There are quaternions and octonions and even sextonions but what about trinonions, quinonions and septonions. Are there 3, 5, and 7 dimensional algebras which could be called trinonions, quinonions and septonions?
The term sextonions is used in
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Bruce W. Westbury, Sextonions and the magic square http://arxiv.org/abs/math/0411428
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J.M. Landsberg, L. Manivel, The sextonions and $E_{7\frac{1}{2}}$ http://arxiv.org/abs/math/0402157
but this 6-dimensional algebra had been studied earlier in
- R.H. Jeurissen, The automorphism groups of octave algebras, Doctoral dissertation, University of Utrecht, 1970.
- E. Kleinfeld, On extensions of quaternions, Indian J. Math. 9 (1968) 443–446.
I encountered the term sextonions at http://en.wikipedia.org/wiki/E7.5 following a link from http://cameroncounts.wordpress.com/2013/09/03/e7-5/.
They are also mentioned in http://math.ucr.edu/home/baez/week260.html
Best Answer
The main thing to know in this area is Hurwitz theorem and Frobenius theorem which are the results that characterize $\Bbb R$ algebras (actually more) with certain properties.
The former says that composition algebras have dimension 1,2,4 or 8, resulting in the reals, complexes, quaternions and octonions respectively. The latter is a subcase asking about division algebras instead, and it says the only dimensions are 1,2,4, (no octonions.)
Beyond that, if you're not asking for any special properties, there are $\Bbb R$ algebras of every dimension, and people are going to name them whatever they please. I know for sure about the sedenions, which are arrived at by a process analogous to passing up through the chain $R\subseteq C\subseteq H\subseteq O$. This process is known as the Cayley-Dickson construction and can be continued indefinitely producing (nonassociative) algebras of dimension $2^n$.
In Lemma 3.3 of this paper I see they construct sextonions in what looks like a modified Cayley-Hamilton way, so this might be a generalization that yields the pattern you are looking for. This could indeed be a method for producing such a family of algebras, but the naming convention is entirely arbitrary.
I haven't had the pleasure of meeting the sextonions, but they sound... attractive.
As I remember someone once saying on this site: inventing and naming algebras was a popular sport in the 19th century.