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 -