Quaternionic roots of SO(8), SO(9), F4 and the related Weyl groups

Mehmet Koca, Ramazan Koç, Muataz Al Barwani

Research output: Contribution to journalArticlepeer-review


The root systems of SO(8), SO(9) and F4 are constructed by quaternions. Triality manifests itself as permutations of pure quaternion units e1, e2 and e3. It is shown that the automorphism groups of the associated root systems are the finite subgroups of O(4) generated by left-right actions of unit quaternions on the root systems. The relevant finite groups of quaternions, the binary tetrahedral and binary octahedral groups, play essential roles in the construction of the Weyl groups and their conjugacy classes. The relations between the Dynkin indices, standard orthogonal vector and the quaternionic weights are obtained.

Original languageEnglish (US)
Pages (from-to)3123-3140
Number of pages18
JournalJournal of Mathematical Physics
Issue number7
StatePublished - Jul 1 2003
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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