9. Zorn Max Zorn, Theorie der alternativen Ringe, Abh. Math. Sem. Univ. Hamburg 8 (1930), 123–147. Zorn2 Max Zorn, Alternativkörper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), 395–402.

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Additional Information

John C. Baez
Affiliation: Department of Mathematics, University of California, Riverside, CA 92521

Received by editor(s): May 31, 2001
Received by editor(s) in revised form: August 2, 2001
Published electronically: December 21, 2001
Article copyright: © Copyright 2001 John C. Baez