### Abstract

The octonionic root system of the exceptional Lie algebra E_{8} has been constructed from the quaternionic roots of F_{4} using the Cayley-Dickson doubling procedure where the roots of E_{7} correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E_{7} is the adjoint Chevalley group G_{2}(2) of order 12096. One of the four maximal subgroups of G_{2}(2) of order 192 preserves the quaternion subalgebra of the E_{7} root system. The other three maximal subgroups of orders 432; 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E_{6} × U(1), SU(2) × SO(12) and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

Original language | English |
---|---|

Pages (from-to) | 808-823 |

Number of pages | 16 |

Journal | Linear Algebra and Its Applications |

Volume | 422 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Apr 15 2007 |

### Fingerprint

### Keywords

- Group structure
- M-Theory
- Quaternions
- Subgroup structure

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis

### Cite this

_{2}(2) of order 12096 and the octonionic root system of E

_{7}

*Linear Algebra and Its Applications*,

*422*(2-3), 808-823. https://doi.org/10.1016/j.laa.2006.12.011

**The Chevalley group G _{2}(2) of order 12096 and the octonionic root system of E_{7}
.** / Koca, Mehmet; Koç, Ramazan; Koca, Nazife Ö.

Research output: Contribution to journal › Article

_{2}(2) of order 12096 and the octonionic root system of E

_{7}',

*Linear Algebra and Its Applications*, vol. 422, no. 2-3, pp. 808-823. https://doi.org/10.1016/j.laa.2006.12.011

_{2}(2) of order 12096 and the octonionic root system of E

_{7}Linear Algebra and Its Applications. 2007 Apr 15;422(2-3):808-823. https://doi.org/10.1016/j.laa.2006.12.011

}

TY - JOUR

T1 - The Chevalley group G2(2) of order 12096 and the octonionic root system of E7

AU - Koca, Mehmet

AU - Koç, Ramazan

AU - Koca, Nazife Ö

PY - 2007/4/15

Y1 - 2007/4/15

N2 - The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 root system. The other three maximal subgroups of orders 432; 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E6 × U(1), SU(2) × SO(12) and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

AB - The octonionic root system of the exceptional Lie algebra E8 has been constructed from the quaternionic roots of F4 using the Cayley-Dickson doubling procedure where the roots of E7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E7 is the adjoint Chevalley group G2(2) of order 12096. One of the four maximal subgroups of G2(2) of order 192 preserves the quaternion subalgebra of the E7 root system. The other three maximal subgroups of orders 432; 192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E6 × U(1), SU(2) × SO(12) and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension.

KW - Group structure

KW - M-Theory

KW - Quaternions

KW - Subgroup structure

UR - http://www.scopus.com/inward/record.url?scp=33847369353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847369353&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2006.12.011

DO - 10.1016/j.laa.2006.12.011

M3 - Article

VL - 422

SP - 808

EP - 823

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 2-3

ER -