### 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 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Mathematical Physics*,

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

**Quaternionic roots of SO(8), SO(9), F4 and the related Weyl groups.** / Koca, Mehmet; Koç, Ramazan; Al-Barwani, Muataz.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 44, no. 7, pp. 3123-3140. https://doi.org/10.1063/1.1578177

}

TY - JOUR

T1 - Quaternionic roots of SO(8), SO(9), F4 and the related Weyl groups

AU - Koca, Mehmet

AU - Koç, Ramazan

AU - Al-Barwani, Muataz

PY - 2003/7/1

Y1 - 2003/7/1

N2 - The root systems of SO(8), SO(9) and F4 are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e1, e2 and e3. 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.

AB - The root systems of SO(8), SO(9) and F4 are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e1, e2 and e3. 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.

UR - http://www.scopus.com/inward/record.url?scp=0037661016&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037661016&partnerID=8YFLogxK

U2 - 10.1063/1.1578177

DO - 10.1063/1.1578177

M3 - Article

AN - SCOPUS:0037661016

VL - 44

SP - 3123

EP - 3140

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 7

ER -