Automorphisms Of The Nilpotent Subalgebras Generated By The Positive Root Vectors Of Chevalley Algebras Of Some Types Over Communicative Rings (for Case That There Is Only The Trivial Graph Automorphism)

Let R be a communicative ring with identity 1. Let L be complex simple Lie algebras that have only the trivial graph automorphism. Let N be nilpotent subalgebra generated by positive root vectors of the Chevalley algebra LR over R. In this paper, we discuss the automorphism group of N.If the root system is of type Bn (n 2), E7, E8, let 2 be in R* and if the root system is of type Cn (n 3), F4, G2, let 2,3 be in R*. Any automorphism of N can be uniquely expressed as = dx b c , where dx, b, c and cr are diagonal, extremal, central and inner automorphisms, respectively, of N, and the automorphism group Aut(N) = , where and 3 are the diagonal, extremal, central and inner automorphism groups, respectively, of N.For the cases that the root system of type B2 and B3, we also determine the automorphism group of N.