### Abstract

The root systems of SO(8), SO(9) and F_{4} are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e_{1}, e_{2} and e_{3}. It is shown that the automorphism groups of the associated root systems are the finite subgroups of O(4) generated by left-right actions of unit quaternions on the root systems. The relevant finite groups of quaternions, the binary tetrahedral and binary octahedral groups, play essential roles in the construction of the Weyl groups and their conjugacy classes. The relations between the Dynkin indices, standard orthogonal vector and the quaternionic weights are obtained.

Original language | English |
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Pages (from-to) | 3123-3140 |

Number of pages | 18 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 2003 |

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

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## Cite this

Koca, M., Koç, R., & Al-Barwani, M. (2003). Quaternionic roots of SO(8), SO(9), F4 and the related Weyl groups.

*Journal of Mathematical Physics*,*44*(7), 3123-3140. https://doi.org/10.1063/1.1578177