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

U2 - 10.1142/9789812770523_0004

DO - 10.1142/9789812770523_0004

M3 - Conference contribution

AN - SCOPUS:84894292082

SN - 9812705910

SN - 9789812705914

T3 - Mathematical Physics - Proceedings of the 12th Regional Conference

SP - 25

EP - 30

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

PB - World Scientific Publishing Co. Pte Ltd

T2 - 12th Regional Conference on Mathematical Physics

Y2 - 27 March 2006 through 1 April 2006

ER -