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References
Books on the Octonions:
- Geoffrey M. Dixon, Division Algebras: Octonions, Quaternions, Complex Numbers and the
Algebraic Design of Physics, Kluwer Academic Publishers, Boston, 1994.
- Feza Gürsey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebras in Particle
Physics, World Scientific, Singapore, 1996.
- S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in
Physics, Cambridge University Press, Cambridge, 1995.
Other related books:
- Clifford Algebras with Numeric and Symbolic Computations, eds. Rafa\l Ab\l amowicz, Pertti Lounesto, Josep M. Parra, Birkhäuser, Boston, 1996.
- Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, New York, 1995.
- Emil Artin, Geometric Algebra, John Wiley & Sons, New York, 1957 & 1988.
- William E. Baylis, Electrodynamics: a Modern Geometrical Approach, Birkhäuser Boston, Cambridge, 1999.
- Michael J. Crowe, A History of Vector Analysis, Dover, Mineola, NY, 1984 (originally published 1967).
- M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Cambridge University Press, Cambridge, 1987.
- F. Reese Harvey, Spinors and Calibrations, Academic Press, Boston, 1990.
- Nathan Jacobson, Structure and Representations of Jordan Algebras, Amer. Math. Soc. Colloq. Publ. 39, American Mathematical Society, Providence, 1968.
- P. Lounesto, Clifford Algebras and Spinors, Cambridge University Press, Cambridge, 1997.
- Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Cambridge University Press, Cambridge, 1984 & 1986.
- Boris Rosenfeld, Geometry of Lie Groups, Kluwer, Dordrecht, 1997.
- Richard D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966 & Dover, Mineola NY, 1995.
Papers by our group:
- David B. Fairlie and Corinne A. Manogue, Lorentz Invariance and the Composite String, Phys. Rev. D34, 1832–1834 (1986).
- David B. Fairlie and Corinne A. Manogue, A Parameterization of the Covariant Superstring, Phys. Rev. D36, 475–479 (1987).
- Corinne A. Manogue and Anthony Sudbery, General Solutions of Covariant Superstring Equations of Motion, Phys. Rev. D40, 4073–4077 (1989).
- Corinne A. Manogue and Jörg Schray, Finite Lorentz transformations, automorphisms, and division algebras, J. Math. Phys. 34, 3746–3767 (1993).
- Jörg Schray, Octonions and Supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University, 1994.
- Jörg Schray, The General Classical Solution of the Superparticle, Class. Quant. Grav. 13, 27 (1996).
- Jörg Schray & Corinne A. Manogue, Octonionic Representations of Clifford Algebras and Triality, Foundations of Physics, 26 17–70 (1996).
- Tevian Dray and Corinne A. Manogue, The Octonionic Eigenvalue Problem, Adv. Appl. Clifford Algebras 8, 341–364 (1998).
- Tevian Dray and Corinne A. Manogue, Finding Octonionic Eigenvectors Using {\slshape Mathematica}, Comput. Phys. Comm. 115, 536–547 (1998).
- Corinne A. Manogue and Tevian Dray, Dimensional Reduction, Mod. Phys. Lett. A14, 93–97 (1999).
- Corinne A. Manogue and Tevian Dray, Octonionic Möbius Transformations, Mod. Phys. Lett. A14, 1243–1255 (1999).
- Tevian Dray and Corinne A Manogue The Exceptional Jordan Eigenvalue Problem, Internat. J. Theoret. Phys. 38, 2901–2916 (1999).
- Tevian Dray and Corinne A. Manogue, Quaternionic Spin, in Clifford Algebras and their Applications in Mathematical Physics, eds. Rafa\l Ab\l amowicz and Bertfried Fauser, Birkhäuser, Boston, 2000, pp. 29–46.
- Tevian Dray, Jason Janesky, and Corinne A. Manogue, Octonionic Hermitian Matrices with Non-Real Eigenvalues Adv. Appl. Clifford Algebras 10, 193–216 (2000).
- Tevian Dray, Jason Janesky, and Corinne A. Manogue, Some Properties of $3\times3$ Octonionic Hermitian Matrices with
Non-Real Eigenvalues, Oregon State University, 2000, 12 pages.
- Tevian Dray, Corinne A. Manogue, and Susumu Okubo, Orthonormal Eigenbases over the Octonions, Algebras Groups Geom. (to appear).
Other related papers:
- A. Adrian Albert, On a Certain Algebra of Quantum Mechanics, Ann. Math. 35, 65–73 (1934).
- Claude Chevalley and R. D. Schafer, The Exceptional Simple Lie Algebras $F_4$ and $E_6$, Proc. Nat. Acad. Sci. U.S.A. 36, 137–141 (1950).
- K. W. Chung and A. Sudbery, Octonions and the Lorentz and Conformal Groups of Ten-Dimensional
Space-Time, Phys. Lett. B198, 161 (1987).
- L. E. Dickson, Ann. Math. 20, 155 (1919).
- P. A. M. Dirac, Proc. Roy. Irish Acad., Sect. A, Vol. L, 261 (1945).
- Freeman J. Dyson, Quaternion Determinants, Helv. Phys. Acta 45, 289–302 (1972).
- Hans Freudenthal, Oktaven, Ausnahmegruppen, und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht, 1951 (mimeographed); new revised edition, 1960; reprinted as Geom. Dedicata 19, 1–63 (1985).
- Hans Freudenthal, Zur Ebenen Oktavengeometrie, Proc. Kon. Ned. Akad. Wet. A56, 195–200 (1953).
- Hans Freudenthal, Lie Groups in the Foundations of Geometry, Adv. Math. 1, 145–190 (1964).
- H. H. Goldstine & L. P. Horwitz, On a Hilbert Space with Nonassociative Scalars, Proc. Nat. Aca. 48, 1134 (1962).
- P. Jordan, Über die Multiplikation quantenmechanischer Größen, Z. Phys. 80, 285–291 (1933).
- P. Jordan, J. von Neumann, and E. Wigner, On an Algebraic Generalization of the Quantum Mechanical Formalism, Ann. Math. 35, 29–64 (1934).
- T. Kugo and P. Townsend, Supersymmetry and the division algebras, Nucl. Phys. B221, 357 (1983).
- О. В. Огиевецкий, Характеристическое Урабнение для Матриц $3\times3$ над Октавами, Uspekhi Mat. Nauk 36, 197–198 (1981); translated in: O. V. Ogievetskii, The Characteristic Equation for
$3\times3$
Matrices over Octaves, Russian Math. Surveys 36, 189–190 (1981).
- Susumu Okubo, Eigenvalue Problem for Symmetric $3\times3$ Octonionic Matrix, Adv. Appl. Clifford Algebras 9, 131–176 (1999).
- A. Sudbery, J. Phys. A17, 939 (1984).