Bulletin of the American Mathematical Society

Author(s): John C. Baez
Journal: Bull. Amer. Math. Soc. 39 (2002), 145-205.
MSC (2000): Primary 17-02, 17A35, 17C40, 17C90, 22E70
Posted: December 21, 2001
Errata: Bull. Amer. Math. Soc. (N.S.) 42 (2005), 213
Retrieve article in: PDF DVI PostScript

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.

1.: John F. Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. 72 (1960), 20-104. MR 25:4530
2.: John F. Adams, Lectures on Exceptional Lie Groups, eds. Zafer Mahmud and Mamoru Mimira, University of Chicago Press, Chicago, 1996. MR 98b:22001
3.: Michael Atiyah and Friedrich Hirzebruch, Bott periodicity and the parallelizability of the spheres. Proc. Cambridge Philos. Soc. 57 (1961), 223-226. MR 23:A3578
4.: Helena Albuquerque and Shahn Majid, Quasialgebra structure of the octonions, J. Algebra 220 (1999), 188-224. MR 2000h:16048
5.: Chris H. Barton and Anthony Sudbery, Magic squares of Lie algebras, preprint available as math.RA/0001083.
6.: Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp. 313-316. MR 88f:53087
7.: F. van der Blij, History of the octaves, Simon Stevin 34 (1961), 106-125. MR 24:A149
8.: F. van der Blij and Tonny A. Springer, Octaves and triality, Nieuw Arch. v. Wiskunde 8 (1960), 158-169. MR 23:A947
9.: Dominik Boos, Ein tensorkategorieller Zugang zum Satz von Hurwitz, Diplomarbeit, ETH Zurich, March 1998.
10.: Armand Borel, Le plan projectif des octaves et les sphéres commes espaces homogènes, Compt. Rend. Acad. Sci. 230 (1950), 1378-1380. MR 11:640c
11.: Raoul Bott and John Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89. MR 21:1590
12.: Robert B. Brown, Groups of type ${E}_7$ , Jour. Reine Angew. Math. 236 (1969), 79-102. MR 40:1439
13.: Élie Cartan, Sur la structure des groupes de tranformations finis et continus, Thèse, Paris, Nony, 1894.
14.: Élie Cartan, Les groupes réels simples finis et continus, Ann. Sci. École Norm. Sup. 31 (1914), 255-262.
15.: Élie Cartan, Nombres complexes, in Encyclopédie des sciences mathématiques, 1, ed. J. Molk, 1908, 329-468.
16.: Élie Cartan, Le principe de dualité et la théorie des groupes simple et semi-simples, Bull. Sci. Math. 49 (1925), 361-374.
17.: Arthur Cayley, On Jacobi's elliptic functions, in reply to the Rev. B. Bronwin; and on quaternions, Philos. Mag. 26 (1845), 208-211.
18.: 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.
19.: Sultan Catto, Carlos J. Moreno and Chia-Hsiung Tze, Octonionic Structures in Physics, to appear.
20.: 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. MR 11:577b
21.: Yvonne Choquet-Bruhat and Cécile DeWitt-Morette, Analysis, Manifolds and Physics, part II, Elsevier, Amsterdam, 2000, pp. 263-274. MR 91e:58001
22.: William K. Clifford, Applications of Grassmann's extensive algebra, Amer. Jour. Math. 1 (1878), 350-358.
23.: Frederick R. Cohen, On Whitehead squares, Cayley-Dickson algebras and rational functions, Bol. Soc. Mat. Mexicana 37 (1992), 55-62. MR 95m:55018
24.: E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and the superstring, Comm. Math. Phys. 122 (1989), 393-410. MR 90j:81183
25.: Harold Scott MacDonald Coxeter, Integral Cayley numbers, Duke Math. Jour. 13 (1946), 561-578. MR 8:370b
26.: Michael J. Crowe, A History of Vector Analysis, University of Notre Dame Press, Notre Dame, 1967. MR 37:5070
27.: 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.
28.: Pierre Deligne et al., eds., Quantum Fields and Strings: A Course for Mathematicians, 2 volumes, Amer. Math. Soc., Providence, Rhode Island, 1999. MR 2000e:81010
29.: Leonard E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. Math. 20 (1919), 155-171.
30.: Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the Algebraic Design of Physics, Kluwer, Dordrecht, 1994. MR 96e:17004
31.: M. J. Duff, ed., The World in Eleven Dimensions: Supergravity, Supermembranes and M-Theory, Institute of Physics Publishing, Bristol, 1999. MR 2001b:81002
32.: Noam Elkies and Benedict H. Gross, The exceptional cone and the Leech lattice, Internat. Math. Res. Notices 14 (1996), 665-698. MR 97g:11070
33.: Gerard G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, New York, 1972.
34.: J. M. Evans, Supersymmetric Yang-Mills theories and division algebras, Nucl. Phys. B298 (1988), 92-108. MR 89f:81096
35.: John R. Faulkner, A construction of Lie algebras from a class of ternary algebras, Trans. Amer. Math. Soc. 155 (1971), 397-408. MR 45:3494
36.: John R. Faulkner and Joseph C. Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 (1977), 1-35. MR 56:3079
37.: 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. MR 92k:17041
38.: Hans Freudenthal, Oktaven, Ausnahmegruppen und Oktavengeometrie, mimeographed notes, 1951. Also available in Geom. Dedicata 19 (1985), 7-63. MR 86k:17018b
39.: Hans Freudenthal, Zur ebenen Oktavengeometrie, Indag. Math. 15 (1953), 195-200. MR 15:56f
40.: Hans Freudenthal, Beziehungen der ${\mathfrak {e}}_7$ und ${\mathfrak {e}}_8$ zur Oktavenebene, I, II, Indag. Math. 16 (1954), 218-230, 363-368. MR 16:108b; MR 16:900d; III, IV, Indag. Math. 17 (1955), 151-157, 277-285. MR 16:900e; MR 16:900f; V-IX, Indag. Math. 21 (1959), 165-201, 447-474; X, XI, Indag. Math. 25 (1963), 457-487. MR 29:506
41.: Hans Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1 (1964), 145-190.
42.: Hans Freudenthal, Bericht über die Theorie der Rosenfeldschen elliptischen Ebenen, in Raumtheorie, Wege Der Forschung, CCLXX, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978, pp. 283-286.
43.: Lynn E. Garner, An Outline of Projective Geometry, North Holland, New York, 1981. MR 82j:51001
44.: Robert Perceval Graves, Life of Sir William Rowan Hamilton, 3 volumes, Arno Press, New York, 1975.
45.: Michael B. Green, John H. Schwarz and Edward Witten, Superstring Theory, volume 1, Cambridge University Press, Cambridge, 1987, pp. 344-349. MR 89f:81001a
46.: 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. MR 87b:53048
47.: Murat Günaydin, Generalized conformal and superconformal group actions and Jordan algebras, Mod. Phys. Lett. A 8 (1993), 1407-1416. MR 94c:17048
48.: Murat Günaydin, Kilian Koepsell, and Hermann Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups, Comm. Math. Phys. 221 (2001), 57-76 CMP 2001:16
49.: Murat Günaydin, C. Piron and H. Ruegg, Moufang plane and octonionic quantum mechanics, Comm. Math. Phys. 61 (1978), 69-85. MR 58:19906
50.: Feza Gürsey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebras in Particle Physics, World Scientific, Singapore, 1996. MR 99g:81061
51.: 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.
52.: Thomas L. Hankins, Sir William Rowan Hamilton, Johns Hopkins University Press, Baltimore, 1980. MR 82h:01051
53.: F. Reese Harvey, Spinors and Calibrations, Academic Press, Boston, 1990. MR 91e:53056
54.: Adolf Hurwitz, Über die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Göttingen (1898), 309-316.
55.: Dale Husemoller, Fibre Bundles, Springer, Berlin, 1994. MR 94k:55001
56.: Pascual Jordan, Über eine Klasse nichtassociativer hyperkomplexer Algebren, Nachr. Ges. Wiss. Göttingen (1932), 569-575.
57.: Pascual Jordan, Über eine nicht-desarguessche ebene projektive Geometrie, Abh. Math. Sem. Hamburg 16 (1949), 74-76. MR 11:50h
58.: Pascual Jordan, John von Neumann, Eugene Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29-64.
59.: Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford U. Press, Oxford, 2000. MR 2001k:53093
60.: I. L. Kantor and A. S. Solodovnikov, Hypercomplex Numbers -- an Elementary Introduction to Algebras, Springer, Berlin, 1989. MR 90b:16001
61.: Michel Kervaire, Non-parallelizability of the sphere for , Proc. Nat. Acad. Sci. USA 44 (1958), 280-283.
62.: 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.
63.: T. Kugo and P.-K. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B 221 (1983), 357-380. MR 85e:81054
64.: Greg Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180 (1996), 109-151. MR 97f:17005
65.: J. M. Landsberg and L. Manivel, The projective geometry of Freudenthal's magic square, J. Algebra 239 (2001), 477-512.
66.: Jaak Lohmus, Eugene Paal, and Leo Sorgsepp, Nonassociative Algebras in Physics, Hadronic Press, Palm Harbor, Florida, 1994. MR 97d:81095
67.: Corinne A. Manogue and Tevian Dray, Octonionic Möbius transformations, Mod. Phys. Lett. A 14 (1999), 1243-1256. MR 2000g:17003
68.: Corinne A. Manogue and Jörg Schray, Finite Lorentz transformations, automorphisms, and division algebras, Jour. Math. Phys. 34 (1993), 3746-3767. MR 94h:81041
69.: Corinne A. Manogue and Jörg Schray, Octonionic representations of Clifford algebras and triality, Found. Phys. 26 (1996), 17-70. MR 97d:15035
70.: Kevin McCrimmon, Jordan algebras and their applications, Bull. Amer. Math. Soc. 84 (1978), 612-627. MR 57:6115
71.: K. Meyberg, Eine Theorie der Freudenthalschen Tripelsysteme, I, II, Indag. Math. 30 (1968), 162-190. MR 37:1429
72.: R. Guillermo Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, preprint available at q-alg/9710013.
73.: Ruth Moufang, Alternativkörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207-222.
74.: Susumu Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, 1995. MR 96j:81052
75.: A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras III, Springer, Berlin, 1991, pp. 167-178. MR 96d:22001
76.: Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, 2 volumes, Cambridge U. Press, Cambridge, 1985-86. MR 88h:83009; MR 89d:83010
77.: Ian R. Porteous, Topological Geometry, Cambridge U. Press, 1981. MR 82c:51018
78.: Boris A. Rosenfeld, Geometrical interpretation of the compact simple Lie groups of the class (Russian), Dokl. Akad. Nauk. SSSR 106 (1956), 600-603.
79.: Boris A. Rosenfeld, Geometry of Lie Groups, Kluwer, Dordrecht, 1997. MR 98i:53002
80.: Markus Rost, On the dimension of a composition algebra, Doc. Math. 1 (1996), 209-214. MR 97c:17004
81.: Helmut Salzmann et al., Compact Projective Planes: With an Introduction to Octonion Geometry, de Gruyter, Berlin, 1995. MR 97b:51009
82.: Richard D. Schafer, On algebras formed by the Cayley-Dickson process, Amer. Jour. of Math. 76 (1954), 435-446. MR 15:774d
83.: Richard D. Schafer, Introduction to Non-Associative Algebras, Dover, New York, 1995. MR 96j:17001
84.: Jörg Schray, Octonions and Supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University, 1994.
85.: 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. MR 88h:81059
86.: Tonny A. Springer, The projective octave plane, I-II, Indag. Math. 22 (1960), 74-101. MR 23:A3492
87.: Tonny A. Springer, Characterization of a class of cubic forms, Indag. Math. 24 (1962), 259-265. MR 25:2104
88.: Tonny A. Springer, On the geometric algebra of the octave planes, Indag. Math. 24 (1962), 451-468. MR 25:5439
89.: Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan Algebras and Exceptional Groups, Springer, Berlin, 2000. MR 2001f:17006
90.: Frederick W. Stevenson, Projective Planes, W. H. Freeman and Company, San Francisco, 1972. MR 49:9734
91.: Anthony Sudbery, Octonionic description of exceptional Lie superalgebras, Jour. Math. Phys. 24 (1983), 1986-1988. MR 85c:17011
92.: Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and spinors, Jour. Phys. A17 (1984), 939-955. MR 85i:22042
93.: Jacques Tits, Le plan projectif des octaves et les groupes de Lie exceptionnels, Bull. Acad. Roy. Belg. Sci. 39 (1953), 309-329. MR 14:947f
94.: 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. MR 16:119
95.: Jacques Tits, Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Indag. Math. 28 (1966), 223-237. MR 36:2658
96.: Jacques Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, Vol. 386, Springer, Berlin, 1974. MR 57:9866
97.: V. S. Varadarajan, Geometry of Quantum Theory, Springer-Verlag, Berlin, 1985. MR 87a:81009
98.: E. B. Vinberg, A construction of exceptional simple Lie groups (Russian), Tr. Semin. Vektorn. Tensorn. Anal. 13 (1966), 7-9.
99.: Max Zorn, Theorie der alternativen Ringe, Abh. Math. Sem. Univ. Hamburg 8 (1930), 123-147.
100.: Max Zorn, Alternativkörper und quadratische Systeme, Abh. Math. Sem. Univ. Hamburg 9 (1933), 395-402.

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 17-02, 17A35, 17C40, 17C90, 22E70

John C. Baez
Affiliation: Department of Mathematics, University of California, Riverside, CA 92521
Email: baez@math.ucr.edu

DOI: 10.1090/S0273-0979-01-00934-X
PII: S 0273-0979(01)00934-X
Received by editor(s): May 31, 2001, and in revised form August 2, 2001
Posted: December 21, 2001
Copyright of article: Copyright 2001, John C. Baez

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-9485(e) ISSN 0273-0979(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues: 1891 - Present Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS