Derivations of Certain Nilpotent Lie Algebras Over Commutative Rings

摘要

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 ?_Z L_Z be a Chevalley algebra of type L over a commutative ring R. Let N(R) be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. In this article, we give an explicit description of any derivation of N(R). We prove that under some conditions for R, any derivation of N(R) can be expressed as a sum of inner, central, diagonal, extremal derivations of N(R).