### Abstract

The Cayley-Dickson procedure has been used to construct the root systems of SO(8), SO(9) and the exceptional Lie algebra F4 in terms of quaternions. The Aut(F_{4}) has a simple presentation of the form (O,O) ⊕ (O,O)* where 0 represents the binary octahedral group. Two sets of quaternionic root system of F_{4}, symbolically written [F_{4}, F_{4}] = F_{4} + e_{7}F_{4}, describe the octonionic root system of E_{7} where the root system of the exceptional Lie algebra E_{7} are represented by the imaginary octonions. The automorphism group of the octonionic root system of E_{7} preserving the octonion algebra is the Chevalley group G_{2}(2) where the maximal subgroups of G_{2}(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E_{6}, SU(2) x SO(12) and SU(8) of E_{7}. Another pairing of the quaternionic roots of the form [F _{4}, F_{4}] = F_{4} + σF_{4}, constitutes the root system of E_{8} in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H_{4}. The relevance of H_{4} to E_{8} and the maximal subgroups of H _{4} have been discussed. Polyhedral structures obtained from the quaternionic root systems of H_{4} are described as the orbits of H _{3}.

Original language | English |
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Title of host publication | Mathematical Physics - Proceedings of the 12th Regional Conference |

Pages | 25-30 |

Number of pages | 6 |

Publication status | Published - 2007 |

Event | 12th Regional Conference on Mathematical Physics - Islamabad, Pakistan Duration: Mar 27 2006 → Apr 1 2006 |

### Other

Other | 12th Regional Conference on Mathematical Physics |
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Country | Pakistan |

City | Islamabad |

Period | 3/27/06 → 4/1/06 |

### Fingerprint

### Keywords

- Polyhedra
- Quaternions and octonions

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

### Cite this

*Mathematical Physics - Proceedings of the 12th Regional Conference*(pp. 25-30)

**Quaternionic and octonionic structures of the exceptional lie algebras.** / Koca, Mehmet.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Mathematical Physics - Proceedings of the 12th Regional Conference.*pp. 25-30, 12th Regional Conference on Mathematical Physics, Islamabad, Pakistan, 3/27/06.

}

TY - GEN

T1 - Quaternionic and octonionic structures of the exceptional lie algebras

AU - Koca, Mehmet

PY - 2007

Y1 - 2007

N2 - The Cayley-Dickson procedure has been used to construct the root systems of SO(8), SO(9) and the exceptional Lie algebra F4 in terms of quaternions. The Aut(F4) has a simple presentation of the form (O,O) ⊕ (O,O)* where 0 represents the binary octahedral group. Two sets of quaternionic root system of F4, symbolically written [F4, F4] = F4 + e7F4, describe the octonionic root system of E7 where the root system of the exceptional Lie algebra E7 are represented by the imaginary octonions. The automorphism group of the octonionic root system of E7 preserving the octonion algebra is the Chevalley group G2(2) where the maximal subgroups of G2(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E6, SU(2) x SO(12) and SU(8) of E7. Another pairing of the quaternionic roots of the form [F 4, F4] = F4 + σF4, constitutes the root system of E8 in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H4. The relevance of H4 to E8 and the maximal subgroups of H 4 have been discussed. Polyhedral structures obtained from the quaternionic root systems of H4 are described as the orbits of H 3.

AB - The Cayley-Dickson procedure has been used to construct the root systems of SO(8), SO(9) and the exceptional Lie algebra F4 in terms of quaternions. The Aut(F4) has a simple presentation of the form (O,O) ⊕ (O,O)* where 0 represents the binary octahedral group. Two sets of quaternionic root system of F4, symbolically written [F4, F4] = F4 + e7F4, describe the octonionic root system of E7 where the root system of the exceptional Lie algebra E7 are represented by the imaginary octonions. The automorphism group of the octonionic root system of E7 preserving the octonion algebra is the Chevalley group G2(2) where the maximal subgroups of G2(2) are the automorphism groups of the octonionic root systems of the maximal Lie algebras E6, SU(2) x SO(12) and SU(8) of E7. Another pairing of the quaternionic roots of the form [F 4, F4] = F4 + σF4, constitutes the root system of E8 in terms of icosians where half of the roots describe the roots of the noncrystallographic Coxeter group H4. The relevance of H4 to E8 and the maximal subgroups of H 4 have been discussed. Polyhedral structures obtained from the quaternionic root systems of H4 are described as the orbits of H 3.

KW - Polyhedra

KW - Quaternions and octonions

UR - http://www.scopus.com/inward/record.url?scp=84894292082&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84894292082&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84894292082

SN - 9812705910

SN - 9789812705914

SP - 25

EP - 30

BT - Mathematical Physics - Proceedings of the 12th Regional Conference

ER -