DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score (2023): 0.6

SJR (2023): 0.214

SNIP (2023): 0.604

Index Copernicus (2022): 121.02

H-Index: 5

Discussiones Mathematicae - General Algebra and Applications

PDF

Discussiones Mathematicae General Algebra and Applications 24(2) (2004) 251-265
DOI: https://doi.org/10.7151/dmgaa.1088

ON THE STRUCTURE AND ZERO DIVISORS OF THE CAYLEY-DICKSON SEDENION ALGEBRA

Raoul E. Cawagas

SciTech R and D Center, OVPRD,
Polytechnic University of the Philippines, Manila

e-mail: raoulec@yahoo.com

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 [O~] called the quasi-octonions that contains the zero-divisors of S.

Keywords: sedenions, subalgebras, zero divisors, octonions, quasi-octonions, quaternions, Cayley-Dickson process, Fenyves identities.

2000 Mathematics Subject Classification: 20N05, 17A45.

References

[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), 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.

Received 19 May 2004
Revised 25 July 2004
Revised 30 December 2004


Close