The algebras ℂ (complex numbers), ℍ (quaternions), and 𝕆 (octonions) are real division algebras obtained from the real numbers ℝ by a doubling procedure called the Cayley-Dickson Process. By doubling ℝ (dim 1), we obtain ℂ (dim 2), then ℂ produces ℍ (dim 4), and ℍ yields 𝕆 (dim 8). The next doubling process applied to 𝕆 then yields an algebra 𝕊 (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra 𝕊 and its zero divisors. In particular, it shows that 𝕊 has subalgebras isomorphic to ℝ, ℂ, ℍ, 𝕆, and a newly identified algebra 𝕆̃ called the quasi-octonions that contains the zero-divisors of 𝕊.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1088,
author = {Raoul E. Cawagas},
title = {On the structure and zero divisors of the Cayley-Dickson sedenion algebra},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {24},
year = {2004},
pages = {251-265},
zbl = {1102.17001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1088}
}
Raoul E. Cawagas. On the structure and zero divisors of the Cayley-Dickson sedenion algebra. Discussiones Mathematicae - General Algebra and Applications, Tome 24 (2004) pp. 251-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1088/
[000] [1] J. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2) (2001), 145-205. | Zbl 1026.17001
[001] [2] R.E. Cawagas, FINITAS - A software for the construction and analysis of finite algebraic structures, PUP Jour. Res. Expo., 1 No. 1, 1st Semester 1997.
[002] [3] R.E. Cawagas, Loops embedded in generalized Cayley algebras of dimension 2 r, r ł 2, Int. J. Math. Math. Sci. 28 (2001). doi: 181-187
[003] [4] J.H. Conway and D.A. Smith, On Quaternions and Octonions: Their Geometry and Symmetry, A.K. Peters Ltd., Natik, MA, 2003. | Zbl 1098.17001
[004] [5] K. Imaeda and M. Imaeda, Sedenions: algebra and analysis, Appl. Math. Comput. 115 (2000), 77-88. | Zbl 1032.17003
[005] [6] R.P.C. de Marrais, The 42 assessors and the box-kites they fly: diagonal axis-pair systems of zero-divisors in the sedenions'16 dimensions, http://arXiv.org/abs/math.GM/0011260 (preprint 2000).
[006] [7] G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 (1998), 13-28. | Zbl 1006.17005
[007] [8] S. Okubo, Introduction to Octonions and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge 1995. | Zbl 0841.17001
[008] [9] J.D. Phillips and P. Vojtechovsky, The varieties of loops of the Bol-Moufang type, submitted to Algebra Universalis. | Zbl 1102.20054
[009] [10] R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York 1966. | Zbl 0145.25601
[010] [11] J.D.H. Smith, A left loop on the 15-sphere, J. Algebra 176 (1995), 128-138. | Zbl 0841.17004
[011] [12] J.D.H, Smith, New developments with octonions and sedenions, Iowa State University Combinatorics/Algebra Seminar. (January 26, 2004), http://www.math.iastate.edu/jdhsmith/math/JS26jan4.htm.
[012] [13] T. Smith, Why not SEDENIONS?, http://www.innerx.net/personal/tsmith/sedenion.html.
[013] [14] J.P. Ward, Quaternions and Cayley Numbers, Kluwer Academic Publishers, Dordrecht 1997. | Zbl 0877.15031