## Journal of Geometry and Symmetry in Physics

### Composition Algebras, Exceptional Jordan Algebra and Related Groups

#### Abstract

Normed division rings are reviewed in the more general framework of composition algebras that include the split (indefinite metric) case. The Jordan - von Neumann - Wigner classification of finite dimensional Jordan algebras is outlined with special attention to the 27 dimensional exceptional Jordan algebra $\frak{J}$. The automorphism group $\rm{F}_4$ of $\frak{J}$ and its maximal Borel-de~Siebenthal subgroups $\frac{\rm SU(3)\times \rm SU(3)}{\mathbb{Z}_3}$ and ${\rm Spin}(9)$ are studied in some detail with an eye to possible applications to the fundamental fermions in the Standard Model of particle physics.

#### Article information

Source
J. Geom. Symmetry Phys., Volume 46 (2017), 59-93.

Dates
First available in Project Euclid: 14 February 2018