On the structure and zero divisors of the Cayley-Dickson sedenion algebra

Raoul E. Cawagas

Discussiones Mathematicae - General Algebra and Applications (2004)

  • Volume: 24, Issue: 2, page 251-265
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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Raoul E. Cawagas. "On the structure and zero divisors of the Cayley-Dickson sedenion algebra." Discussiones Mathematicae - General Algebra and Applications 24.2 (2004): 251-265. <http://eudml.org/doc/287717>.

@article{RaoulE2004,
abstract = {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.},
author = {Raoul E. Cawagas},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {sedenions; subalgebras; zero divisors; octonions; quasi-octonions; quaternions; Cayley-Dickson process; Fenyves identities},
language = {eng},
number = {2},
pages = {251-265},
title = {On the structure and zero divisors of the Cayley-Dickson sedenion algebra},
url = {http://eudml.org/doc/287717},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Raoul E. Cawagas
TI - On the structure and zero divisors of the Cayley-Dickson sedenion algebra
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2004
VL - 24
IS - 2
SP - 251
EP - 265
AB - 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.
LA - eng
KW - sedenions; subalgebras; zero divisors; octonions; quasi-octonions; quaternions; Cayley-Dickson process; Fenyves identities
UR - http://eudml.org/doc/287717
ER -

References

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  2. [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. 
  3. [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 
  4. [4] J.H. Conway and D.A. Smith, On Quaternions and Octonions: Their Geometry and Symmetry, A.K. Peters Ltd., Natik, MA, 2003. Zbl1098.17001
  5. [5] K. Imaeda and M. Imaeda, Sedenions: algebra and analysis, Appl. Math. Comput. 115 (2000), 77-88. Zbl1032.17003
  6. [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). 
  7. [7] G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 (1998), 13-28. Zbl1006.17005
  8. [8] S. Okubo, Introduction to Octonions and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge 1995. Zbl0841.17001
  9. [9] J.D. Phillips and P. Vojtechovsky, The varieties of loops of the Bol-Moufang type, submitted to Algebra Universalis. Zbl1102.20054
  10. [10] R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York 1966. Zbl0145.25601
  11. [11] J.D.H. Smith, A left loop on the 15-sphere, J. Algebra 176 (1995), 128-138. Zbl0841.17004
  12. [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. 
  13. [13] T. Smith, Why not SEDENIONS?, http://www.innerx.net/personal/tsmith/sedenion.html. 
  14. [14] J.P. Ward, Quaternions and Cayley Numbers, Kluwer Academic Publishers, Dordrecht 1997. Zbl0877.15031

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