Automorphisms Of The Maximal Nilpotent Subalgebra Of Type D_n Lie Algebras

In Reference [3], Cao and Tan described the automorphisms of Lie algebra of the strictly upper triangular matrices over R which is a maximal nilpotent subalgebra of the A_n type classical simple Lie algebra. It is natural to consider this problem for other types of Lie algebras. In the present paper we consider this problem for type D_n Lie algebras.Let n ≥ 4 be a fixed integer, R be a unital commutative ring of characteristic not 2, and D_n(R) be a D_n-type classical Lie algebra over R. Set N be a maximal nilpotent subalgebra of D_n(R). In this paper, we determine all automorphisms of N. Since it is easy to determine the generators of the maximal subalgebra N, we only need to figure out the image of the automorphism φ which applies on these generators.Our main result is that for any fixed φ ∈ Aut(N) , it can be expressed aswhere ω_ε, η_h, σ_u,(?)_ν, v_g and μ_f are graph, diagonal, inner, adjoint, second central and central automorphisms respectively. All this automorphisms are called standard automorphisms in this paper.