In this thesis, we consider a general construction of dihedral subgroups D_n, in the auto-morphism group of a complex finite-dimensional simple Lie algebra g. Our main results are based on Kac's classification results of finite order automorphisms on g and a classic result on classification of Z-gradings on g. A dihedral subgroup D_n in the automorphism group of g can be defined by an order n automorphism σ and an order 2 automorphism r of g satisfying the relation rσ = σ^-1r. The key point to find the generators (σ, r) of the dihedral subgroups is to fix the order n automorphism σ and extend an involution r₀ on the fixed point of σ to r on the whole Lie algebra g. Finally, We give a complete list of classification of (σ, r) generating D₃-subgroups in the automorphism group of g....[ Read more ]

In this thesis, we consider a general construction of dihedral subgroups D_n, in the auto-morphism group of a complex finite-dimensional simple Lie algebra g. Our main results are based on Kac's classification results of finite order automorphisms on g and a classic result on classification of Z-gradings on g. A dihedral subgroup D_n in the automorphism group of g can be defined by an order n automorphism σ and an order 2 automorphism r of g satisfying the relation rσ = σ^-1r. The key point to find the generators (σ, r) of the dihedral subgroups is to fix the order n automorphism σ and extend an involution r₀ on the fixed point of σ to r on the whole Lie algebra g. Finally, We give a complete list of classification of (σ, r) generating D₃-subgroups in the automorphism group of g.