### Abstract

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP(2)≈SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut (F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4. The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU(3)XSU(3), and SP(3)×SU(2) have been constructed as the subgroups of Aut (F4). We have also classified the other non-parabolic subgroups of Aut (F4) which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1, e2, e3 is used to show that SO(7) and SO(9) can be embedded, triply symmetric way in SO(8) and F4 in respectively.

Original language | English (US) |
---|---|

Article number | 043507 |

Journal | Journal of Mathematical Physics |

Volume | 47 |

Issue number | 4 |

DOIs | |

State | Published - May 15 2006 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*47*(4), [043507]. https://doi.org/10.1063/1.2190334

**Quaternionic root systems and subgroups of the Aut (F4).** / Koca, Mehmet; Al Barwani, Muataz; Ko̧, Ramazan.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 47, no. 4, 043507. https://doi.org/10.1063/1.2190334

}

TY - JOUR

T1 - Quaternionic root systems and subgroups of the Aut (F4)

AU - Koca, Mehmet

AU - Al Barwani, Muataz

AU - Ko̧, Ramazan

PY - 2006/5/15

Y1 - 2006/5/15

N2 - Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP(2)≈SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut (F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4. The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU(3)XSU(3), and SP(3)×SU(2) have been constructed as the subgroups of Aut (F4). We have also classified the other non-parabolic subgroups of Aut (F4) which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1, e2, e3 is used to show that SO(7) and SO(9) can be embedded, triply symmetric way in SO(8) and F4 in respectively.

AB - Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions, and octonions. Starting with the roots and weights of SU(2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP(2)≈SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the groups structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut (F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in four-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4. The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU(3)XSU(3), and SP(3)×SU(2) have been constructed as the subgroups of Aut (F4). We have also classified the other non-parabolic subgroups of Aut (F4) which are not Weyl groups. Two subgroups of orders 192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1, e2, e3 is used to show that SO(7) and SO(9) can be embedded, triply symmetric way in SO(8) and F4 in respectively.

UR - http://www.scopus.com/inward/record.url?scp=33646427187&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646427187&partnerID=8YFLogxK

U2 - 10.1063/1.2190334

DO - 10.1063/1.2190334

M3 - Article

AN - SCOPUS:33646427187

VL - 47

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

M1 - 043507

ER -