COMMUTING DERIVATIONS AND AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

Let L be a finite-dimensional complex simple Lie algebra, L-Z be the Z-span of a Chevalley basis of L, and L-R=R circle times L-Z(Z) be a Chevalley algebra of type L over a commutative ring R. Let ?(R) be the nilpotent subalgebra of L-R spanned by the root vectors associated with positive roots. A map phi of ?(R) is called commuting if [phi(x), x]=0 for all x ?(R). In this article, we prove that under some conditions for R, if phi is not of type A(2), then a derivation (resp., an automorphism) of ?(R) is commuting if and only if it is a central derivation (resp., automorphism), and if phi is of type A(2), then a derivation (resp., an automorphism) of ?(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).