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2004 | 24 | 2 | 251-265
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
24
Numer
2
Strony
251-265
Opis fizyczny
Daty
wydano
2004
otrzymano
2004-05-19
poprawiono
2004-07-25
poprawiono
2004-12-30
Twórcy
  • SciTech R and D Center, OVPRD, Polytechnic University of the Philippines, Manila
Bibliografia
  • [1] J. Baez, The octonions, Bull. Amer. Math. Soc. 39 (2) (2001), 145-205.
  • [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] 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] J.H. Conway and D.A. Smith, On Quaternions and Octonions: Their Geometry and Symmetry, A.K. Peters Ltd., Natik, MA, 2003.
  • [5] K. Imaeda and M. Imaeda, Sedenions: algebra and analysis, Appl. Math. Comput. 115 (2000), 77-88.
  • [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] G. Moreno, The zero divisors of the Cayley-Dickson algebras over the real numbers, Bol. Soc. Mat. Mexicana (3) 4 (1998), 13-28.
  • [8] S. Okubo, Introduction to Octonions and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge 1995.
  • [9] J.D. Phillips and P. Vojtechovsky, The varieties of loops of the Bol-Moufang type, submitted to Algebra Universalis.
  • [10] R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York 1966.
  • [11] J.D.H. Smith, A left loop on the 15-sphere, J. Algebra 176 (1995), 128-138.
  • [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] T. Smith, Why not SEDENIONS?, http://www.innerx.net/personal/tsmith/sedenion.html.
  • [14] J.P. Ward, Quaternions and Cayley Numbers, Kluwer Academic Publishers, Dordrecht 1997.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1088
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