About the Group

Our interest is the symmetries of the problems of high energy physics and cosmology. The group theoretical applications such as GUT’s, relevance of quaternions and octonions to exceptional Lie algebras are among the main theme. We work on the finite subgroups of quaternions and relate them to the root systems in 4-dimensions. We construct the Weyl-Coxeter groups of the roots of concern by quaternion actions on the lattice vectors. In this context we obtain the polyhedral structures in three dimensions using the orbits of the subgroups acting in 3- dimensions of the Weyl-Coxeter groups in 4-dimensions. We also work on the project where a Lie group can be broken to its finite subgroups by Higg’s mechanism. Description of the root system of the exceptional group E8 by octonions and its automorphism group and the relevant subgroups have been worked out for some time. In particular, the Chevalley group G2(2) reveals itself as the automorphism group of the octonionic root system of E7. Description of the lattice structure of E8 with icosians and its relation to the Coxeter group H4 is also among our current interest.

Our interest extends to the description of the universe with a manifold S3/ Γ where S3 is the group of quaternions isomorphic to SU(2) and Γ is one of the finite subgroups of SO(4). We work on a project whether one can reproduce the experimental data on the temperature distribution by using one of those polyhedra in 3-dimension.