Quaternionic and octonionic structures of the exceptional lie algebras

Mehmet Koca*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution


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.

Original languageEnglish
Title of host publicationMathematical Physics - Proceedings of the 12th Regional Conference
Number of pages6
Publication statusPublished - 2007
Event12th Regional Conference on Mathematical Physics - Islamabad, Pakistan
Duration: Mar 27 2006Apr 1 2006


Other12th Regional Conference on Mathematical Physics


  • Polyhedra
  • Quaternions and octonions

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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