Quaternions and Octonions

By James McCusker, University of Adelaide

Quaternions and octonions are the two largest of the four normed division algebras that extend the more familiar concepts of real and complex numbers. However these higher dimensional number systems possess quirks unlike their real and complex counterparts – the quaternions being noncommutative, and octonions even being nonassociative. Despite these quirks, they both continue to find a great number of uses in many areas of research, both purely theoretical and applied in nature. For example, both can be used in a topological construction of the famed Hopf fibration that involves projective geometry. The octonions can also be used to realize the smallest of the exceptional Lie groups, G2, as this can be defined as the automorphism group of the octonions. This object is naturally of great importance, as it has applications not only in differential geometry, but also in mechanics, where it appears as the symmetry group of certain kinematic systems, such as two spheres rolling onto each other without slipping or twisting.

While these applications are of course of great interest in their own right, the very fact that there exist only four normed division algebras, and that we seem to continue to lose properties as we go into higher dimensions leads us to pose the natural question – why is this so? An answer to this question was not known until over 50 years after the quaternions and subsequently octonions were discovered in 1843, when a theorem by Hurwitz was published in 1898 that completely classified the normed division algebras. The result that there are only four normed division algebras, and that these occur in dimensions 1, 2, 4 and 8, unsurprisingly has a great deal of significance in a number of areas of mathematics.

To explore these applications and Hurwitz’ answer to this question was the object of my summer research. In doing so, I was able to engage with the fields of topology, differential geometry, and algebra and some of their fundamental constructions in a very direct way. The process of research was extremely rewarding and challenging, and I found that investigating these abstract areas of mathematics with which I was not particularly familiar helped me to gain a greater appreciation for the ideas involved, and has instilled a desire to continue with mathematical research.

James McCusker was a recipient of a 2018/19 AMSI Vacation Research Scholarship.