# Quaternionic Root Systems and Subgroups of the $AutF {4}$

Quaternionic Root Systems and Subgroups of the $AutF {4}$ - Download this document for free, or read online. Document in PDF available to download.

Download or read this book online for free in PDF: **
Quaternionic Root Systems and Subgroups of the $AutF {4}$**

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU2 expressed as the real numbers one can construct the root systems of the Lie algebras of SO4,SP2= SO5,SO8,SO9,F {4} and E {8} in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E {8} which form a closed octonion algebra. The automorphism group AutF {4} of the Dynkin diagram of F {4} of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F {4}.The Weyl groups of many Lie algebras, such as, G {2},SO7,SO8,SO9,SU3XSU3 and SP3X SU2 have been constructed as the subgroups of AutF {4}. We have also classified the other non-parabolic subgroups of AutF {4} which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group $G {2}$ of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO8 manifesting itself as the cyclic symmetry of the quaternionic imaginary units e {1},e {2},e {3} is used to show that SO7 and SO9 can be embedded triply symmetric way in SO8 and F {4} respectively.

Author: **Mehmet Koca; Ramazan Koc; Muataz Al-Barwani**

Source: https://archive.org/